Geometry and measures worksheet — GCSE Higher

1.A radar station is at the origin of a grid, where each unit represents 1 km. A boat is detected at the point (7, 24). Work out the boat's distance from the station, then work out how many hours it will take the boat to reach the station travelling directly towards it at 5 km per hour.

2.A pop-up canopy has two sloping supports that meet at the top. Each support makes an angle of 30° with the ground and is 6 m long. Using the exact value of cos 30°, work out the total width of the canopy's base.

3.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.

4.A rectangular notice board measures 150 cm by 80 cm. Work out its area in m².

5.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.

6.A circle has diameter 10 cm. Using π = 3.14, work out the circumference of the circle.

7.A solid has two parallel, congruent polygon faces joined by flat rectangles, and its cross-section is the same shape all the way along its length. Which solid is it?

8.Two bronze statues are geometrically similar and made from the same solid bronze. Their heights are 30 cm and 45 cm. The smaller statue has a mass of 12 kg. Work out the mass of the larger statue, in kg.

9.Triangle ABC is translated by the column vector with top number 3 and bottom number −5 to form triangle A′B′C′. Triangle A′B′C′ is then translated by the column vector with top number −7 and bottom number 2 to form triangle A″B″C″. Work out the single column vector that translates triangle ABC directly to triangle A″B″C″.

10.A trapezium-shaped allotment plot has two parallel sides that face each other, and two sloping sides that are equal in length (an isosceles trapezium). One of the angles next to the shorter parallel side is 118°. Work out the angle next to the other end of the same shorter parallel side.

11.A shop sells a scale model of a van. The model and the real van are similar shapes, with lengths in the ratio 1 : 24. The model is 18.5 cm long. Work out the length of the real van, giving your answer in metres.

12.Two straight roads cross at a junction. The angle between them, measured with a protractor, is 118°. What is the size of the angle vertically opposite to it?

13.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.

14.In triangle ABC, angle ABC = 90° and angle BAC = 30°. The hypotenuse AC = 12 cm. Work out the exact length of AB.

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