Hemisphere Volume Calculator

Calculate the exact volume, curved dome area, and total surface boundary of a hemisphere from its radius with complete step-by-step mathematical working.

½ Hemisphere Volume Calculator

Formula: V = (2/3)πr³
Radius of the hemisphere base
Display precision for result
Calculated Hemisphere Volume
261.7994 cm³
Step-by-Step Mathematical Breakdown:
1. Radius \( r = 5\text{ cm} \)
2. Volume Formula: \( V = \frac{2}{3}\pi r^3 \)
3. Cube radius: \( r^3 = 5^3 = 125 \)
4. Volume Calculation: \( V = \frac{2}{3} \times \pi \times 125 = \frac{250}{3}\pi \approx \mathbf{261.7994\text{ cm}^3} \)
5. Total Surface Area (Curved + Flat Base): \( A = 3\pi r^2 = 75\pi \approx \mathbf{235.6194\text{ cm}^2} \)

Mathematical Derivation of Hemisphere Formulas

Because a hemisphere is formed by cutting a sphere exactly across its equatorial plane:

V_{\text{hemisphere}} = \frac{1}{2} V_{\text{sphere}} = \frac{1}{2} \left(\frac{4}{3}\pi r^3\right) = \frac{2}{3}\pi r^3

Notice that while volume is simply cut in half, the total surface area includes an additional circular boundary (\(\pi r^2\)):

Atotal = Acurved + Abase = 2πr² + πr² = 3πr²

Visit the main Sphere Volume Formula page to compare hemisphere and full sphere mathematics.

Frequently Asked Questions

What is the formula for the volume of a hemisphere?

The volume of a hemisphere is exactly half of a full sphere: V = (2/3)πr³, where r is the radius.

What is the total surface area of a closed hemisphere?

A closed hemisphere includes both the curved dome (2πr²) and the flat circular bottom base (πr²), yielding total surface area A = 3πr².

How does a hemisphere relate to a sphere?

A hemisphere is created by bisecting any sphere through a plane that passes through its geometric center.

Related Calculators

V
Full Sphere Volume
Calculate volume of the complete sphere: V = (4/3)πr³.
Open Calculator →
A
Sphere Surface Area
Full sphere surface boundary: A = 4πr².
Open Calculator →
Σ
All Sphere Formulas
Comprehensive reference sheets for spheres, hemispheres, and caps.
View Formulas →