# Hemisphere: Definition, Volume, Surface Area Formulas, and Examples

## What is a Hemisphere?
A **hemisphere** is a three-dimensional solid that is **exactly half of a sphere**, formed by cutting a sphere along a plane that passes through its centre. The cut produces two identical halves. Each half has two surfaces: a **curved surface** (the dome, which is half the sphere's outer skin) and a **flat circular base** (the new circle exposed by the cut). The radius **r** of the hemisphere is the same as the radius of the original sphere.

The flat base is a full circle of radius **r**, so its area is πr². Because the base is flat and the dome is curved, a hemisphere has **two faces, one curved edge** (the rim where they meet), **and no vertices** — there is no sharp corner anywhere on it.

A hemisphere belongs to the family of curved solids alongside the [cone](/content/math/geometry/cone/index.html) and the cylinder — solids with at least one curved surface, unlike the flat-faced prisms and pyramids.

## Volume of A Hemisphere
The **volume** of a hemisphere is exactly half the volume of the sphere it came from.

A full **sphere** of radius **r** has volume \(V = \frac{4}{3}\pi r^3\). Cut it in half and each piece holds half of that:

\[ V = \frac{1}{2} \times \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3 \]

**Variable glossary:** \(V\) is the volume, **r** is the radius, and \(\pi \approx 3.14159\). Volume comes out in cubic units (cm³, m³).

## Surface Area of A Hemisphere
A hemisphere has two surfaces, and the two surface-area formulas count different things.

**Curved surface area (CSA)** — the dome only:
\[ \text{CSA} = 2\pi r^2 \]

**Total surface area (TSA)** — the dome plus the flat circular base:
\[ \text{TSA} = 2\pi r^2 + \pi r^2 = 3\pi r^2 \]

**Variable glossary:** \(\text{CSA}\) is the curved surface area (dome only), \(\text{TSA}\) is the total surface area (dome plus base).

| Quantity | Formula | Units |
| --- | --- | --- |
| Volume | \( V = \frac{2}{3} \pi r^3 \) | cubic |
| Curved surface area | \( \text{CSA} = 2 \pi r^2 \) | square |
| Total surface area | \( \text{TSA} = 3 \pi r^2 \) | square |

## Examples of the Hemisphere

### Example 1
**Find the volume of a hemisphere with radius 3 cm.**
\[ V = \frac{2}{3}\pi r^3 \]
\[ V = \frac{2}{3} \times 3.14 \times 3^3 = \frac{2}{3} \times 3.14 \times 27 \approx 56.52 \text{ cm}^3 \]

### Example 2
**A hemisphere has radius 7 cm. Find the correct total surface area.**
\[ \text{TSA} = 3\pi r^2 = 3 \times \frac{22}{7} \times 7^2 = 462 \text{ cm}^2 \]

### Example 3
**Find the curved surface area of a hemisphere with radius 5 cm.**
\[ \text{CSA} = 2\pi r^2 = 2 \times 3.14 \times 25 = 157 \text{ cm}^2 \]

### Example 4
**Find the total surface area of a hemisphere with radius 10 cm.**
\[ \text{TSA} = 3\pi r^2 = 3 \times 3.14 \times 100 = 942 \text{ cm}^2 \]

### Example 5
**A solid hemisphere has volume 18π cm³. Find its radius.**
Starting from the volume formula:
\[ 18\pi = \frac{2}{3}\pi r^3 \]
This gives \( r = 3\) cm.

### Example 6
**A bowl is a hollow hemisphere of inner radius 6 cm. How much water can it hold, in litres?**
\[ V = \frac{2}{3}\pi r^3 = \frac{2}{3} \times 3.14 \times 6^3 = 0.45 	ext{ litre} \]

## Why the Half-Sphere Shows Up Everywhere
The hemisphere earns its place wherever a structure needs to span space without a flat lid.

A dome is a hemisphere doing structural work in architecture, allowing for less material and greater strength.

## Where Students Trip Up On Hemispheres
### Mistake 1: Forgetting the flat base in total surface area
### Mistake 2: Confusing curved surface area with total area
### Mistake 3: Using the diameter as the radius

## Conclusion
- A **hemisphere** is exactly half a sphere, with one curved dome surface and one flat circular base.
- **Volume** is 23πr³ — cleanly half the sphere's volume.
- **Curved surface area** is 2πr²; **total surface area** is 3πr².