Geometry and measures worksheet — GCSE Higher

1.A triangular solar panel has two edges of length 1.8 m and 1.3 m, with an angle of 72° between them. Manufacturing the panel costs £85 per square metre. Work out the total manufacturing cost. Give your answer to the nearest penny.

2.Triangle T has a vertex A at (4, 6). It is enlarged by a scale factor of −1/2, centre (2, 2). Work out the coordinates of the image of point A.

3.A triangle has one unknown side or angle, marked x. In which one of these situations should the sine rule, rather than the cosine rule, be used to find x?

4.From an external point P, two tangents PA and PB touch a circle with centre O at A and B. Angle AOB = 112°. Work out the size of angle APB.

5.A chocolate bar is a prism. Its cross-section is a triangle with a base of 6 cm and a perpendicular height of 5 cm, and the bar is 12 cm long. Work out the volume of the bar.

6.A parallelogram has an area of 84 cm² and a base of 12 cm. Work out the perpendicular height of the parallelogram.

7.AB is a diameter of a circle with centre O, and C is a point on the circle. A student writes four statements to prove that angle ACB = 90°. Statement 1: OA = OC, since both are radii, so triangle OAC is isosceles with angle OAC = angle OCA. Statement 2: OB = OC, since both are radii, so triangle OBC is isosceles with angle OBC = angle OCB. Statement 3: in triangle ABC, the three angles sum to 360°, so angle OAC + angle OBC + angle ACB = 360°, meaning 2 × angle ACB = 360° and angle ACB = 180°. Statement 4: A, O and B lie on a straight line, since AB is a diameter through the centre O. Which one of these four statements is mathematically incorrect?

8.A triangular sail for a small boat has two sides of 4.2 m and 3.6 m, with an angle of 115° between them. One litre of waterproofing paint covers 3 m² of sail. Work out the least number of whole tins of paint (1 litre each) needed to cover the sail.

9.Shape S has an area of 48 cm². It is enlarged by a scale factor of 1/4 to give shape T. Work out the area of shape T.

10.In triangle XYZ, the side opposite vertex X is labelled using the standard lowercase-letter convention. Which label is correct?

11.A scale drawing shows a room that is 5 cm wide. The real room is 15 m wide. Write the scale of the drawing in the form 1 : n.

12.A solid has 5 faces, 9 edges and 6 vertices. Two of its faces are triangles and the other three are rectangles. Which solid is it?

13.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?

14.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.