Math
Hemisphere Calculator
A hemisphere is half a sphere, and the one thing students get wrong is its surface area: the curved dome is 2πr², but the total surface adds the flat circular base for 3πr².
The hemisphere calculator reports both, plus the volume (2/3)πr³, all exact in terms of π, from the radius, diameter, volume, curved area, or total area.
Areas in square units, volume in cubic units, the rest are lengths.
Pick the unit your measurements use. Area is reported in square units and volume in cubic units.
Try an example
Result
Volume
—
- Volume (exact)
- —
- Curved surface area
- —
- Total surface area (with base)
- —
- Flat base area
- —
- Radius / diameter
- —
- Capacity
- —
Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What the hemisphere calculator solves
A hemisphere is half a sphere, and the one thing students get wrong is its surface area: the curved dome is 2πr², but the total surface adds the flat circular base for 3πr². The hemisphere calculator reports both, plus the volume (2/3)πr³, all exact in terms of π, from the radius, diameter, volume, curved area, or total area.
Formula
How to use it
- Choose the measurement you know: radius, diameter, volume, curved area, or total area.
- Enter it and pick a unit.
- Read the volume and both surface areas; use the total area for painting a dome including its floor, the curved area for the dome alone.
How to read the answer
The volume is exactly half a sphere's, (2/3)πr³. The curved surface is half a sphere's surface, 2πr². The total surface area adds the flat circular cut face, πr², for 3πr² — one and a half times the curved area.
Common mistakes and edge cases
- Reporting 2πr² as the total surface area. That is only the dome; the total includes the base and is 3πr².
- Halving the sphere's surface area and calling it the total hemisphere area (that gives the curved area only).
- Using the diameter as the radius, which inflates the volume 8×.
- Zero or negative inputs cannot form a hemisphere.
Worked examples
Radius 3 (volume 18π)
Volume
56.5487
Diameter 10
Volume
261.7994
Volume 100 → radius
Volume
100
Curved area 50
Volume
47.0158
Total area 75 (same hemisphere as curved 50)
Volume
47.0158
Decimal radius 1.5
Volume
7.0686
Negative radius (impossible)
Volume
Error
Frequently asked questions
What is the volume of a hemisphere?+
Half a sphere: V = (2/3)πr³. Radius 3 gives 18π ≈ 56.55 cubic units.
What is the surface area of a hemisphere?+
Curved surface: 2πr². Total surface including the flat base: 3πr². Be sure which one the question wants.
Why is the total surface area 3πr²?+
Half the sphere's 4πr² is 2πr² for the dome, plus the circular base πr², giving 3πr².
How do I find the radius from the volume of a hemisphere?+
r = cbrt(3V / (2π)). Volume 100 gives r ≈ 3.628.
How do I find the radius from the curved surface area?+
r = sqrt(A / (2π)). Curved area 50 gives r ≈ 2.821.
Is a dome the same as a hemisphere?+
Only if the dome is exactly half a sphere. Shallower domes are spherical caps and need the cap height as well as the radius.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026