How is the Volume of a Hemisphere Related to Its Surface Area?

The volume and surface area of a hemisphere are mathematically linked because both depend on the radius (r) of the same 3-D shape.

Formulas Recap

• Volume (V): (2/3) × π × r³
• Curved Surface Area (CSA): 2 × π × r²
• Total Surface Area (TSA): 3 × π × r² (includes circular base)

Mathematical Relationship

Dividing volume by surface area gives a measure of the hemisphere’s mean “depth” relative to its radius:

V / CSA = [(2/3) × π × r³] / [2 × π × r²] = r / 3

This means the volume of a hemisphere grows proportionally to the cube of the radius, while its surface area grows proportionally to the square of the radius.

Doubling the radius increases volume eightfold (2³), but surface area only fourfold (2²). This principle is important in physics and engineering where volume-to-surface ratios affect cooling, buoyancy, and fluid capacity.