Geometry and measures worksheet — GCSE Higher

1.A circle has centre O and radius 13 cm. AB is a chord of the circle, and M is the midpoint of AB. OM is drawn, and OM = 5 cm. Work out the length of AB.

2.From an external point P, two tangents PA and PB touch a circle with centre O at the points A and B. Angle APB = 50°. Work out the size of angle OAB.

3.In triangle ABC, AB = 8 cm, BC = 11 cm and AC = 6 cm. Work out the size of angle ABC. Give your answer to 1 decimal place.

4.A plan of a school hall uses a scale of 1 : 250. A wall is drawn 3.4 cm long on the plan. Sam wants to know the wall's real length in metres. What is it?

5.A treasure map has a scale of 1 cm to 4 m. A path from the start to a rock is drawn as two straight sections, measuring 3 cm and 2.5 cm. Work out the total real distance from the start to the rock, in metres.

6.Four angles meet at a point. Three of them measure 82°, 105° and 96°. Work out the size of the fourth angle.

7.Lines JK and LM are parallel. A straight line crosses JK at point P and crosses LM at point Q. Angle KPQ = 118°. Angle KPQ and angle MQP are co-interior (allied) angles. Work out angle MQP.

8.A tangent to a circle with centre O touches the circle at point P, where OP = 5 cm. Point Q lies on the tangent so that PQ = 12 cm. Using the fact that a tangent is perpendicular to the radius at the point of contact, work out the length OQ.

9.A solid has 4 triangular faces, 4 vertices and 6 edges. What is the name of this solid?

10.Work out the exact value of (sin 30°)² + (cos 30°)².

11.A sector of a circle has radius 6 cm and an arc length of 15.7 cm. Using π = 3.14, work out the angle of the sector.

12.Two triangles are similar. The sides of the first triangle are 3 cm, 4 cm and 5 cm. The shortest side of the second triangle is 9 cm. Work out the scale factor from the first triangle to the second.

13.A line joins the points A(0, 0, 0) and B(9, 12, 8) in a three-dimensional coordinate system in which the z-axis is vertical. Work out the angle this line makes with the horizontal (the xy-plane). Give your answer correct to 1 decimal place.

14.A pinhole camera is set up at a pitch-side stand, with its aperture at the point O = (2, 1) on a grid. Light from the corner F of a triangular flag, at the point (4, 2), passes through the aperture and forms an inverted image on the film behind it. The projection is an enlargement of scale factor −1.5 centred at O. Work out the coordinates of the image of corner F.

15.A wooden cube has edges of length 4 cm. Work out the total surface area of the cube.