Surface area and volume of spheres and hemispheres

Learning outcome

Calculate sphere and hemisphere areas and volumes, separating a curved surface from a flat circular face or an open mouth.

Concepts and assumptions

A sphere’s surface lies at distance r from its centre. Its diameter is 2 × r. It has no flat base: its entire surface is curved.

A geometric surface comparison gives the same area as the curved wall of a cylinder of radius r and height 2 × r. Thus sphere area = 4 × π × r². Comparing circular slices shows its volume is two-thirds of that enclosing cylinder: sphere volume = (4/3) × π × r³.

Cutting through the centre produces two hemispheres. Each has half the sphere’s volume and half its curved area:

Hemisphere volume = (2/3) × π × r³; curved area = 2 × π × r².

A solid hemisphere also has the circular cut face, area π × r². Its total area is therefore 3 × π × r². A bowl’s open mouth is not a surface to coat: its inner curved area excludes that disk. A solid hemisphere resting flat on a surface has only its curved area exposed if the base is hidden.

Use π = 22/7 as the prescribed approximation. Dimensions are exact; bowl capacity uses internal radius and negligible wall thickness. Keep intermediate fractions unrounded. Round only the explicitly requested final capacity to the nearest 0.001 L; 1000 cm³ = 1 L.

Worked examples

Example 1 — Sphere. A sphere has radius 21 cm. Area = 4 × (22/7) × 21² = 5544 cm². Volume = (4/3) × (22/7) × 21³ = 38808 cm³. Area is measured in square units, while volume is measured in cubic units.

Example 2 — Solid hemisphere. Radius is 10.5 cm. The circular face has area (22/7) × 10.5² = 346.5 cm². Curved area = 2 × 346.5 = 693 cm²; total area = 693 + 346.5 = 1039.5 cm². Volume = (2/3) × 346.5 × 10.5 = 2425.5 cm³.

Example 3 — Recover bowl capacity. A hemispherical bowl has inner curved area 1232 cm². Find its radius and brimful capacity to the nearest 0.001 L. From 2 × (22/7) × r² = 1232, r² = 196 and r = 14 cm. Capacity = (2/3) × (22/7) × 14³ = 17248/3 cm³ = 17248/3000 L ≈ 5.749 L. Do not round before converting units.

Common mistakes

A hemisphere’s total area is not half a sphere’s area: the cut face is extra. Do not coat an imaginary disk across a bowl’s opening or substitute diameter for radius.

Practice questions

1. A sphere has radius 10.5 cm. Find surface area and volume.
2. A solid hemisphere has radius 7 cm. Find curved and total areas.
3. An open hemispherical bowl has internal diameter 42 cm. Find its capacity in litres.
4. A sphere with surface area 616 cm² is cut into two equal solid hemispheres. Find the sum of their total surface areas.

Worked solutions

1. Area = 4 × (22/7) × 10.5² = 1386 cm². Volume = (4/3) × (22/7) × 10.5³ = 4851 cm³.
2. Circular face area = (22/7) × 49 = 154 cm². Curved area = 308 cm²; total area = 308 + 154 = 462 cm².
3. Radius = 42 ÷ 2 = 21 cm. Capacity = (2/3) × (22/7) × 21³ = 19404 cm³ = 19.404 L.
4. Each new circular face has area 616 ÷ 4 = 154 cm². The original curved surface is retained, so combined area = 616 + 2 × 154 = 924 cm².