Geometry and measures worksheet — GCSE Higher

1.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?

2.A circle has two straight radii drawn from the centre to the edge, together with the arc between them. Write down the name of the region enclosed by these two radii and the arc.

3.A triangle has vertices X, Y and Z, with a right angle at Y. Side XY is twice as long as side YZ. Which of the following correctly describes this triangle?

4.In triangle OAB, OA = a and OB = b. P is the point on OA such that OP = (1/3)a, and Q is the point on OB such that OQ = (1/3)b. Express the vector PQ in terms of a and b.

5.A solid is built from centimetre cubes standing on a table. Seen from the front, the left-hand column is 2 cubes high, the middle column is 2 cubes high and the right-hand column is 1 cube high. Work out how many squares make up the front elevation.

6.A cuboid has three different edge lengths: a length, a width and a height, all different from each other. How many rectangles make up its net in total?

7.The diagram shows a solid made from two cuboids joined together. Which shape is the front elevation of this solid?

8.A cube has edges of length 5 cm. Work out the length of the diagonal that runs from one corner of the cube to the opposite corner. Give your answer as a surd in its simplest form.

9.Four angles meet at a point. Three of them measure 82°, 105° and 96°. Work out the size of the fourth angle.

10.Point P has coordinates (4, 2). P is rotated 90° clockwise about the point (1, 1), and the image is then reflected in the line x = 1. Work out the coordinates of the final image of P.

11.A, B, C and D are points on a circle with centre O, placed in that order around the circle so that ABCD is a cyclic quadrilateral. B lies on the major arc AC and D lies on the minor arc AC. Angle AOC = 104°. Work out the size of angle ADC.

12.A ship's radio can be heard up to 30 km from the ship. A lighthouse's light can be seen up to 20 km from the lighthouse. The ship and the lighthouse are 40 km apart along the coast. Describe the region where BOTH the radio can be heard AND the light can be seen.

13.Two spheres are similar solids. Their radii are in the ratio 3 : 4. Write down the ratio of their surface areas.

14.A, B and C are points on a circle with centre O. B is on the minor arc AC. Angle ABC = 100°. Work out the size of the non-reflex angle AOC.

15.Two triangles are proved congruent using the ASA condition. What can be concluded about the two remaining pairs of corresponding sides that were not part of the original ASA facts?