Start from the sphere
Every hemisphere formula is a sphere formula, halved. A sphere's surface area is 4πr² and its volume is ⁴⁄₃πr³. Cut the sphere through its center and you get a hemisphere — a dome with one flat circular face. Halve each formula and you are nearly done: the curved surface area is 2πr² and the volume is ⅔πr³. The twist is the flat base. A sphere has no base, but a hemisphere does — a circle of area πr² — and whether you add it depends on what you are actually measuring.
The two area formulas (pick the right one)
There are two "hemisphere area" answers, and textbooks test whether you know which one the question wants. The curved surface area, 2πr², covers only the dome — use it for an open bowl, a dome skylight seen from outside, or paint on a planetarium roof. The total surface area, 3πr², adds the flat base (πr²) — use it for a closed dome, a solid half-ball, or any shape where the bottom face counts as surface. When a question just says "surface area of a hemisphere," it usually means the total, 3πr² — but check whether the base is part of the shape in the problem.
Worked example 1: radius given
Find the total surface area and volume of a hemisphere with radius 6 cm. Curved area: 2 × π × 6² = 2 × π × 36 = 72π ≈ 226.19 cm². Base: π × 36 = 36π ≈ 113.10 cm², so total area = 72π + 36π = 108π ≈ 339.29 cm² (exactly 3πr²). Volume: ⅔ × π × 6³ = ⅔ × π × 216 = 144π ≈ 452.39 cm³. Notice the pattern: keep π symbolic as long as possible (72π, 108π, 144π) and only reach for 3.14159 at the last step — cleaner work, fewer rounding errors.
Worked example 2: diameter given
A hemispherical bowl has a diameter of 14 inches. What is its capacity? First halve the diameter: r = 14 ÷ 2 = 7 inches — this is the step most people skip, and skipping it quadruples the area answer. Capacity is volume: ⅔ × π × 7³ = ⅔ × π × 343 = 686π/3 ≈ 718.38 cubic inches. In kitchen terms, a cubic inch is about 0.069 cups, so this bowl holds roughly 49.6 cups — a serious mixing bowl. If you also wanted the inside surface (for a non-stick coating estimate), use the curved area only — the bowl is open: 2 × π × 49 = 98π ≈ 307.88 in².
The three classic mistakes
1. Using the diameter as the radius. The formulas want r, and area scales with r² — plugging in d instead of d÷2 makes the answer four times too big (and volume eight times). Always halve first. 2. Answering 2πr² when the base counts. If the shape is closed or the question says "total surface area," add the πr² base for 3πr². 3. Squaring before multiplying by π — in the wrong order. 2πr² means 2 × π × (r²), not (2πr)². Square the radius first, then multiply. For r = 5: 2 × π × 25 = 157.08, not (31.42)².
Quick-reference table
| Measure | Formula | r = 5 example |
|---|---|---|
| Curved surface area | 2πr² | 157.08 |
| Base area | πr² | 78.54 |
| Total surface area | 3πr² | 235.62 |
| Volume | ⅔πr³ | 261.80 |
| Radius from diameter | r = d ÷ 2 | d = 10 → r = 5 |
Practice set
Try these, then check yourself on the hemisphere calculator: (1) Total surface area, r = 4 — 150.80 (3 × π × 16). (2) Volume, d = 12 — halve to r = 6, then 452.39. (3) Curved area, r = 10 — 628.32. (4) A dome with total area 1,000 m²: r = √(1000 ÷ 3π) ≈ 10.30 m. The method never changes: halve the sphere formula, then decide about the base.