Geometry and measures worksheet — GCSE Higher

1.A footpath is 3750 m long. Work out this distance in kilometres.

2.A trapezium has parallel sides of length 6 cm and 10 cm, and a perpendicular height of 4 cm. Work out its area.

3.Triangle PQR has vertices P(0, 0), Q(6, 0) and R(0, 8). Work out the perimeter of the triangle.

4.In triangle ABC, AB = 12 cm, AC = 7 cm and the area of the triangle is 33 cm². Given that angle BAC is obtuse, work out the size of angle BAC. Give your answer to 1 decimal place.

5.A mast stands vertically on level ground and its top is 60 m above the ground. Amelia and Noah stand on the ground on the same side of the mast, in line with its foot. The angle of elevation of the top of the mast is 30° from where Amelia stands and 60° from where Noah stands. Work out the distance between Amelia and Noah. Give your answer to 1 decimal place.

6.A ship's radar can detect other ships up to 20 km away, over a sector-shaped area with angle 90°. A monitoring company charges £0.50 per square kilometre covered by one radar sweep. Using π = 3.14, work out the total charge for one sweep.

7.In a kite, one pair of opposite angles are equal in size. In kite WXYZ, angle X = 40° and angle Z = 100° are the two angles that are NOT equal to each other. The other two angles, W and Y, are equal to each other. Work out the size of angle W.

8.A circle has circumference 31.4 cm. Using π = 3.14, work out the radius of the circle.

9.A picture frame is a rhombus with a diagonal of 16 cm and a diagonal of 12 cm. The two diagonals meet at right angles at their midpoints. Work out the length of one side of the rhombus.

10.A locus consists of all the points that are the same perpendicular distance from two parallel lines 6 cm apart. Describe this locus.

11.From an external point P, two tangents PA and PB touch a circle with centre O at points A and B. Angle APB = 44°. C is a point on the major arc AB. Using the fact that PA = PB, and the alternate segment theorem, work out the size of angle ACB.

12.A garden plan is drawn to a scale of 1 cm : 1.8 m. A path on the plan measures 4.5 cm. What is the real length of the path, in metres, to 1 decimal place?

13.A hiker walks on a bearing of 065°. She then turns clockwise through 90° and continues walking in a straight line. What bearing is she now walking on?

14.A builder props a straight plank against a vertical wall to reach a window ledge. The foot of the plank is 2.1 m from the base of the wall, and the plank is 3.5 m long. Work out how high up the wall the plank reaches.

15.In triangle ABC the angle at C is 90°, AC = 8 cm and the angle at A is 30°. Work out the length of BC. Give your answer to 1 decimal place.