Hemisphere Calculator
Find the volume and surface area of a solid hemisphere from the radius or diameter, or recover the radius from a known volume or total surface. The formulas are V = (2/3)πr³, curved area 2πr², and total surface 3πr².
Curved 157.079632679 · total S = 235.619449019
- Diameter is twice the radius: d = 2r = 2 × 5 = 10.
- Volume V = (2/3)πr³ = (2/3) × 3.14159265359 × 5³ = 261.799387799.
- Curved surface = 2πr² = 157.079632679. Total surface (including the base) = 3πr² = 235.619449019.
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How to Use the Hemisphere Calculator
Choose the one measurement you know — radius, diameter, volume, or total surface — then Calculate. The others come from V = (2/3)πr³, curved area 2πr², and total surface 3πr².
- Select From radius, From diameter, From volume, or From total surface.
- Enter a non-negative number. Zero is a point with volume 0 and surface 0.
- Read the volume (or the radius, when you started from volume), plus the curved dome and the total including the base.
A full ball belongs on the Sphere Calculator (V = (4/3)πr³). A shallower or deeper slice belongs on the Spherical Cap Calculator (V = (1/3)πh²(3R − h)). A band with two circular cuts belongs on the Spherical Zone Calculator. The flat face alone belongs on the Circle Calculator (A = πr²). Converting cubic units belongs on the Volume Converter.
Volume and Surface
A hemisphere is half a sphere, cut through a great circle. Volume is therefore half of (4/3)πr³, which is (2/3)πr³. The curved dome is half of 4πr², which is 2πr². The cut leaves a flat circular face of area πr², so the total surface of the solid is 2πr² + πr² = 3πr².
Solving for r from volume takes a cube root: r = ³√(3V / 2π). Solving from total surface takes a square root: r = √(S / 3π). Height of a hemisphere standing on its flat face is just r — it is not an independent length the way a cylinder’s h is.
Hemisphere Calculator Examples
Radius 5
d = 10. V = (2/3)π × 125 = 250π/3 ≈ 261.799387799. Curved = 50π ≈ 157.079632679. Total = 75π ≈ 235.619449019.
Radius 3
V = (2/3)π × 27 = 18π. Curved = 2π × 9 = 18π. Volume and curved area match numerically (different units) only at this radius. Total surface is 27π ≈ 84.8230016469.
Diameter 10
r = 5, the same hemisphere as the first example.
Volume of the r = 5 hemisphere
Entering V ≈ 261.799387799 and solving for r returns 5. The cube-root step undoes (2/3)πr³.
A point hemisphere
r = 0 gives V = 0 and both surfaces 0. Nothing is enclosed and there is no dome.
A negative radius
A length cannot be negative. The calculator rejects −1 instead of using |r|.
Two Surface Areas
Homework often asks for “the surface area of a hemisphere” without saying which one. The curved dome alone is 2πr² — think of a bowl’s outside, or a planetarium. The solid including the flat circular lid is 3πr². This page reports both. A thin hollow shell has negligible volume; only the curved 2πr² remains.
At r = 3, (2/3)πr³ and 2πr² are both 18π. Cancel 2π (for r ≠ 0): (1/3)r³ = r², so r = 3. The units are still different — cubic versus square — so they are not the same quantity. The full sphere has the same coincidence for volume and its entire skin at r = 3; here it is volume versus the dome only.
Common Hemisphere Mistakes
Using (4/3)πr³ for a dome
That is a full sphere, on the Sphere Calculator. A hemisphere is half of that: (2/3)πr³.
Using πr³ as the volume
That is a cylinder of height r. A hemisphere is (2/3)πr³, which is smaller.
Reporting 2πr² when the problem wants the total
2πr² is the dome. Add the base πr², or use 3πr², when the solid includes the flat face.
Using diameter as if it were r
d = 10 means r = 5. Plugging 10 into (2/3)πr³ gives eight times the volume of the r = 5 hemisphere.
Treating a spherical cap as a hemisphere
A hemisphere is the special cap whose height equals the radius. A shallower or deeper slice belongs on the Spherical Cap Calculator (V = (1/3)πh²(3R − h)).
Hemisphere Calculator FAQs
What is the volume of a hemisphere?
V = (2/3)πr³. For radius 5, V = 250π/3 ≈ 261.799387799 cubic units.
What is the surface area of a hemisphere?
The curved dome is 2πr². The total including the circular base is 3πr². For radius 5 those are 50π ≈ 157.080 and 75π ≈ 235.619.
How do you find the radius from the volume?
r = ³√(3V / 2π). Take three-halves of V over π, then the cube root.
Why are volume and curved area equal at r = 3?
Because (2/3)π(27) and 2π(9) are both 18π. The units still differ. It is a numerical coincidence, not a conversion.
Is this the same as the Sphere Calculator or a cylinder of height r?
No. A sphere is the full ball, V = (4/3)πr³. A cylinder with h = r has volume πr³. This page is half a ball plus the flat face.
Are my numbers sent to a server?
No. The calculation runs locally in your browser.
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Reviewed by Prof. Dr. Khalil Mudassar (PhD) · Last updated: [mc_modified_date]
Sources
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Disclaimer
This calculator is intended for general educational and practical use. Review important results before relying on them in professional work.