Geometry and measures worksheet — GCSE Higher

1.A path goes from A(1, 1) to B(1, 5), then from B to C(6, 5). Work out the total length of the path from A to C.

2.A cone is cut by a flat plane parallel to its circular base, partway up between the base and the apex. What shape is the cross-section?

3.In quadrilateral ABCD the two sides AB and BC are each 6 cm long and meet each other at B. The two sides CD and DA are each 9 cm long and meet each other at D. No side of ABCD is parallel to any other side. Write down the mathematical name of this quadrilateral.

4.A right-angled triangle has two sides of length 5 cm and 12 cm, and the angle between those two sides is 90°. Work out the length of the hypotenuse.

5.A rotation of 180° about the point (3, 2) can be written as a reflection in the line x = 3, followed by a reflection in a second line, for every point in the plane. Which line is the second line?

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

7.In triangle ABC the angle at A is 90° and BC = 10 cm. For the angle at B, sin B = 3/5. Work out the length of AC.

8.A shape is reflected in the line x = 1, and the image is then reflected in the line x = 5. Which single transformation is equivalent to this combination, for every point?

9.The diagram shows the plan of the base layer of a solid built from identical cubes. A second layer of cubes is added on top, filling the entire base layer to a height of 2 cubes everywhere. Work out how many squares are visible in the plan view of this solid.

10.A hexagonal prism has 8 faces and 12 vertices. Using F + V − E = 2, work out how many edges it has.

11.A warehouse stores identical cube-shaped crates. Its plan view is a 2 by 4 rectangle of crate positions, and every position is filled to a height of 3 crates, except one corner position, which has only 2 crates stacked on it because a delivery was incomplete. How many crates are there in total?

12.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.

13.The diagram shows a solid resting on its side on a table. Work out which shape is its plan view, the view looking straight down from above.

14.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.

15.A parallelogram has an area of 54 cm² and a base of 9 cm. Work out its perpendicular height.