Geometry and measures worksheet — GCSE Higher

1.Work out the exact value of sin 45° × cos 45°.

2.A circular table mat has diameter 20 cm. Yasmin has a 70 cm length of ribbon and sews it around the edge of the mat with no overlap. Using π = 3.14, work out how much ribbon is left over after going around the mat once.

3.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.

4.Two similar ponds have perimeters in the ratio 4 : 11. The perimeter of the larger pond is 88 m. Work out the perimeter of the smaller pond.

5.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?

6.A landscaper marks out two triangular flower beds using stakes and a tape measure. Bed 1 has two edges of 5 m and 7 m, meeting at a corner with a marked angle of 60° between them. Bed 2 has edges of 5 m and 7 m, meeting at a corner also marked at 60° between them. The landscaper wants to check the two beds will be exactly the same size and shape using only these three measurements. Which condition proves the two beds are congruent?

7.A tent has the cross-section of a right-angled triangle. The sloping side of the cross-section is 10 m long, and it makes an angle of 60° with the horizontal ground. Using the exact value of cos 60°, work out the horizontal distance covered by the sloping side.

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.A regular polygon has an exterior angle of 45°. Work out the number of sides of the polygon.

10.Point Q has coordinates (3, 5). It is translated by the vector (1, −2), and the image is then enlarged by scale factor 3, centre the origin. Work out the coordinates of the final image of Q.

11.A cylindrical candle and a cone-shaped candle have the same base radius and the same height. State the cone's volume as a fraction of the cylinder's volume.

12.A, B, C and D are points on a circle with centre O, in that order around the circle. B lies on the major arc AC, and angle AOC = 152°. In triangle ACD, angle ACD = 30°. Work out the size of angle DAC.

13.A triangle has vertices A(0, 0), B(10, 0) and C(5, 12). Work out the area of the triangle.

14.OAB is a triangle, with OA = a and OB = b. E lies on OA produced beyond A, such that A is the midpoint of OE. F lies on AB such that FB is twice AF. G is the midpoint of OB. By finding the vectors EF and EG, show that E, F and G are collinear, and give the scalar k such that EF = k × EG.

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