Last updated: 2026-06-05

What is a hemisphere?

A hemisphere is exactly half a sphere — a three-dimensional solid bounded by a flat circular face (the base) and a curved dome-shaped surface. The base circle shares its radius with the parent sphere. Everyday examples include bowls, igloo structures, and hemispherical domes.

Formulas

For a hemisphere with radius r:

Property	Formula	Notes
Volume	$V = \dfrac{2}{3}\pi r^3$	Half the volume of a full sphere
Curved surface area	$A_c = 2\pi r^2$	Dome only, no base
Base area	$A_b = \pi r^2$	Flat circular bottom
Total surface area	$A = 3\pi r^2$	Dome + base

Derivations

Volume

The volume of a full sphere of radius $r$ is $\frac{4}{3}\pi r^3$. A hemisphere is exactly half a sphere:

$V = \frac{1}{2} \cdot \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3$

The same result follows from integrating circular cross-sections of radius $\sqrt{r^2 - h^2}$ from height $h = 0$ to $h = r$:

$V = \int_0^r \pi(r^2 - h^2)\,dh = \pi\!\left[r^2 h - \frac{h^3}{3}\right]_0^r = \frac{2}{3}\pi r^3$

Curved surface area

The full sphere surface area is $4\pi r^2$. The dome is half of this:

$A_c = \frac{1}{2} \cdot 4\pi r^2 = 2\pi r^2$

Base area and total surface area

The flat bottom is a circle of radius $r$:

$A_b = \pi r^2$

Total surface area combines both:

$A = A_c + A_b = 2\pi r^2 + \pi r^2 = 3\pi r^2$

Hemisphere vs. full sphere

Property	Hemisphere	Full sphere	Ratio
Volume	$\frac{2}{3}\pi r^3$	$\frac{4}{3}\pi r^3$	1/2
Curved surface	$2\pi r^2$	$4\pi r^2$	1/2
Total surface	$3\pi r^2$	$4\pi r^2$	3/4

The total surface area of a hemisphere is three-quarters of the full sphere's — not one-half — because the flat base contributes an additional $\pi r^2$. This is the most common source of error when splitting a sphere calculation in two.

Worked example

A hemispherical bowl has a radius of 10 cm (0.10 m).

Volume: $V = \frac{2}{3}\pi (0.10)^3 = \frac{2}{3}\pi \times 0.001 \approx 0.002094 \text{ m}^3 = 2{,}094 \text{ cm}^3$

Curved surface area: $A_c = 2\pi (0.10)^2 = 2\pi \times 0.01 \approx 0.0628 \text{ m}^2 = 628.3 \text{ cm}^2$

Base area: $A_b = \pi (0.10)^2 = \pi \times 0.01 \approx 0.0314 \text{ m}^2 = 314.2 \text{ cm}^2$

Total surface area: $A = 3\pi (0.10)^2 = 3\pi \times 0.01 \approx 0.0942 \text{ m}^2 = 942.5 \text{ cm}^2$

Radius scaling

Because volume grows as $r^3$ and surface area as $r^2$, scaling the radius has an outsized effect on volume:

Radius factor	Volume factor	Surface area factor
×2	×8	×4
×3	×27	×9
×½	×⅛	×¼

See also: Sphere Volume & Surface Area Calculator for the parent shape, and Cone Volume & Surface Area Calculator for another common solid with a circular base.

Frequently Asked Questions (FAQ)

What is the formula for the volume of a hemisphere?

The volume of a hemisphere is V = (2/3)πr³ — exactly half the volume of a full sphere (4/3)πr³. For a hemisphere with r = 5 cm: r³ = (0.05 m)³ = 0.000125 m³, so V = (2/3) × π × 0.000125 ≈ 0.000262 m³ ≈ 261.8 cm³.

Does the total surface area of a hemisphere include the flat base?

Yes. Total surface area = curved surface (2πr²) + flat base (πr²) = 3πr². If you only need the dome surface, use curved area 2πr² alone. For r = 5 cm: curved = 2π × 25 ≈ 157.1 cm², base = π × 25 ≈ 78.5 cm², total ≈ 235.6 cm².

How does a hemisphere compare to a full sphere?

A hemisphere has half the volume of a full sphere of the same radius (V = (2/3)πr³ vs (4/3)πr³). Its curved surface area (2πr²) is also half the sphere's surface area (4πr²). However, the total surface area (3πr²) is three-quarters of the full sphere's (4πr²), because the flat base adds an extra πr².

What is the formula for just the curved surface of a hemisphere?

The curved (dome) surface area is A_c = 2πr², not including the flat base. For r = 5 cm: A_c = 2 × π × 25 ≈ 157.1 cm². Adding the flat base (πr² ≈ 78.5 cm²) gives total surface area 3πr² ≈ 235.6 cm².