Questions: Volume of Spheres

Question 1

A sphere has radius 3 cm. You double the radius to 6 cm. By what factor does the volume increase?

• A. 2 — the volume doubles because the radius doubled
• B. 4 — the volume quadruples because area scales as r²
• C. 6 — the factor matches the new radius value
• D. 8 — the volume increases 8-fold because volume scales as r³

Answer: (D)

Volume scales as r³. When radius doubles, volume increases by 2³ = 8. This is the most counterintuitive property of sphere volume: a seemingly modest linear change produces a dramatic volumetric change. Answer A would be correct if volume scaled linearly; answer B would be correct if volume scaled as r² (like area does). Recognizing that volume involves r³ while area involves r² is the key dimensional check for all 3D formulas.

Question 2

A sphere is inscribed in a cylinder such that the sphere's diameter equals both the cylinder's diameter and height. What fraction of the cylinder's volume does the sphere occupy?

• A. 1/3
• B. 1/2
• C. 2/3
• D. 3/4

Answer: (C)

This is Archimedes' proportion. The cylinder has volume πr² × 2r = 2πr³. The sphere has volume (4/3)πr³. Dividing: (4/3)πr³ ÷ 2πr³ = (4/3) ÷ 2 = 2/3. The sphere occupies exactly 2/3 of its circumscribed cylinder. The remaining 1/3 is the gap between sphere and cylinder. This 2/3 relationship is part of the elegant 1:2:3 ratio between cone, sphere, and cylinder sharing the same radius and height equal to the diameter.

Question 3

The volume formula V = (4/3)πr³ and the surface area formula SA = 4πr² involve the same variables, so either could be used to find volume depending on which is easier to remember.

• T. True
• F. False

Answer: False

These formulas measure fundamentally different things and cannot be substituted for each other. Volume involves r³ and is measured in cubic units (cm³, m³); surface area involves r² and is measured in square units (cm², m²). This dimensional difference is a built-in error check: if you compute 'volume' and get an answer involving r², you've used the wrong formula. The structural distinction between r² (area) and r³ (volume) is more important than the specific coefficients.

Question 4

When solving for the radius of a sphere given its volume, you must take the cube root of a rearranged expression — not the square root.

• T. True
• F. False

Answer: True

From V = (4/3)πr³, solving for r gives r³ = 3V/(4π), so r = ∛(3V/4π). Students who reach automatically for a square root are applying the logic of 2D area formulas, where r = √(A/π) for a circle. That logic does not transfer to volume. The cube root step is the one most commonly skipped under exam pressure — and it produces an answer in the wrong units, which dimensional analysis would catch.

Question 5

Explain the '1:2:3 ratio' that Archimedes discovered, and explain why it is useful as a practical computational shortcut.

Model answer: For a cone, sphere, and cylinder that all share radius r and height 2r (the sphere's diameter), their volumes are in a 1:2:3 ratio. The cone has volume (2/3)πr³, the sphere (4/3)πr³, and the cylinder 2πr³ — ratios of 1:2:3. Once you know any one of these volumes, you can find the others by multiplying or dividing by 2, without recomputing from the formulas. The sphere is always 2/3 of the circumscribed cylinder; the cone is always 1/3.

Archimedes valued this result so highly that he asked for a diagram of the sphere inscribed in a cylinder to be carved on his tomb. The practical value is as a cross-check: the 1:2:3 ratio is clean enough that a deviation from it signals a calculation error. It also provides intuition about scale — a sphere is surprisingly large relative to its enclosing cylinder, occupying 2/3 of it — which is useful for estimation problems and for building 3D spatial intuition.