3D geometry: distance, midpoint, volume, surface area practice questions.

Calculate the distance between the points A(1, −2, 3) and B(4, 2, 3).

• 7
• 5
• 25
• √7

Answer: 5

d = √((4−1)² + (2−(−2))² + (3−3)²) = √(9 + 16 + 0) = √25 = 5. Remember to square the differences before adding, then take the square root.

A sphere has radius 2. Determine its volume.

• 32π/3
• 16π
• 8π/3
• 32π

Answer: 32π/3

V = (4/3)πr³ = (4/3)π(8) = 32π/3. Note that 16π is the surface area 4πr², a common mix-up.

A right cone has base radius 6 and vertical height 8. Determine its curved surface area.

• 48π
• 96π
• 60π
• 80π

Answer: 60π

First find the slant height: l = √(6² + 8²) = 10. Curved surface area = πrl = π(6)(10) = 60π. Using the vertical height instead of the slant height gives the wrong 48π.

A vertical pole of height 5 m stands at point B on horizontal ground. Point A lies on the ground with AB = 5√3 m. Determine the angle of elevation of the top of the pole from A.

• 60°
• 45°
• 20°
• 30°

Answer: 30°

tan θ = opposite/adjacent = 5/(5√3) = 1/√3, so θ = 30°. Choosing 60° comes from inverting the ratio.

A solid hemisphere has radius 2. Determine its total surface area, including the flat circular face.

• 8π
• 12π
• 16π
• 24π

Answer: 12π

Curved part: (1/2)(4πr²) = 8π; flat circular face: πr² = 4π. Total = 12π. Forgetting the flat face gives 8π.

A cylinder and a cone have equal base radii and equal heights. State the ratio of the volume of the cylinder to the volume of the cone.

• 2 : 1
• 3 : 2
• 3 : 1
• π : 1

Answer: 3 : 1

V(cylinder) = πr²h and V(cone) = (1/3)πr²h, so the ratio is 1 : 1/3 = 3 : 1.