Geometry and measures worksheet — GCSE Higher

1.Put sin 30°, tan 30° and cos 30° in order of size, starting with the smallest.

2.Lines AB and CD are parallel. A straight line EF crosses AB at point P and crosses CD at point Q. At P, angle APE = 65°. Angle APE and angle BPQ are vertically opposite. Angle BPQ and angle PQD are co-interior (allied) angles. Work out angle PQD.

3.A sector of a circle has radius 9 cm and angle 40°. Work out the area of the sector, in terms of π.

4.A sector of a circle has an angle of 90° at the centre. Write down what fraction of the whole circle this sector represents.

5.Triangle PQR has angle P = 90°. The side PQ = 9 cm and the hypotenuse QR = 15 cm. Work out the length of PR.

6.In a kite, one pair of opposite angles are equal in size. In kite WXYZ, angle X = 40° and angle Z = 100° are the two angles that are NOT equal to each other. The other two angles, W and Y, are equal to each other. Work out the size of angle W.

7.A birthday cake is a cylinder of radius 10 cm and height 8 cm, with a thin ribbon fixed exactly once around the curved side, at half the height, and joined with no overlap. Work out the length of ribbon needed. Use π = 3.14.

8.Point P has coordinates (1, 3). P is reflected in the line x = 4, then the image is reflected in the line y = 1, and then that image is translated by the vector (2, −3). Work out the coordinates of the final image of P.

9.A textbook contains the instruction 'Draw AB' and, on the next line, the statement 'AB = 5 cm'. Which statement about the notation AB is correct?

10.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.

11.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.

12.Shape T has a vertex at (5, 3). T is enlarged by scale factor −1, centre (2, 1). Work out the coordinates of the image of the vertex (5, 3).

13.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?

14.A composite solid is made from a cylinder of radius 3 cm and height 10 cm, with a cone of the same radius and vertical height 4 cm fixed on top, apex upward. Work out the total volume of the solid. Use π = 3.14 and give your answer correct to 1 decimal place.

15.Two sides of a triangle are 9 cm and 15 cm long. Write down which one of these lengths is possible for the third side.