Geometry and measures worksheet — GCSE Higher

1.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.

2.A straight line crosses a pair of parallel lines. One of the co-interior (allied) angles is 118°. Work out the size of the other co-interior angle.

3.A circular tabletop has a radius of 15 cm. Work out the area of the tabletop. Use π = 3.14. Give your answer to 1 decimal place.

4.A tank holds 3.2 m³ of water. Work out this volume in litres.

5.Rectangle ABCD has vertices A(−1, 2), B(4, 2), C(4, 5) and D(−1, 5). The rectangle is translated so that C moves to (0, −1). Write down the coordinates of the image of A.

6.A solid metal sphere has a radius of 5 cm. It is melted down and recast into a solid cylinder with the same radius, 5 cm. Work out the height of the cylinder, so that its volume equals the volume of the sphere. Use π = 3.14 and give your answer correct to 1 decimal place.

7.Work out the exact value of sin 45° × cos 45°.

8.A sign-maker is told a metal bracket outline must have two pairs of adjacent sides equal in length (not opposite pairs), exactly one pair of opposite angles equal, and exactly one line of symmetry. Which quadrilateral matches this description?

9.A rhombus has all four sides equal in length. Which statement about a rhombus is correct?

10.Triangle ABC is extended so that side BC is extended beyond C to a point D, forming exterior angle ACD. Angle BAC = 48° and angle ABC = 67°. Which single angle fact gives angle ACD = 115° in one step, without first working out the interior angle at C?

11.Triangles ABE and CDE share the vertex E, where lines AC and BD cross at E. AE equals CE, and BE equals DE. Angle AEB and angle CED are formed as vertically opposite angles where the lines cross. Which condition proves that triangle ABE is congruent to triangle CDE?

12.Two similar triangular tiles have lengths in the ratio 3 : 5. The longest side of the smaller tile is 9 cm. Work out the length of the longest side of the larger tile.

13.A sector of a circle has radius 9 cm and angle 40°. Work out the area of the sector, in terms of π.

14.AT is a diameter of a circle with centre O. PT is a tangent to the circle at the point T, and P is a point on this tangent such that A, T and P form a triangle. Angle PAT = 28°. Work out the size of angle APT.

15.A solid pyramid has a square base of side 10 cm and four identical triangular faces. Its apex is directly above the centre of the base, at a vertical height of 12 cm, and the slant height of each triangular face — the distance from the apex to the midpoint of a base edge — is 13 cm. Work out the total surface area of the pyramid.