Geometry and measures worksheet — GCSE Higher

1.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.

2.A ship's radar can detect other ships up to 20 km away, over a sector-shaped area with angle 90°. A monitoring company charges £0.50 per square kilometre covered by one radar sweep. Using π = 3.14, work out the total charge for one sweep.

3.Point A and point B are 5 cm apart. How many points are exactly 4 cm from A and exactly 3 cm from B?

4.A regular pentagon has centre O. Each vertex is 8 cm from O, and each of the 5 triangles formed by joining O to the vertices is isosceles with an angle of 72° at O. Work out the area of the pentagon. Give your answer to 1 decimal place.

5.Triangle ABC has AB = (2x + 3) cm, BC = 11 cm and angle B = 90°. Triangle DEF has DE = 13 cm, EF = 11 cm and angle E = 90°. Given that the two triangles are congruent by SAS, work out the value of x.

6.In triangle ABC and triangle DEF, AB = DE, AC = DF, and the angle at A equals the angle at D. Write down the congruence condition that proves the two triangles are congruent.

7.A rectangular photograph has an area of 96 cm². One of its sides is 8 cm long. Work out the length of the other side.

8.To prove that the angle at the centre is twice the angle at the circumference, a student draws radii OA and OC, where A and C are points on the circle and O is the centre. The student's first step is: 'Since OA = OC, both being radii of the circle, triangle OAC is isosceles, so angle OAC = angle OCA.' Is this first step correct?

9.A regular nonagon has 9 sides. Work out the sum of its interior angles.

10.A plumber charges a £45 call-out fee, plus £28 for each hour worked. She works on a job for 3 hours 30 minutes, and her hours are billed by rounding up to the next whole hour. Work out the total amount she charges.

11.A garden is made of two triangular flower beds, ABC and ACD, joined along the edge AC. In triangle ABC, angle ABC = 90°, AB = 5 m and BC = 12 m. In triangle ACD, AD = 9 m and angle CAD = 40°, the angle between AC and AD. Work out the area of flower bed ACD. Give your answer to 1 decimal place.

12.Triangle T has a vertex A at (3, 1). It is enlarged by a scale factor of −2, centre (1, 1). Work out the coordinates of the image of point A.

13.Shape S has an area of 5 cm². Shape S is enlarged by a scale factor of −3 to give shape T. Work out the area of shape T.

14.A window is made from a rectangle with a semicircle on top. The rectangle is 40 cm wide and 60 cm tall. The semicircle has the same width as the rectangle. Work out the total area of the window. Use π = 3.14. Give your answer to the nearest whole number.

15.In triangle OAB, OA = a and OB = b. M is the midpoint of OA, and N is the midpoint of OB. Express the vector MN in terms of a and b.