Geometry and measures worksheet — GCSE Higher

1.A children's slide is built as a right-angled triangle: the vertical ladder side is 2.4 m, the horizontal base along the ground is 3.2 m, and the sloping slide is the third side. Work out the exact length of the sloping slide.

2.A ramp rises at 30° to the horizontal. Its sloping surface is 4.8 m long. Safety rules say the vertical rise of a ramp must be no more than 2.5 m. Work out how far below that limit the rise of this ramp is.

3.A cheese counter sells cheddar at £8.40 per kilogram. Work out the cost of a 350 g piece of cheddar.

4.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.

5.Two vertical masts stand on level ground, 40 m apart. One mast is 30 m tall and the other is 50 m tall. Work out the straight-line distance between the tops of the two masts. Give your answer to 1 decimal place.

6.A circle has diameter 10 cm. Using π = 3.14, work out the circumference of the circle.

7.Three of these are rational numbers and one is irrational. Write down the one that is irrational.

8.A circle has centre O. Radii OA and OB are drawn, together with the arc AB, enclosing a sector. The straight chord AB is also drawn. Write down the name of the region enclosed between the chord AB and the arc AB.

9.A solid metal sphere has a radius of 5 cm. It is melted down and recast into a solid cylinder with the same radius, 5 cm. Work out the height of the cylinder, so that its volume equals the volume of the sphere. Use π = 3.14 and give your answer correct to 1 decimal place.

10.A straight line touches the edge of a circle at exactly one point and does not cross into the circle at all. Write down the term for this line.

11.Triangle ABC has AB = 8 cm, BC = 10 cm and CA = 6 cm. Triangle LMN has LM = 10 cm, MN = 6 cm and NL = 8 cm. The two triangles are congruent by SSS. Which of the following correctly matches the corresponding vertices?

12.A rectangular name card has a perimeter of 30 cm. The card is 9 cm long. Work out the width of the card.

13.Point E is at (4, 2). It is rotated 180° about the point (1, 1). Work out the coordinates of the image of point E.

14.A locus consists of all the points that are the same perpendicular distance from two parallel lines 6 cm apart. Describe this locus.

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