Geometry and measures worksheet — GCSE Higher

1.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.

2.A trundle wheel has a diameter of 0.5 m. It is rolled along the ground and makes 20 complete turns. Using π = 3.14, work out the total distance rolled, in metres.

3.A roof has two triangular sections, P and Q. Section P has a sloping edge of 3.6 m and a base edge of 2.4 m, with an angle of 70° between them. Section Q has a sloping edge of 3.6 m and a base edge of 2.4 m, with an angle of 70° between them, arranged the same way round. A roofer wants to cut identical triangular felt panels for both sections without measuring Section Q separately. Which condition confirms that the two sections are congruent?

4.A triangular sail for a small boat has a base of 2.4 m and a height of 1.75 m. Sailcloth costs £12.50 per m². Work out the total cost of the sailcloth needed for the sail.

5.A map has a scale of 1 : 25000. A distance between two points measures 2 cm on the map. What is the real distance, in kilometres?

6.In triangle ABC, AB = 6 cm, angle ABC = 40° and AC = 9 cm. Using the sine rule, sin(ACB) = AB × sin(ABC) ÷ AC, which gives two possible values for angle ACB: 25.4° or 154.6° (each to 1 decimal place). By considering the angle sum of the triangle, decide which of these two values angle ACB can actually take, and hence work out angle BAC. Give your answer to 1 decimal place.

7.OAB is a triangle, with OA = a and OB = b. E lies on OA produced beyond A, such that A is the midpoint of OE. F lies on AB such that FB is twice AF. G is the midpoint of OB. By finding the vectors EF and EG, show that E, F and G are collinear, and give the scalar k such that EF = k × EG.

8.A square-based pyramid has a square base and four identical triangular sloping faces, with its apex directly above the centre of the base. How many planes of symmetry does it have?

9.In triangle ABC, AB = x cm, AC = (x + 3) cm, BC = 7 cm and angle BAC = 60°. Work out the value of x.

10.In parallelogram PQRS, angle P = 65°. Which reason correctly explains why angle R = 65°?

11.A bead starts at position (2, −1) on a grid, in centimetres. It is moved by the column vector u, with top number 3 and bottom number 5, and then moved by the column vector v, with top number −7 and bottom number 2. Work out the coordinates of the bead's final position.

12.A wooden cube has edges of length 4 cm. Work out the total surface area of the cube.

13.Triangle PQR has angle P = 90°. The side PQ = 9 cm and the hypotenuse QR = 15 cm. Work out the length of PR.

14.A, B, C and D are points on a circle, with B and D both on the same major arc AC. E lies on the straight line through A and B, beyond B, so that angle CBE = 145°. Work out the size of angle ADC.

15.A sector of a circle has radius 8 cm and takes up three-quarters of the full circle. Work out the perimeter of the sector, in terms of π.