Questions: Volume of Pyramids and Cones

Question 1

A cone and a cylinder share the same circular base (radius 3 cm) and the same height (7 cm). What is the volume of the cone?

• A. 63π cm³
• B. 21π cm³
• C. 9π cm³
• D. 3π cm³

Answer: (B)

V = (1/3)πr²h = (1/3) × π × 9 × 7 = 21π cm³. Option A (63π) is the cylinder's volume — the most common error is omitting the 1/3 factor. The cone holds exactly one-third as much as the matching cylinder, which can be verified by filling the cone with water and pouring it into the cylinder three times to fill it completely.

Question 2

A right square pyramid has a base side of 6 cm and a slant height of 5 cm. What is its volume?

• A. 60 cm³
• B. 48 cm³
• C. 72 cm³
• D. 36 cm³

Answer: (B)

The slant height (5 cm) is not the perpendicular height. The distance from the base center to the midpoint of an edge is half the side length = 3 cm. Using the Pythagorean theorem: perpendicular height = √(5² − 3²) = √16 = 4 cm. Volume = (1/3)(6²)(4) = (1/3)(36)(4) = 48 cm³. Option A (60 cm³) is the error from substituting slant height directly: (1/3)(36)(5) = 60.

Question 3

A cone and a cylinder with the same base and height have equal volumes.

• T. True
• F. False

Answer: False

The cone's volume is exactly one-third the cylinder's. The cylinder uses V = B × h; the cone uses V = (1/3) × B × h. The 1/3 factor arises because the cone's horizontal cross-sections taper toward the apex, unlike the cylinder's uniform cross-sections. Filling the cone three times fills the matching cylinder exactly.

Question 4

The slant height of a right cone is always greater than its perpendicular height.

• T. True
• F. False

Answer: True

The slant height is the hypotenuse of a right triangle whose legs are the perpendicular height and the base radius. Since a hypotenuse is always longer than either leg, the slant height must exceed the perpendicular height. This is why substituting slant height into V = (1/3)Bh always overestimates the volume.

Question 5

Explain in your own words why the volume of a cone is one-third the volume of a cylinder with the same base and height, rather than one-half.

Model answer: As you move upward through a cone, horizontal cross-sections shrink as the square of the fraction of height remaining — a cross-section at height z has area B × ((h−z)/h)². Integrating these shrinking sections from base to apex gives (1/3)Bh. The 1/3 comes from integrating a squared function, not a linear one, which would give 1/2.

The cross-sections shrink in two dimensions simultaneously (both width and depth taper), so their area shrinks as the square of the linear scale factor. Integrating x² from 0 to 1 gives 1/3, not 1/2. A result of 1/2 would arise only if the cross-sections shrank in one dimension, but area scales as the square of linear dimensions — which is why the factor is precisely 1/3.