Geometry and measures

In a right-angled triangle the side opposite angle θ is 4 cm and the hypotenuse is 9 cm. Calculate θ to 1 decimal place.

A cuboid measures 15 cm by 8 cm by 6 cm. Calculate the length of the longest diagonal to 1 decimal place.

Triangle ABC has AC = 12 cm, BC = 15 cm and angle ACB = 37°. (a) Calculate AB to 3 significant figures. (b) Calculate the area of the triangle to 3 significant figures.

In triangle PQR, angle P = 64°, angle Q = 49° and QR = 18 cm. Calculate PR to 3 significant figures.

A solid cone has radius 7 cm and perpendicular height 18 cm. Calculate (a) its volume and (b) its total surface area, both to 3 significant figures. (V = ⅓πr²h, curved surface area = πrl)

Two mathematically similar solids have lengths in the ratio 1 : 5. The smaller has surface area 18 cm² and volume 7 cm³. Find the surface area and volume of the larger solid.

A sector has radius 10 cm and angle 84°. Work out (a) its area and (b) its perimeter, to 1 decimal place.

A garden is a trapezium with parallel sides 14 m and 20 m and perpendicular height 6 m. A circular pond of radius 1.2 m is dug in it. Turf costs £5.25 per m². Work out the cost of turfing the rest of the garden.

A sphere has radius 6 cm. Work out its volume and surface area to 3 significant figures. (V = 4/3πr³, A = 4πr²)

Work out the exact value of tan 30° × cos 30° + sin 45° × cos 45°.

A, B and C are points on a circle with centre O. Angle AOC = 144°. B is on the major arc. (a) Find angle ABC. (b) D is on the minor arc AC. Find angle ADC. Give a reason for each answer.

Describe fully the single transformation that maps the triangle with vertices (1, 2), (3, 2), (3, 5) onto the triangle with vertices (2, 1), (2, 3), (5, 3).

OPQR is a trapezium with OP = 2a, OR = b and RQ = a. X is the midpoint of PQ. Find OX in terms of a and b, and explain whether OX is parallel to RQ.

A regular polygon has 9 sides. Work out the size of each interior angle, and explain why regular polygons with interior angle 100° cannot exist.

Two parallel lines are crossed by a transversal. One of the angles formed is 117°. Work out the size of the co-interior angle and the alternate angle, giving reasons.

In a right-angled triangle the side opposite angle θ is 13 cm and the hypotenuse is 21 cm. Calculate θ to 1 decimal place.

A cuboid measures 9 cm by 9 cm by 4 cm. Calculate the length of the longest diagonal to 1 decimal place.

Triangle ABC has AC = 5.2 cm, BC = 7.9 cm and angle ACB = 99°. (a) Calculate AB to 3 significant figures. (b) Calculate the area of the triangle to 3 significant figures.

A solid cone has radius 6.5 cm and perpendicular height 15 cm. Calculate (a) its volume and (b) its total surface area, both to 3 significant figures. (V = ⅓πr²h, curved surface area = πrl)

Two mathematically similar solids have lengths in the ratio 1 : 2. The smaller has surface area 72 cm² and volume 45 cm³. Find the surface area and volume of the larger solid.

A sector has radius 14 cm and angle 50°. Work out (a) its area and (b) its perimeter, to 1 decimal place.

Two parallel lines are crossed by a transversal. One of the angles formed is 124°. Work out the size of the co-interior angle and the alternate angle, giving reasons.

A regular polygon has 18 sides. Work out the size of each interior angle, and explain why regular polygons with interior angle 100° cannot exist.

In triangle PQR, angle P = 38°, angle Q = 81° and QR = 9.5 cm. Calculate PR to 3 significant figures.

A cylindrical tank has radius 25 cm and height 60 cm. Water flows in at 8 litres per minute. How long does it take to fill the tank? Give your answer in minutes to 1 decimal place. (1 litre = 1000 cm³)