acalculator

How big is my sphere?

Type any one of the radius, diameter, circumference, surface area, or volume, and the sphere calculator finds the other four as you type.

Your numbers

Units
Diameter
10 in

A sphere of radius 5 in has a surface area of 314.159265 in² and a volume of 523.598776 in³.

Circumference
31.415927 in
Surface area
314.159265 in²
Volume
523.598776 in³

Diameter: 10 in. A sphere of radius 5 in has a surface area of 314.159265 in² and a volume of 523.598776 in³.

How to calculate

Finds the radius, diameter, circumference, surface area, and volume of a sphere from any one of them, with A = 4πr² and V = 4/3 πr³.

Example with the default inputs (Radius 5 in): A sphere of radius 5 in has a surface area of 314.159265 in² and a volume of 8,580.246646 cm³.

Formula: d = 2r, C = 2πr, A = 4πr², and V = 4/3 πr³, where r is the radius.

  • π is the full-precision value 3.141592653589793…, not 3.14 or 22/7.
  • The sphere is perfect: every point of the surface is the same distance r from the centre.
  • Each box has its own unit; each value must be greater than 0.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. Radius 196.850394 in gives Diameter 393.700787 in, Circumference 1,236.847501 in, Surface area 486,947.835202 in², Volume 523,598,775.598299 cm³.Source: OpenStax, Prealgebra 2e, §9.6, sphere volume and surface area (https://openstax.org/books/prealgebra-2e/pages/9-6-solve-geometry-applications-volume-and-surface-area)
  2. Diameter 472.440945 in gives Radius 236.220472 in, Surface area 701,204.882691 in², Volume 904,778,684.233861 cm³.
  3. Volume 1,000,000,000 cm³ gives Radius 244.232477 in, Surface area 749,577.757773 in².
  4. Surface area 155,000.310001 in² gives Radius 111.060942 in, Volume 94,031,597.257959 cm³.
  5. Circumference 1,577,756,562.440945 in gives Radius 251,107,755.908153 in.

How it works

A sphere is set by one number: its radius r, the distance from the centre to any point of the surface. Every other measure follows from r:

  • Diameter: d = 2r
  • Circumference of a great circle (the widest circle on the sphere): C = 2πr
  • Surface area: A = 4πr²
  • Volume: V = 4/3 πr³

Type any one of the five values. The calculator finds the radius first, then the other three:

  • from the diameter: r = d ÷ 2
  • from the circumference: r = C ÷ 2π
  • from the surface area: r = √(A ÷ 4π)
  • from the volume: r = ∛(3V ÷ 4π)

Two checks tie the answers together: V = A × r ÷ 3, and C = πd.

Assumptions

  • π is the full-precision value 3.141592653589793…, not 3.14 or 22/7.
  • Each value is greater than 0. The radius can be at most 10¹⁰⁰ m, and the others at most what that radius gives (diameter 2 × 10¹⁰⁰ m, circumference 7 × 10¹⁰⁰ m, surface area 1.3 × 10²⁰¹ m², volume 4.2 × 10³⁰⁰ m³); a value typed or solved beyond these has no answer.
  • Each box has its own unit. Lengths are converted through metres, areas through square metres, and volumes through cubic metres, with 1 in = 0.0254 m exactly. The formulas give the same numbers in any one unit: a radius of 5 in gives a volume of 523.6 in³, and a radius of 5 m gives 523.6 m³.
  • Answers show at most 6 decimals; a value below 0.0001 shows 6 significant figures. Rounding is half up on the decimal value.

Worked examples by hand

Radius 5. d = 2 × 5 = 10. C = 2π × 5 = 10π = 31.4159. A = 4π × 5² = 100π = 314.1593. V = 4/3 × π × 5³ = 500π ÷ 3 = 523.5988.

Diameter 12. r = 12 ÷ 2 = 6. A = 4π × 36 = 144π = 452.3893. V = 4/3 × π × 216 = 288π = 904.7787.

Volume 1,000. r = ∛(3 × 1,000 ÷ 4π) = ∛238.7324 = 6.2035. A = 4π × 6.2035² = 483.5976.

Surface area 100. r = √(100 ÷ 4π) = √7.9577 = 2.8209. V = 4/3 × π × 2.8209³ = 94.0316.

The Earth as a sphere. The WGS 84 equator is 40,075,016.686 m long (2π times the equatorial radius). As a great circle, r = 40,075,016.686 ÷ 2π = 6,378,137.000067 m, which matches the WGS 84 equatorial radius of 6,378,137 m.

Other questions people ask

What is the formula for the volume of a sphere?

V = 4/3 πr³, where r is the radius. A ball of radius 5 cm holds 4/3 × π × 125 = 523.6 cm³. If you know the diameter d, the same formula is V = πd³ ÷ 6.

What is the formula for the surface area of a sphere?

A = 4πr². The surface of a sphere is exactly four times the area of a circle with the same radius (a great circle). A sphere of radius 5 cm has a surface area of 100π = 314.16 cm².

How do I find the radius of a sphere from its volume?

Reverse V = 4/3 πr³: r = ∛(3V ÷ 4π). A volume of 1,000 cm³ gives r = ∛(3,000 ÷ 12.566) = ∛238.73 = 6.2035 cm.

How do I find the radius from the surface area?

Reverse A = 4πr²: r = √(A ÷ 4π). A surface area of 100 in² gives r = √(100 ÷ 12.566) = 2.8209 in.

What is the circumference of a sphere?

It is the length of a great circle, the biggest circle you can draw on the sphere (like the equator on a globe): C = 2πr = πd. It is what a tape measure reads when you wrap it around a ball at its widest.

Why does doubling the radius make the volume eight times as large?

The volume grows with the cube of the radius, and 2³ = 8. The surface area grows with the square, so it becomes 2² = 4 times as large. This is why big balls hold much more than their surface suggests.

Can I use different units for each box?

Yes. Each box has its own unit switch. Lengths are converted through metres, areas through square metres, and volumes through cubic metres, so a radius in inches and a volume in litres give the same sphere.