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 $\pi r^2$ — the same circle you meet when you study the circumference of a circle. 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. The whole world map uses this same word: the northern and southern halves of the Earth are its two hemispheres.
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. You can see how it fits the wider family in the guide to 3D geometry shapes.
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 $\frac{4}{3}\pi r^3$. Cut it in half and each piece holds half of that:
$$V = \frac{1}{2} \times \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3$$
Where this comes from: the sphere volume $\frac{4}{3}\pi r^3$ is the established result; a hemisphere is literally half a sphere, so you halve it. Unlike surface area, volume does split cleanly in half — the cut adds no new "inside", it only exposes a face.
Variable glossary: $V$ is the volume, $r$ is the radius (centre of the base to the rim, and also up to the top of the dome), 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:
$$\text{CSA} = 2\pi r^2$$
Where this comes from: the full sphere has surface area $4\pi r^2$. The curved dome is exactly half that skin, so $\frac{1}{2} \times 4\pi r^2 = 2\pi r^2$.
Total surface area (TSA) — the dome plus the flat circular base:
$$\text{TSA} = 2\pi r^2 + \pi r^2 = 3\pi r^2$$
Where this comes from: add the curved dome ($2\pi r^2$) to the flat base, which is a circle of area $\pi r^2$. This is the step students miss. Halving a sphere's surface gives only the dome; the flat face is an extra $\pi r^2$ that the cut created. So the total is $3\pi r^2$, not $2\pi r^2$.
The clearest way to see the two pieces is to imagine the hemisphere's "net" — flatten the dome and lay the circular base beside it. The dome contributes $2\pi r^2$, the base contributes $\pi r^2$, and together they make $3\pi r^2$.
Quantity | Formula | Units |
|---|---|---|
Volume | V = ⅔ π r³ | cubic |
Curved surface area | CSA = 2 π r² | square |
Total surface area | TSA = 3 π r² | square |
Variable glossary: CSA is the curved surface area (dome only), TSA is the total surface area (dome plus base), $r$ is the radius. Surface area comes out in square units (cm², m²).
Examples of the Hemisphere
For consistency, every example below uses centimetres and takes $\pi \approx 3.14$.
Example 1
Find the volume of a hemisphere with radius 3 cm.
$$V = \frac{2}{3}\pi r^3$$
$$V = \frac{2}{3} \times 3.14 \times 3^3$$
$$V = \frac{2}{3} \times 3.14 \times 27$$
$$V = \frac{2}{3} \times 84.78$$
Final answer: $V \approx 56.52$ cm³
Example 2
A hemisphere has radius 7 cm. A student finds the total surface area by halving the sphere's surface area. Find the correct total surface area. (Use $\pi \approx \frac{22}{7}$.)
Take the wrong path first, because halving the sphere is the classic hemisphere error.
Wrong attempt: the student writes "half a sphere, so half the surface area" and computes $\frac{1}{2} \times 4\pi r^2 = 2\pi r^2$.
$$2 \times \frac{22}{7} \times 7^2 = 2 \times 22 \times 7 = 308 \text{ cm}^2$$
The break: that result is only the curved dome. Cutting the sphere created a new flat circle that the halving never counted. The answer is missing the base.
Correct method: add the flat base.
$$\text{TSA} = 3\pi r^2 = 3 \times \frac{22}{7} \times 7^2 = 3 \times 22 \times 7$$
Final answer: $\text{TSA} = 462$ cm²
Example 3
Find the curved surface area of a hemisphere with radius 5 cm.
$$\text{CSA} = 2\pi r^2$$
$$\text{CSA} = 2 \times 3.14 \times 5^2$$
$$\text{CSA} = 2 \times 3.14 \times 25$$
Final answer: $\text{CSA} = 157$ cm²
Example 4
Find the total surface area of a hemisphere with radius 10 cm.
$$\text{TSA} = 3\pi r^2$$
$$\text{TSA} = 3 \times 3.14 \times 10^2$$
$$\text{TSA} = 3 \times 3.14 \times 100$$
Final answer: $\text{TSA} = 942$ cm²
Example 5
A solid hemisphere has volume $18\pi$ cm³. Find its radius.
Start from the volume formula and solve for $r$.
$$V = \frac{2}{3}\pi r^3$$
$$18\pi = \frac{2}{3}\pi r^3$$
Divide both sides by $\pi$.
$$18 = \frac{2}{3} r^3$$
Multiply both sides by $\frac{3}{2}$.
$$r^3 = 27$$
Final answer: $r = 3$ cm
Example 6
A bowl is a hollow hemisphere of inner radius 6 cm. How much water can it hold, in litres? (Use $\pi \approx 3.14$; 1000 cm³ = 1 litre.)
The water it holds is the inside volume of the hemisphere.
$$V = \frac{2}{3}\pi r^3 = \frac{2}{3} \times 3.14 \times 6^3 = \frac{2}{3} \times 3.14 \times 216$$
$$V = \frac{2}{3} \times 678.24 = 452.16 \text{ cm}^3$$
Convert to litres: $452.16 \div 1000 \approx 0.45$ litre.
Final answer: about $0.45$ 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. The reason cathedral domes, planetarium roofs, and stadium covers are hemispherical is that the curved shell carries load outward to the rim instead of sagging in the middle — a flat ceiling of the same span would need far more support. The volume formula matters the moment you fill one: a hemispherical bowl, a domed grain silo, or the rounded end-cap of a pressurised gas tank all need $\frac{2}{3}\pi r^3$ to size their contents. Get the surface area wrong and you under-order the material to clad the dome. Engineers also lean on the fact that, for a fixed rim radius, a hemisphere encloses the most volume of any dome you could build over that circle — the shape is doing real optimisation work, not just looking neat. You can read how the dome idea scales up to planet-sized halves in the language of geography, where "hemisphere" still means exactly half a sphere.
Where Students Trip Up On Hemispheres
Mistake 1: Forgetting the flat base in total surface area
Where it slips in: any "total surface area" question, especially right after studying spheres.
Don't do this: write $\text{TSA} = 2\pi r^2$ by halving the sphere's $4\pi r^2$. That is only the curved dome.
The correct way: add the flat circular base. $\text{TSA} = 2\pi r^2 + \pi r^2 = 3\pi r^2$. The student who rushes straight from the sphere formula carries over "just halve it" and loses the base every time — the cut creates a new face that halving never sees.
Mistake 2: Confusing curved surface area with total surface area
Where it slips in: questions that say "surface area" without specifying which, or describe an open bowl.
Don't do this: report $3\pi r^2$ for an open hemispherical bowl, or $2\pi r^2$ for a solid closed half-ball.
The correct way: read whether the flat face is part of the object. A solid hemisphere or a closed dome has both surfaces, so use $3\pi r^2$. An open bowl has only the curved shell, so use $2\pi r^2$. The second-guesser who can never decide should ask one question: is the flat top there or not?
Mistake 3: Using the diameter as the radius
Where it slips in: problems that give the diameter or the width across the base.
Don't do this: plug the full width straight in as $r$. Because the formulas cube and square the radius, a doubled value throws the answer out by a factor of 8 (volume) or 4 (area).
The correct way: halve the diameter first. If the base is 14 cm across, then $r = 7$ cm. The memorizer who recalls the formula but not what each letter means makes this slip — define $r$ as the radius before substituting anything.
Conclusion
A hemisphere is exactly half a sphere, with one curved dome surface and one flat circular base, sharing the sphere's radius $r$.
Volume is $\frac{2}{3}\pi r^3$ — cleanly half the sphere's volume, because the cut adds no inside.
Curved surface area is $2\pi r^2$ (half the sphere's skin); total surface area is $3\pi r^2$, which adds the flat base.
The flat base is the single most-missed piece: halving a sphere's surface gives only the dome.
Always work from the radius, not the diameter, and read whether a problem wants the open (curved) or closed (total) surface.
Practice And Next Steps
Work through these problems to solidify your understanding, then check each answer against the formulas above.
Find the volume of a hemisphere with radius 6 cm ($\pi \approx 3.14$).
Find the total surface area of a hemisphere with radius 7 cm ($\pi \approx \frac{22}{7}$).
A hemispherical dome has curved surface area $200\pi$ m². Find its radius.
To build solid geometry with a teacher who explains why each formula works rather than asking you to memorise it, explore Bhanzu's geometry tutor, our high school math tutor, or math classes online. Want a live Bhanzu trainer to walk through more hemisphere problems? Book a free demo class.
Read More
Sphere — the full solid a hemisphere is exactly half of.
What is a polyhedron — the flat-faced solids, contrasted with curved ones like the hemisphere.
Geometric shapes — the full 2D and 3D shape family in one place.
Area of a circle — the πr² that gives a hemisphere's flat base its area.
What is volume — the space-measuring idea behind every solid's volume formula.
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