Geometry and measures worksheet — GCSE Higher

1.At a cycling event, a circular track has centre O. Two flags, F and M, are fixed on the track. A course marker P is also on the track, positioned so that angle FOP = 55° and angle POM = 43°, with P between F and M as seen from O. A spectator S stands on the major arc FM, on the opposite side of the track from P. The event programme needs angle FSM. Work out angle FSM.

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

3.What does the statement AB = CD mean, where A, B, C and D are all points?

4.A patio plan states: 'Side AB of the patio is parallel to side DC, and AB is longer than DC. Sides BC and AD are not parallel to each other and are not equal in length.' The four sides measure AB = 5.1 m, BC = 3.2 m, DC = 2.8 m and AD = 4.5 m. The tiler needs to order edging strip only for the sides that are not parallel to any other side. What total length of edging strip, in metres, should the tiler order?

5.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?

6.Triangle DEF is isosceles, with DE = DF. Angle E = 58°. Work out angle F.

7.A chord divides a circle into two segments of different sizes. Write down the name given to the smaller of the two segments.

8.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.

9.A fan-shaped rug is in the shape of a sector of a circle with radius 4 m and angle 180°. It costs £4 per square metre to make. Using π = 3.14, work out the total cost of the rug.

10.A, B, C and D are points on a circle, and ABCD is a cyclic quadrilateral with the vertices in that order around the circle. Angle ABC = 108° and angle ADC = 72°. Which circle theorem is the reason that angle ABC + angle ADC = 180°?

11.Two cylindrical storage tins are geometrically similar. The smaller tin has height 4 cm and the larger tin has height 6 cm. Work out the ratio of the surface area of the smaller tin to the surface area of the larger tin, giving your answer in the form a : b in its simplest form.

12.A quadrilateral has all four sides equal in length and all four interior angles equal to 90°. Its diagonals are equal in length and bisect each other at right angles. Which quadrilateral matches this description exactly?

13.Work out the distance between the points (−3, 4) and (5, −2).

14.A garden designer describes a flower bed as a shape with four straight sides: exactly one pair of sides is parallel, those two parallel sides are different lengths, and the other two sides are equal in length to each other but not parallel to each other. Which shape should the designer draw?

15.A rectangular gate is braced with a diagonal strut. The gate is 1.2 m wide and 0.9 m tall. Work out the exact length of the diagonal strut.