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Hemisphere Calculator

Volume equals two-thirds pi r cubed

Solution

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Calculate Hemisphere Volume from Radius

Use this form when the radius is known. The hemisphere holds exactly half the volume of a full sphere with the same radius.

V = (2/3) π r³

Calculate Hemisphere Total Surface Area

Use this form for the curved hemispherical surface PLUS the flat circular base. Common for closed-bottom hemispherical tanks and domes with floors.

S = 3π r²

Calculate Hemisphere Curved Surface Area

Use this form for the curved dome surface only (excluding the flat base). Common for open hemispheres like protective domes and bowls.

S_curved = 2π r²

Calculate Hemisphere Radius from Volume

Use this rearrangement when the volume is known and you need the radius.

r = ∛(3V / (2π))

How It Works

This hemisphere (half sphere) calculator solves V = (2/3)πr³ for volume, S = 3πr² for total surface area (curved + flat base), and S_curved = 2πr² for the curved dome surface alone. The inverse r = ∛(3V/(2π)) recovers the radius from the volume. Pick the unknown with the solve-for toggle and enter the remaining values. Base disk area (πr²) and base circumference (2πr) are always shown as supplementary outputs.

Example Problem

A hemispherical glass dome covers an exhibit with a 3 m radius. What is the dome's volume, curved surface area, and total surface area?

  1. Knowns: r = 3 m
  2. Volume: V = (2/3) · π · r³ = (2/3) · π · 27 = 18π ≈ 56.549 m³
  3. Curved surface area (the dome's outside): S_curved = 2π r² = 2π · 9 = 18π ≈ 56.549 m²
  4. Total surface area (curved + flat floor): S = 3π r² = 3π · 9 = 27π ≈ 84.823 m²
  5. Base disk area (floor footprint): π r² = 9π ≈ 28.274 m²
  6. Sanity check (inverse): from V = 18π, r = ∛(3·18π/(2π)) = ∛27 = 3 m, recovering the radius.

At r = 3, the numeric value of V and curved SA happen to coincide (both equal 18π) — a numerical curiosity, not a structural relationship. They have different units (m³ vs m²).

When to Use Each Variable

  • Solve for Volume — when the radius is known and you need the enclosed half-sphere volume — domes, bowls, hemispherical tanks.
  • Solve for Total Surface Area — when you need the entire exterior including the flat base — closed hemispherical containers.
  • Solve for Curved Surface Area — when only the dome surface matters — paint or coating on a dome with an open base.
  • Solve for Radius — when the volume is known and you need the radius.

Key Concepts

A hemisphere is exactly half of a sphere, cut by a plane through the center. The flat side is a circular disk (the equator of the original sphere). All hemisphere formulas derive from sphere formulas by halving: V_sphere = (4/3)πr³ → V_hemi = (2/3)πr³; A_sphere = 4πr² → A_hemi_curved = 2πr². The total surface area of a hemisphere adds the flat base (πr²) to the curved half, giving 3πr².

Applications

  • Architecture: hemispherical domes (St. Peter's, the Pantheon, observatory domes)
  • Storage tanks: hemispherical end caps on pressure vessels, propane tanks
  • Bowls, cups, helmets: any vessel with a hemispherical interior
  • Astronomy: planetary observatory domes, radio dish dimensions

Common Mistakes

  • Forgetting whether you need the curved-only (2πr²) or total (3πr²) surface area — open hemispheres use curved, closed ones use total
  • Using sphere V = (4/3)πr³ when you mean hemisphere V = (2/3)πr³ — the hemisphere is half, not full
  • Confusing 'base' (the flat circular disk) with 'curved surface' — the base is a flat πr² circle, not a curved area
  • Mixing units between radius and volume — keep length and volume units consistent

Frequently Asked Questions

How do you calculate the volume of a hemisphere?

Use V = (2/3) π r³. The hemisphere holds half the volume of the full sphere. For r = 3 m, V = 18π ≈ 56.549 m³.

What is the surface area of a hemisphere?

Total surface area is S = 3πr² (curved 2πr² plus the flat circular base πr²). Curved surface alone is 2πr². For r = 3 m: total ≈ 84.82 m², curved ≈ 56.55 m².

How is a hemisphere different from a sphere?

A hemisphere is exactly half of a sphere, formed by cutting the sphere through its center with a plane. Its volume is half the sphere's; its curved surface area is half the sphere's. The total surface area adds a flat circular base that the sphere doesn't have.

How do you find the radius of a hemisphere given the volume?

Rearrange V = (2/3)πr³ to r = ∛(3V/(2π)). For V = 56.549 m³, r = ∛(169.65 / 6.2832) = ∛27 = 3 m.

Why does a hemisphere have THREE pi r squared as its total surface area?

The curved half-sphere surface is 2πr². The flat circular base contributes another πr². Total: 2πr² + πr² = 3πr². Half a sphere has more surface than a quarter sphere because it includes the new flat face that wasn't there in the full sphere.

What is the difference between hemisphere total and curved surface area?

Total surface area (S = 3πr²) includes both the curved dome and the flat circular base. Curved surface area (2πr²) is just the dome — useful when the base is open or already painted.

How much paint does an open-bottom hemispherical dome need?

Use the curved surface area 2πr² (not 3πr²) because there's no base to paint. For a 5 m radius open dome, 2π·25 = 50π ≈ 157.1 m² of paint coverage.

Can I use this calculator for a half cylinder or half cone?

No — those have different formulas. A half cylinder has V = (1/2)πr²h; a half cone has V = (1/6)πr²h. A hemisphere is specifically half of a sphere — use the sphere or this hemisphere calculator.

Reference: Weisstein, Eric W. "Hemisphere." MathWorld — A Wolfram Web Resource. https://mathworld.wolfram.com/Hemisphere.html

Worked Examples

Dome

How much enclosed space does a 3 m hemisphere dome have?

A glass exhibit dome is a hemisphere with 3 m radius. Compute its volume to estimate interior air conditioning load.

  • Knowns: r = 3 m
  • Formula: V = (2/3) π r³
  • V = (2/3) π · 27 = 18π ≈ 56.549 m³

Volume ≈ 56.5 m³ (about 14,930 US gallons)

Half the volume of a 3 m sphere (113 m³). Useful for scaling estimates between full-sphere and dome geometries.

Open Dome

How much paint covers a 5 m radius observatory dome?

An astronomy observatory has an open-bottom hemispherical dome with radius 5 m. Compute the curved surface area to estimate paint.

  • Knowns: r = 5 m
  • Formula: S_curved = 2π r²
  • S = 2π · 25 = 50π ≈ 157.08 m²

Curved surface area ≈ 157.1 m² (≈ 1,690 ft²)

Open-bottom domes need only the curved-area formula. If the dome had a floor, use S = 3πr² ≈ 235.6 m² instead.

Inverse Solve

What radius does a 100 L hemispherical bowl need?

A salad bowl must hold 100 L (0.1 m³). The bowl is hemispherical. Find the required radius.

  • Knowns: V = 0.1 m³
  • Formula: r = ∛(3V / (2π))
  • r = ∛(0.3 / 6.2832) ≈ ∛0.04775 ≈ 0.3628 m

Radius ≈ 0.363 m (≈ 36.3 cm)

The diameter is twice that — about 0.73 m (29 inches). A surprisingly large bowl for 100 liters because hemispherical capacity is half that of a sphere.

Hemisphere Formulas

A hemisphere (half sphere) is defined by a single dimension: the radius r. Volume, both surface-area variants, and the base disk all derive from it.

Volume equals two-thirds pi r cubedTotal surface area equals three pi r squaredCurved surface area equals two pi r squaredRadius equals the cube root of three V over two pi
Hemisphere with labeled radius rr

Where:

  • V — enclosed half-sphere volume
  • S_total — total surface area: curved dome plus flat base disk
  • S_curved — curved hemispherical surface only
  • r — radius from center of the flat base to the curved surface

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