Sphere Calculator
Find a sphere’s volume and surface area from the radius or diameter, or recover the radius from a known volume or surface area. The formulas are V = (4/3)πr³ and S = 4πr².
r = 5 · S = 314.159265359
- Diameter is twice the radius: d = 2r = 2 × 5 = 10.
- Volume V = (4/3)πr³ = (4/3) × 3.14159265359 × 5³ = 523.598775598.
- Surface area S = 4πr² = 4 × 3.14159265359 × 5² = 314.159265359.
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How to Use the Sphere Calculator
Choose the one measurement you know — radius, diameter, volume, or surface area — then Calculate. The other three come from V = (4/3)πr³ and S = 4πr².
- Select From radius, From diameter, From volume, or From surface area.
- Enter a non-negative number. Zero is a point with volume 0 and surface area 0.
- Read the volume (or the radius, when you started from volume), plus diameter and surface area.
A flat disk belongs on the Circle Calculator (A = πr², C = 2πr). A right circular can belongs on the Cylinder Calculator (V = πr²h). A dome belongs on the Hemisphere Calculator (V = (2/3)πr³). A slice that is not a hemisphere belongs on the Spherical Cap Calculator (V = (1/3)πh²(3R − h)). A band with two circular cuts belongs on the Spherical Zone Calculator (V = (1/6)πh(3a² + 3b² + h²)). Converting the cubic units after you have V belongs on the Volume Converter.
Volume and Surface Area
A sphere is the set of points at a fixed distance r from a center in space. The diameter is a straight cut through that center, so d = 2r. Volume scales with the cube of the radius: double r and you get eight times the volume. Surface area scales with the square: double r and you get four times the skin.
Solving for r from volume takes a cube root: r = ³√(3V / 4π). Solving from surface area takes a square root: r = √(S / 4π). Both recover the same r you would have started with.
Sphere Calculator Examples
Radius 5
d = 10. V = (4/3)π × 125 = 500π/3 ≈ 523.598775598. S = 4π × 25 = 100π ≈ 314.159265359.
Radius 3
V = (4/3)π × 27 = 36π. S = 4π × 9 = 36π. Volume and surface area match numerically (different units) only at this radius.
Diameter 10
r = 5, the same sphere as the first example.
Volume of the r = 5 sphere
Entering V ≈ 523.598775598 and solving for r returns 5. The cube-root step undoes (4/3)πr³.
A point sphere
r = 0 gives V = 0 and S = 0. Nothing is enclosed and there is no surface.
A negative radius
A length cannot be negative. The calculator rejects −1 instead of using |r|.
Why r = 3 Is Special
Set (4/3)πr³ = 4πr² and cancel 4π (for r ≠ 0): (1/3)r³ = r², so r = 3. Only that radius makes the volume number equal the surface-area number. The units are still different — cubic versus square — so they are not the same quantity. Homework sometimes asks you to notice the coincidence.
A circle of the same radius is the equator of the sphere. Its area πr² is one-fourth of the sphere’s surface 4πr². That is why S = 4 × (area of the great circle).
Common Sphere Mistakes
Using A = πr² as the volume
That is the area of a circle, on the Circle Calculator. Volume needs r³ and the 4/3.
Forgetting the 4/3
πr³ is not the volume. The 4/3 is part of the formula, not optional rounding.
Using diameter as if it were r
d = 10 means r = 5. Plugging 10 into (4/3)πr³ gives eight times the volume of the r = 5 sphere.
Treating V = S at r = 3 as the same measurement
36π cubic units is not 36π square units. Only the numerals match.
Using (4/3)πr³ for a slice of the ball
That is the whole sphere. A cap of height h belongs on the Spherical Cap Calculator (V = (1/3)πh²(3R − h)).
Converting liters on this page
V is already in the cube of your input unit. Use the Volume Converter for liters, gallons, and cups.
Sphere Calculator FAQs
What is the volume of a sphere?
V = (4/3)πr³. For radius 5, V = 500π/3 ≈ 523.598775598 cubic units.
What is the surface area of a sphere?
S = 4πr². For radius 5, S = 100π ≈ 314.159265359 square units.
How do you find the radius from the volume?
r = ³√(3V / 4π). Take three-halves of V over π, then the cube root.
Why are volume and surface area equal at r = 3?
Because (4/3)π(27) and 4π(9) are both 36π. The units still differ. It is a numerical coincidence, not a conversion.
Is this the same as the Circle Calculator or the Volume Converter?
No. Circle is a 2-D disk. Volume Converter changes liters and gallons. This page computes V and S of a solid ball.
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.