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Hemisphere — Definition, Formula & Examples

A hemisphere is exactly half of a sphere, created by slicing a sphere along a plane that passes through its center. It has a curved surface on top and a flat circular base on the bottom.

Given a sphere of radius rr centered at the origin, a hemisphere is the solid formed by the intersection of the sphere with a closed half-space bounded by a great circle plane. The resulting solid has volume 23πr3\frac{2}{3}\pi r^3 and total surface area 3πr23\pi r^2.

Key Formula

V=23πr3Acurved=2πr2Atotal=3πr2V = \frac{2}{3}\pi r^3 \qquad A_{\text{curved}} = 2\pi r^2 \qquad A_{\text{total}} = 3\pi r^2
Where:
  • VV = Volume of the hemisphere
  • AcurvedA_{\text{curved}} = Curved (lateral) surface area only
  • AtotalA_{\text{total}} = Total surface area including the flat circular base
  • rr = Radius of the hemisphere (and of the original sphere)

How It Works

To work with a hemisphere, you need just one measurement: the radius rr of the original sphere. The volume is exactly half the volume of the full sphere. The total surface area combines two parts: the curved surface (half the sphere's surface area) and the flat circular base. When a problem asks for "surface area" of a hemisphere, check whether it means just the curved part or the total including the base — this distinction matters.

Worked Example

Problem: A hemisphere has a radius of 6 cm. Find its volume and total surface area.
Find the volume: Use the hemisphere volume formula with r = 6.
V=23π(6)3=23π(216)=144π≈452.4 cm3V = \frac{2}{3}\pi (6)^3 = \frac{2}{3}\pi (216) = 144\pi \approx 452.4 \text{ cm}^3
Find the curved surface area: The curved part is half the surface area of a full sphere.
Acurved=2π(6)2=2π(36)=72π≈226.2 cm2A_{\text{curved}} = 2\pi (6)^2 = 2\pi (36) = 72\pi \approx 226.2 \text{ cm}^2
Find the total surface area: Add the flat circular base area to the curved surface area.
Atotal=72π+π(6)2=72π+36π=108π≈339.3 cm2A_{\text{total}} = 72\pi + \pi (6)^2 = 72\pi + 36\pi = 108\pi \approx 339.3 \text{ cm}^2
Answer: The volume is 144π≈452.4144\pi \approx 452.4 cm³ and the total surface area is 108π≈339.3108\pi \approx 339.3 cm².

Another Example

Problem: A hemispherical bowl has a diameter of 20 cm. How much water can it hold?
Find the radius: The diameter is 20 cm, so the radius is half of that.
r=202=10 cmr = \frac{20}{2} = 10 \text{ cm}
Calculate the volume: Substitute r = 10 into the hemisphere volume formula.
V=23π(10)3=23π(1000)=2000π3≈2094.4 cm3V = \frac{2}{3}\pi (10)^3 = \frac{2}{3}\pi (1000) = \frac{2000\pi}{3} \approx 2094.4 \text{ cm}^3
Answer: The bowl can hold approximately 2094.4 cm³ (about 2.09 liters) of water.

Why It Matters

Hemispheres appear throughout middle school and high school geometry courses when studying composite solids — for example, a cylinder topped with a hemisphere. Engineers and architects use hemisphere calculations to design domes, satellite dishes, and storage tanks. Understanding hemispheres also builds the foundation for integration problems involving solids of revolution in calculus.

Common Mistakes

Mistake: Using the full sphere volume formula instead of halving it.
Correction: The hemisphere volume is 23πr3\frac{2}{3}\pi r^3, which is half of the sphere's 43πr3\frac{4}{3}\pi r^3. Always divide by 2.
Mistake: Forgetting to include the flat circular base when finding total surface area.
Correction: A hemisphere has two surfaces: the curved part (2πr22\pi r^2) and the circular base (πr2\pi r^2). If the problem asks for total surface area, add both to get 3πr23\pi r^2.

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