Cylinder, Cone and Sphere (Surface Area and Volume)
350. Half a Sphere's Secrets · Volume and surface area of a hemisphere
The radius r defines the size of the hemisphere.
A hemisphere is half a sphere. For radius r: curved surface area = 2πr², total surface area = 3πr² (the curved part plus the flat circular face), and volume = (2/3)πr³.
What this lesson covers
Try to break it
Drag A around the rim. The radius readout doesn't change — every point on the rim is the same distance r from the centre, because the rim itself is a circle of radius r. Volume = (2/3)πr³ and total surface = 3πr² depend only on r, so they don't shift as A moves around.
How you build it
Draw the hemisphere's flat circular base, then mark the two measurements that define it — the radius r and the height (also r).
- Draw the flat circular face of the hemisphere as an ellipse through B and A.
- Draw the radius OA from the centre O to the rim at A.
- Draw the height OH straight up from O to H.
The proof, step by step
Prove that a hemisphere has curved surface area 2πr² and volume (2/3)πr³.
- Volume of a sphere = 4/3 πr³
- Volume of hemisphere = 1/2 × Volume of sphere
- Volume of hemisphere = 2/3 πr³
- Surface area of sphere = 4πr²
- Curved surface area of hemisphere = 2πr²
- Area of base = πr²
- Total surface area = 3πr²
Worked example
A hemisphere has a radius of 6 cm. What is its volume?
Volume of hemisphere = 2/3 πr³ = 2/3 π(6)³ = 2/3 π(216) = 144π cm³.
- 72π cm³
- 144π cm³ — correct
- 216π cm³
- 288π cm³