Mathematics
Geometry and mensuration — Topics 3-6
- 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 - 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] - 3.
A sector of a circle has radius 8 cm and angle 45°. Calculate the area of the sector.
[3 marks] - 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] - 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] - 6.
A cone has base radius 6 cm and slant height 10 cm. Calculate its curved surface area, using the formula πrl.
[3 marks] - 7.
Calculate the exterior and interior angle of a regular decagon (10-sided polygon).
[3 marks] - 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] - 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] - 10.
A pyramid has a square base of side 6 cm and vertical height 8 cm. Calculate its volume.
[3 marks] - 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 - 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 - 13.
The bearing of point B from point A is 072°. Find the bearing of A from B.
[3 marks] - 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] - 15.
Calculate the volume of a sphere with radius 6 cm, giving your answer to three significant figures. (V = (4/3)πr³)
[4 marks] - 16.
Calculate the surface area of a sphere with radius 6 cm, giving your answer to three significant figures. (A = 4πr²)
[3 marks] - 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] - 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 - 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] - 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] - 21.
Calculate the sum of the interior angles of a regular polygon with 12 sides.
[3 marks] - 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] - 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] - 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] - 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] - 26.
A right-angled triangle has perpendicular sides 9 cm and 12 cm. Find its hypotenuse and area.
[3 marks] · no calculator - 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 - 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] - 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