3D distance/midpoint; volume and surface area of solids; angles practice questions.

A right pyramid has a square base of side 10 cm and vertical height 12 cm. Calculate its volume, in cm³. (Use V = 1/3 × base area × height.)

• 400 cm³
• 1200 cm³
• 600 cm³
• 1000 cm³

Answer: 400 cm³

V = 1/3 × (10×10) × 12 = 1/3 × 100 × 12 = 400 cm³.

In a cuboid room 8 m long, 6 m wide and 3 m high, a wire runs from a bottom corner to the diagonally opposite top corner. Determine the angle this wire makes with the floor, to the nearest degree.

• 17°
• 21°
• 27°
• 31°

Answer: 17°

Floor diagonal = √(8² + 6²) = 10 m. The angle θ with the floor satisfies tan θ = 3/10, so θ = arctan(0.3) = 16.7° ≈ 17°.

A sphere of radius 3 cm and a cylinder of radius 3 cm and height 3 cm are compared. Determine which has the greater volume and by how much (in terms of π).

• Sphere, by 9π cm³
• Cylinder, by 9π cm³
• They are equal
• Sphere, by 27π cm³

Answer: Sphere, by 9π cm³

Sphere V = 4/3·π·3³ = 36π; cylinder V = π·3²·3 = 27π. The sphere is greater by 36π − 27π = 9π cm³.

A square-based right pyramid has base side 8 cm and vertical height 3 cm. Determine the angle between a sloping face and the base, to 3 s.f. (Use the apothem: the distance from base centre to the midpoint of a base edge is 4 cm.)

• 36.9°
• 20.6°
• 27.9°
• 53.1°

Answer: 36.9°

The face makes angle θ with the base where tan θ = height/apothem = 3/4, so θ = arctan(0.75) = 36.9°.

An open-topped cylindrical water butt has radius 0.4 m and height 1.2 m. It is to be painted on the outside, including the base but not the missing lid. Calculate the area to be painted, in m², correct to three significant figures.

• 3.52 m²
• 4.02 m²
• 3.02 m²
• 2.01 m²

Answer: 3.52 m²

One circular base plus the curved surface: πr² + 2πrh = 0.503 + 3.016 = 3.52 m². The value 4.02 counts two circular ends, 3.02 is the curved surface alone and 2.01 comes from omitting the factor 2 in 2πrh.

A monument is a right pyramid on a square base of side 6 m with vertical height 4 m. Determine the angle between a slant edge and the base, correct to three significant figures.

• 43.3°
• 53.1°
• 46.7°
• 36.9°

Answer: 43.3°

A slant edge meets the base at a corner, so the horizontal side of the right-angled triangle is half the base diagonal, (6√2)/2 = 4.243 m, giving tan θ = 4/4.243 and θ = 43.3°. Using the apothem 3 m instead gives 53.1°, which is the angle of a sloping face, and 46.7° comes from inverting the ratio.