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