Sphere and Hemisphere Calculator

How it works

Everything about a sphere follows from its radius: the diameter is twice it, the circumference around the widest part is 2π times it, the surface area is 4πr², and the volume is 4⁄3πr³. Because each of those is a simple function of r, the calculator can start from whichever one you have and work back to the radius: halve a diameter, divide a circumference by 2π, take the square root of a surface area over 4π, or the cube root of three quarters of a volume over π.

A hemisphere is half a sphere with a flat circular face. Its volume and its curved surface are half the sphere's, ⅔πr³ and 2πr², and its total surface area adds the flat face, πr², for 3πr² in all. A hollow sphere, or shell, is a sphere with a smaller concentric sphere removed: its volume is the material in the wall, 4⁄3π(R³ − r³), and its total surface area counts the outside and the inside. When you start from a volume or a surface area, enter the figure for the shape you chose; the calculator inverts the matching formula, so any result it shows can be typed back in and will return the same radius.

The unit you choose labels the lengths, areas and volumes; the numbers are the same whatever the unit. It matters for the capacity line, where the volume in cubic inches, feet, yards, centimetres or metres is converted to US gallons and litres from the legal definitions of those units.

Formula

r = d ÷ 2 = C ÷ 2π = √(SA ÷ 4π) = ∛(3V ÷ 4π)
sphere:      V = 4⁄3 × π × r³         SA = 4 × π × r²         C = 2 × π × r
hemisphere:  V = ⅔ × π × r³          curved = 2πr²   total = 3πr²
shell:       V = 4⁄3 × π × (R³ − r³)   outside = 4πR²  total = 4πR² + 4πr²
capacity: 1 US gal = 231 in³ = 3.785411784 L;  1 ft³ = 7.48052 gal;  1 m³ = 1,000 L

Example

A sphere with a 1 ft radius has a diameter of 2 ft, a circumference of 2π ≈ 6.2832 ft, a surface area of 4π ≈ 12.5664 ft² and a volume of 4⁄3π ≈ 4.1888 ft³, which is 4.1888 × 7.48052 ≈ 31.33 US gallons or 118.61 litres. Entering 2 ft as the diameter, 6.2832 ft as the circumference, 4.1888 ft³ as the volume or 12.5664 ft² as the surface area instead gives the same sphere.

A regulation men's basketball (size 7) is 29.5 in around. Entering that as the circumference in inches gives a radius of 29.5 ÷ 2π ≈ 4.6951 in, a diameter of ≈ 9.3901 in, a surface area of ≈ 277.0092 in² and a volume of ≈ 433.5259 in³, about 1.88 gallons or 7.10 litres of air.

A hemisphere of 1 ft radius has half the sphere's volume, 2π/3 ≈ 2.0944 ft³ (15.67 gallons), a curved area of 2π ≈ 6.2832 ft² and a total surface area of 3π ≈ 9.4248 ft² including the flat face. A hollow ball with an outer radius of 2 ft and an inner radius of 1 ft contains 4⁄3π × (8 − 1) ≈ 29.3215 ft³ of material and has a total surface area, outside and inside, of 16π + 4π = 20π ≈ 62.8319 ft².

Assumptions and limitations

• The solid is a perfect sphere; a hemisphere is exactly half of one with a flat circular face; a shell is two concentric spheres. Real balls, domes and tanks are only approximately spherical.
• For a shell the volume is the material in the wall, and the capacity converts that wall volume. For what a hollow ball or tank holds, use the sphere option with the inside radius.
• A hemisphere's total surface area includes its flat face; the curved area excludes it.
• The gallon is the US liquid gallon of 231 cubic inches, not the imperial gallon.
• Results are rounded for display only.

Frequently asked questions