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Cambridge IGCSE · 0580

Mathematics

Geometry and mensuration — Topics 3-6

Name: ____________________Date: October 10, 2026
  1. 1.

    A right-angled triangle has hypotenuse 13 cm and one shorter side 5 cm. Calculate the length of the third side.

    [3 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Uses Pythagoras: third side² = 13² − 5². Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Obtains third side² = 144. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: States the third side as 12 cm. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The hypotenuse is the longest side, so its square is the starting total.

    Marking points

    • Uses Pythagoras: third side² = 13² − 5².
    • Obtains third side² = 144.
    • States the third side as 12 cm.

    Examiner tip: The hypotenuse is the longest side, so its square is the starting total.

  2. 2.

    A closed cylinder has radius 3.0 cm and height 8.0 cm. Calculate its total surface area in terms of π and as a decimal to three significant figures.

    [4 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Uses total area = 2πr² + 2πrh. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Substitutes r = 3 and h = 8. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Obtains 66π cm². Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Work through this mathematical step: Obtains 207 cm² to three significant figures. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: A closed cylinder has two circular ends.

    Marking points

    • Uses total area = 2πr² + 2πrh.
    • Substitutes r = 3 and h = 8.
    • Obtains 66π cm².
    • Obtains 207 cm² to three significant figures.

    Examiner tip: A closed cylinder has two circular ends.

  3. 3.

    A sector of a circle has radius 8 cm and angle 45°. Calculate the area of the sector.

    [3 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Uses area = (angle/360) × πr². Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Substitutes (45/360) × π × 8². Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Obtains 25.1 cm² to three significant figures. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The fraction of the circle (angle/360) applies equally to both sector area and arc length formulas.

    Marking points

    • Uses area = (angle/360) × πr².
    • Substitutes (45/360) × π × 8².
    • Obtains 25.1 cm² to three significant figures.

    Examiner tip: The fraction of the circle (angle/360) applies equally to both sector area and arc length formulas.

  4. 4.

    Two similar triangles have corresponding sides in the ratio 2:5. The area of the smaller triangle is 12 cm². Calculate the area of the larger triangle.

    [3 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: States that the area ratio is the square of the length ratio: 2² : 5² = 4 : 25. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Sets up 12/4 = area/25. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Obtains area = 75 cm². Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Never apply the linear scale factor directly to area or volume — square it for area, cube it for volume.

    Marking points

    • States that the area ratio is the square of the length ratio: 2² : 5² = 4 : 25.
    • Sets up 12/4 = area/25.
    • Obtains area = 75 cm².

    Examiner tip: Never apply the linear scale factor directly to area or volume — square it for area, cube it for volume.

  5. 5.

    From a point 40 m from the base of a tower, the angle of elevation to the top of the tower is 32°. Calculate the height of the tower.

    [3 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Identifies the right-angled triangle with the 40 m side adjacent to the 32° angle. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Uses height = 40 × tan(32°). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Obtains height = 25.0 m to three significant figures. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Sketch the right-angled triangle first and label the angle of elevation, the known side, and the required side before choosing which trig ratio to use.

    Marking points

    • Identifies the right-angled triangle with the 40 m side adjacent to the 32° angle.
    • Uses height = 40 × tan(32°).
    • Obtains height = 25.0 m to three significant figures.

    Examiner tip: Sketch the right-angled triangle first and label the angle of elevation, the known side, and the required side before choosing which trig ratio to use.

  6. 6.

    A cone has base radius 6 cm and slant height 10 cm. Calculate its curved surface area, using the formula πrl.

    [3 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Uses curved surface area = πrl. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Substitutes π × 6 × 10. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Obtains 188.5 cm² to four significant figures. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Do not confuse the slant height (l) with the vertical (perpendicular) height of the cone — the curved surface area formula uses slant height.

    Marking points

    • Uses curved surface area = πrl.
    • Substitutes π × 6 × 10.
    • Obtains 188.5 cm² to four significant figures.

    Examiner tip: Do not confuse the slant height (l) with the vertical (perpendicular) height of the cone — the curved surface area formula uses slant height.

  7. 7.

    Calculate the exterior and interior angle of a regular decagon (10-sided polygon).

    [3 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Uses exterior angle = 360 ÷ number of sides = 360 ÷ 10. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Obtains exterior angle = 36°. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Uses interior angle = 180 − exterior angle to obtain 144°. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The sum of exterior angles of any convex polygon is always 360°, regardless of the number of sides.

    Marking points

    • Uses exterior angle = 360 ÷ number of sides = 360 ÷ 10.
    • Obtains exterior angle = 36°.
    • Uses interior angle = 180 − exterior angle to obtain 144°.

    Examiner tip: The sum of exterior angles of any convex polygon is always 360°, regardless of the number of sides.

  8. 8.

    A triangle has sides 7 cm and 9 cm with an included angle of 60°. Calculate the area of the triangle, using the formula (1/2)ab sin C.

    [3 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Uses area = (1/2) × 7 × 9 × sin(60°). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Substitutes sin(60°) ≈ 0.866. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Obtains area = 27.3 cm² to three significant figures. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: This area formula requires two sides and the angle directly between (included by) them — not any two sides and any angle.

    Marking points

    • Uses area = (1/2) × 7 × 9 × sin(60°).
    • Substitutes sin(60°) ≈ 0.866.
    • Obtains area = 27.3 cm² to three significant figures.

    Examiner tip: This area formula requires two sides and the angle directly between (included by) them — not any two sides and any angle.

  9. 9.

    A ladder of length 6.5 m leans against a vertical wall, reaching 6.0 m up the wall. Calculate the angle the ladder makes with the ground.

    [3 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Identifies the right-angled triangle with hypotenuse 6.5 m and opposite side 6.0 m. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Uses sin(angle) = 6.0/6.5. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Obtains angle = 67.4° to three significant figures. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Identify which sides are known relative to the required angle (opposite, adjacent, hypotenuse) before choosing sin, cos or tan.

    Marking points

    • Identifies the right-angled triangle with hypotenuse 6.5 m and opposite side 6.0 m.
    • Uses sin(angle) = 6.0/6.5.
    • Obtains angle = 67.4° to three significant figures.

    Examiner tip: Identify which sides are known relative to the required angle (opposite, adjacent, hypotenuse) before choosing sin, cos or tan.

  10. 10.

    A pyramid has a square base of side 6 cm and vertical height 8 cm. Calculate its volume.

    [3 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Uses volume = (1/3) × base area × height. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Calculates base area = 6² = 36 cm². Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Obtains volume = (1/3) × 36 × 8 = 96 cm³. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The 1/3 factor applies to any pyramid or cone volume formula, regardless of the shape of the base.

    Marking points

    • Uses volume = (1/3) × base area × height.
    • Calculates base area = 6² = 36 cm².
    • Obtains volume = (1/3) × 36 × 8 = 96 cm³.

    Examiner tip: The 1/3 factor applies to any pyramid or cone volume formula, regardless of the shape of the base.

  11. 11.

    AB is a diameter of a circle, and C is a point on the circumference. Angle ABC = 35°. Calculate angle ACB and angle BAC, giving a geometric reason for angle ACB.

    [4 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: States angle ACB = 90°, because the angle in a semicircle (subtended by a diameter) is a right angle. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: States that the angles in triangle ABC sum to 180°. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Calculates angle BAC = 180 − 90 − 35. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Work through this mathematical step: States angle BAC = 55°. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Whenever a triangle is drawn using a circle's diameter as one side, the angle at the third point on the circle is automatically 90°.

    Marking points

    • States angle ACB = 90°, because the angle in a semicircle (subtended by a diameter) is a right angle.
    • States that the angles in triangle ABC sum to 180°.
    • Calculates angle BAC = 180 − 90 − 35.
    • States angle BAC = 55°.

    Examiner tip: Whenever a triangle is drawn using a circle's diameter as one side, the angle at the third point on the circle is automatically 90°.

  12. 12.

    Two parallel lines are cut by a transversal. One of the angles formed measures 118°. Using angle properties of parallel lines, find (a) the alternate angle and (b) the co-interior (allied) angle, giving a reason for each.

    [3 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: States the alternate angle = 118°, because alternate angles between parallel lines are equal. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: States that co-interior (allied) angles between parallel lines sum to 180°. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Calculates the co-interior angle = 180 − 118 = 62°. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Alternate angles form a 'Z' shape and are equal; co-interior angles form a 'C' shape and sum to 180° — sketching the transversal helps identify which pattern applies.

    Marking points

    • States the alternate angle = 118°, because alternate angles between parallel lines are equal.
    • States that co-interior (allied) angles between parallel lines sum to 180°.
    • Calculates the co-interior angle = 180 − 118 = 62°.

    Examiner tip: Alternate angles form a 'Z' shape and are equal; co-interior angles form a 'C' shape and sum to 180° — sketching the transversal helps identify which pattern applies.

  13. 13.

    The bearing of point B from point A is 072°. Find the bearing of A from B.

    [3 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Recognises that a back bearing is found by adding 180° when the original bearing is less than 180°. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Calculates 072° + 180°. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: States the bearing of A from B as 252°. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Add 180° for an original bearing under 180°, or subtract 180° for one at or above 180° — the result must always stay within 000° to 360°.

    Marking points

    • Recognises that a back bearing is found by adding 180° when the original bearing is less than 180°.
    • Calculates 072° + 180°.
    • States the bearing of A from B as 252°.

    Examiner tip: Add 180° for an original bearing under 180°, or subtract 180° for one at or above 180° — the result must always stay within 000° to 360°.

  14. 14.

    A cuboid has dimensions 6 cm by 8 cm by 10 cm. Calculate the length of the space diagonal of the cuboid.

    [4 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Uses the 3D Pythagoras' theorem: diagonal² = length² + width² + height². Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Substitutes 6² + 8² + 10². Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Obtains diagonal² = 200. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Work through this mathematical step: States the diagonal as √200 ≈ 14.1 cm. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The space diagonal of a cuboid extends Pythagoras' theorem to three dimensions by adding all three squared side lengths before taking the square root.

    Marking points

    • Uses the 3D Pythagoras' theorem: diagonal² = length² + width² + height².
    • Substitutes 6² + 8² + 10².
    • Obtains diagonal² = 200.
    • States the diagonal as √200 ≈ 14.1 cm.

    Examiner tip: The space diagonal of a cuboid extends Pythagoras' theorem to three dimensions by adding all three squared side lengths before taking the square root.

  15. 15.

    Calculate the volume of a sphere with radius 6 cm, giving your answer to three significant figures. (V = (4/3)πr³)

    [4 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: States the formula V = (4/3)πr³. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Substitutes r = 6. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Calculates V = 288π. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Work through this mathematical step: States V ≈ 905 cm³ to three significant figures. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Cube the radius before multiplying by the constants — a common error is multiplying by 3 instead of cubing.

    Marking points

    • States the formula V = (4/3)πr³.
    • Substitutes r = 6.
    • Calculates V = 288π.
    • States V ≈ 905 cm³ to three significant figures.

    Examiner tip: Cube the radius before multiplying by the constants — a common error is multiplying by 3 instead of cubing.

  16. 16.

    Calculate the surface area of a sphere with radius 6 cm, giving your answer to three significant figures. (A = 4πr²)

    [3 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Substitutes r = 6 into A = 4πr² = 4π(6²). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Calculates A = 144π. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: States A ≈ 452 cm² to three significant figures. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Square the radius, not the diameter, when using this formula directly.

    Marking points

    • Substitutes r = 6 into A = 4πr² = 4π(6²).
    • Calculates A = 144π.
    • States A ≈ 452 cm² to three significant figures.

    Examiner tip: Square the radius, not the diameter, when using this formula directly.

  17. 17.

    Two similar solids have corresponding lengths in the ratio 3 : 4. The volume of the smaller solid is 54 cm³. Calculate the volume of the larger solid.

    [3 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: States that the volume ratio is the cube of the length ratio: 3³ : 4³ = 27 : 64. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Sets up 54/27 = volume/64. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: States the volume of the larger solid as 128 cm³. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Never apply a linear scale factor directly to volume — cube it first, exactly as you would square it for area.

    Marking points

    • States that the volume ratio is the cube of the length ratio: 3³ : 4³ = 27 : 64.
    • Sets up 54/27 = volume/64.
    • States the volume of the larger solid as 128 cm³.

    Examiner tip: Never apply a linear scale factor directly to volume — cube it first, exactly as you would square it for area.

  18. 18.

    A sector of a circle has radius 10 cm and angle 72°. Calculate the arc length of the sector, giving your answer in terms of π.

    [3 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Uses arc length = (angle/360) × 2πr. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Substitutes (72/360) × 2π × 10. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: States the arc length as 4π cm. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The fraction (angle/360) scales the full circumference 2πr down to just the arc, in exactly the same way it scales the full area down to a sector.

    Marking points

    • Uses arc length = (angle/360) × 2πr.
    • Substitutes (72/360) × 2π × 10.
    • States the arc length as 4π cm.

    Examiner tip: The fraction (angle/360) scales the full circumference 2πr down to just the arc, in exactly the same way it scales the full area down to a sector.

  19. 19.

    A shape consists of a rectangle 10 cm by 6 cm with a semicircle of diameter 6 cm attached along one of the shorter sides. Calculate the total area of the shape, giving your answer to three significant figures.

    [4 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Calculates the rectangle's area as 10 × 6 = 60 cm². Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Identifies the semicircle's radius as 3 cm (half the diameter of 6 cm). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Calculates the semicircle's area as (1/2)π(3²) = 4.5π cm². Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Work through this mathematical step: States the total area as 60 + 4.5π ≈ 74.1 cm² to three significant figures. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Break a composite shape into simple shapes whose formulas you know, find each area separately, then add (or subtract, for a cut-out) them.

    Marking points

    • Calculates the rectangle's area as 10 × 6 = 60 cm².
    • Identifies the semicircle's radius as 3 cm (half the diameter of 6 cm).
    • Calculates the semicircle's area as (1/2)π(3²) = 4.5π cm².
    • States the total area as 60 + 4.5π ≈ 74.1 cm² to three significant figures.

    Examiner tip: Break a composite shape into simple shapes whose formulas you know, find each area separately, then add (or subtract, for a cut-out) them.

  20. 20.

    In triangle ABC, AB = 8 cm, BC = 10 cm and CA = 6 cm. Calculate the size of angle B, using the cosine rule.

    [4 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: States the cosine rule rearranged for an angle: cos B = (AB² + BC² − CA²)/(2 × AB × BC). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Substitutes to obtain cos B = (64 + 100 − 36)/160. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Calculates cos B = 0.8. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Work through this mathematical step: States angle B ≈ 36.9° to three significant figures. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: To find an angle, use the two sides adjacent to that angle and the side opposite it — the rearranged cosine rule always needs exactly these three lengths.

    Marking points

    • States the cosine rule rearranged for an angle: cos B = (AB² + BC² − CA²)/(2 × AB × BC).
    • Substitutes to obtain cos B = (64 + 100 − 36)/160.
    • Calculates cos B = 0.8.
    • States angle B ≈ 36.9° to three significant figures.

    Examiner tip: To find an angle, use the two sides adjacent to that angle and the side opposite it — the rearranged cosine rule always needs exactly these three lengths.

  21. 21.

    Calculate the sum of the interior angles of a regular polygon with 12 sides.

    [3 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: States the formula: sum of interior angles = (n − 2) × 180°. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Substitutes n = 12. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: States the sum as 1800°. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: This formula works by dividing the polygon into (n − 2) triangles from a single vertex, each contributing 180°.

    Marking points

    • States the formula: sum of interior angles = (n − 2) × 180°.
    • Substitutes n = 12.
    • States the sum as 1800°.

    Examiner tip: This formula works by dividing the polygon into (n − 2) triangles from a single vertex, each contributing 180°.

  22. 22.

    A trapezium has parallel sides of length 7 cm and 11 cm, and a perpendicular height of 5 cm between them. Calculate its area.

    [3 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: States the formula: area = (1/2)(a + b)h, where a and b are the parallel sides. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Substitutes (1/2)(7 + 11)(5). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: States the area as 45 cm². Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The height in a trapezium is always the perpendicular distance between the two parallel sides, not the length of a slanted side.

    Marking points

    • States the formula: area = (1/2)(a + b)h, where a and b are the parallel sides.
    • Substitutes (1/2)(7 + 11)(5).
    • States the area as 45 cm².

    Examiner tip: The height in a trapezium is always the perpendicular distance between the two parallel sides, not the length of a slanted side.

  23. 23.

    A cylindrical water tank has radius 0.5 m and height 1.2 m. Calculate the capacity of the tank in litres. (1 m³ = 1000 litres)

    [4 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: States the formula V = πr²h. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Substitutes r = 0.5, h = 1.2 to obtain V = π(0.5²)(1.2) = 0.3π m³. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Converts to litres by multiplying by 1000. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Work through this mathematical step: States the capacity as approximately 942 litres. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Calculate the volume in cubic metres first, then convert to litres last — converting the radius or height to different units partway through risks an error.

    Marking points

    • States the formula V = πr²h.
    • Substitutes r = 0.5, h = 1.2 to obtain V = π(0.5²)(1.2) = 0.3π m³.
    • Converts to litres by multiplying by 1000.
    • States the capacity as approximately 942 litres.

    Examiner tip: Calculate the volume in cubic metres first, then convert to litres last — converting the radius or height to different units partway through risks an error.

  24. 24.

    From the top of a cliff 45 m high, the angle of depression to a boat at sea is 20°. Calculate the horizontal distance from the base of the cliff to the boat.

    [4 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: States that the angle of depression from the cliff equals the angle of elevation from the boat (alternate angles between parallel horizontal lines), giving a right-angled triangle with opposite side 45 m and required angle 20°. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Uses tan(20°) = 45/d, where d is the horizontal distance. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Rearranges to d = 45/tan(20°). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Work through this mathematical step: States d ≈ 124 m to three significant figures. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The angle of depression is always measured from the horizontal, and equals the angle of elevation seen from the object being viewed — draw the horizontal line at the top of the cliff to see this alternate-angle relationship clearly.

    Marking points

    • States that the angle of depression from the cliff equals the angle of elevation from the boat (alternate angles between parallel horizontal lines), giving a right-angled triangle with opposite side 45 m and required angle 20°.
    • Uses tan(20°) = 45/d, where d is the horizontal distance.
    • Rearranges to d = 45/tan(20°).
    • States d ≈ 124 m to three significant figures.

    Examiner tip: The angle of depression is always measured from the horizontal, and equals the angle of elevation seen from the object being viewed — draw the horizontal line at the top of the cliff to see this alternate-angle relationship clearly.

  25. 25.

    Two similar shapes have areas 50 cm² and 200 cm². The perimeter of the smaller shape is 20 cm. Calculate the perimeter of the larger shape.

    [3 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: States the area ratio as 50 : 200 = 1 : 4. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: States that the length (and perimeter) ratio is the square root of the area ratio: √1 : √4 = 1 : 2. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: States the perimeter of the larger shape as 20 × 2 = 40 cm. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Going from area ratio to length ratio requires a square root — the reverse of squaring the length ratio to get the area ratio.

    Marking points

    • States the area ratio as 50 : 200 = 1 : 4.
    • States that the length (and perimeter) ratio is the square root of the area ratio: √1 : √4 = 1 : 2.
    • States the perimeter of the larger shape as 20 × 2 = 40 cm.

    Examiner tip: Going from area ratio to length ratio requires a square root — the reverse of squaring the length ratio to get the area ratio.

  26. 26.

    A right-angled triangle has perpendicular sides 9 cm and 12 cm. Find its hypotenuse and area.

    [3 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. The two given sides meet at a right angle, so they are both the Pythagorean legs and a valid base-height pair.
    2. Take the positive square root for a length; calculate area separately using half the product of the perpendicular sides.

    Marking points

    • Hypotenuse squared = 9^2 + 12^2 = 225.
    • Hypotenuse = 15 cm.
    • Area = (1/2)(9)(12) = 54 cm^2.

    Examiner tip: Lengths use cm, while area uses cm^2.

  27. 27.

    Two similar shapes have areas 36 cm^2 and 81 cm^2. The smaller perimeter is 20 cm. Find the linear scale factor from smaller to larger and the larger perimeter.

    [4 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. An enlargement multiplies two independent lengths in each area, so its area multiplier is the square of its length multiplier.
    2. Take the square root before scaling the perimeter; the resulting factor is greater than one as required for the larger shape.

    Marking points

    • Area scale factor = 81/36 = 9/4.
    • Linear scale factor = sqrt(9/4) = 3/2.
    • Perimeter scales linearly, so multiply 20 by 3/2.
    • Larger perimeter = 30 cm.

    Examiner tip: Do not multiply a perimeter directly by an area ratio.

  28. 28.

    From O, a walker travels 8 km east then 6 km north to P. Calculate OP and the three-figure bearing of P from O to the nearest degree.

    [4 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. The displacement forms a right triangle whose east component is opposite the angle measured from north.
    2. The angle from east would be its complement, not the requested bearing. Round only after evaluating the inverse tangent.

    Marking points

    • OP = sqrt(8^2 + 6^2) = 10 km.
    • Measure clockwise from north; tan(theta) = 8/6.
    • theta = 53.13... degrees.
    • The three-figure bearing is 053 degrees.

    Examiner tip: A bearing starts at north, not at the horizontal axis.

  29. 29.

    A solid consists of a cylinder of radius 3 cm and height 8 cm with a hemisphere of the same radius attached on top. Find its volume and exposed surface area in terms of pi. The bottom is exposed; the joining circle is internal.

    [5 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. Volumes add because the interiors do not overlap. Surface areas require identifying which boundaries touch the surrounding air.
    2. The circle at the join belongs to neither exposed boundary. Include the bottom disk once, along with the two curved surfaces.

    Marking points

    • Cylinder volume = pi(3^2)(8) = 72pi cm^3.
    • Hemisphere volume = (2/3)pi(3^3) = 18pi cm^3; total = 90pi cm^3.
    • Cylinder curved area = 2pi(3)(8) = 48pi cm^2.
    • Hemisphere curved area = 2pi(3^2) = 18pi cm^2.
    • Add the exposed base 9pi, not the internal join: total exposed area = 75pi cm^2.

    Examiner tip: Adding full cylinder and hemisphere surface areas double-counts the hidden join.

Original, unofficial practice questions from SubjectScout — not official past-paper questions.