What this calculator does
Measure one thing about a ball, and get every other size you need. A sphere is a perfectly round ball. Every point on its surface is the same distance from the center. That distance is the radius, and it's the only number you need. From the radius come the diameter (the width through the center) and the circumference (the distance around the widest part). The surface area and the volume follow too. Each comes from a fixed formula that uses π (pi, about 3.14159).
Choose the measurement you have and type it in. The calculator works back to the radius, then forward to everything else. It's handy for balls, tanks, domes, beads, planets and droplets. Measure whichever size is easiest, and let the formulas do the rest.
Where it's used
Why this math matters outside the classroom:
- Balloons. A balloon blown up to a 15 cm radius holds (4/3)π × 15³, about 14,137 cm³, or 14.1 liters, of air.
- Ball covers. A basketball with a 12 cm radius needs about 4π × 12² ≈ 1,810 cm² of outer material.
- Water tanks. A spherical tank 4 m across holds (4/3)π × 2³, about 33.5 m³, of water.
- Planet surfaces. Taking Earth's mean radius as 6,371 km gives a surface area of about 510 million km².
- Ice cream scoops. A scoop with a 3 cm radius holds (4/3)π × 27, about 113 cm³, of ice cream.
Grade level and standards: Grades 8–10; Common Core State Standards for Mathematics 8.G.C.9, HSG.GMD.A.3. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
How to use it
- Select what you know: radius, diameter, circumference, surface area or volume.
- Enter its value in any unit. An area must be in the matching square unit, and a volume in the matching cubic unit.
- Read the other four measurements. The main result is the volume, or the radius if you started from the volume.
What the inputs mean
- Circumference (C)
- The distance around the sphere at its widest point, called a great circle. It's what you get by wrapping a tape around the middle of a ball.
- Surface area (A)
- The area of the whole outside surface, in square units of your length unit.
- Volume (V)
- The space inside, in cubic units. 1,000 cm³ is one liter.
Formula and method
d = 2r C = 2πr A = 4πr² V = (4 ÷ 3)πr³ r = C ÷ 2π = √(A ÷ 4π) = ∛(3V ÷ 4π)
In these formulas, r is the radius and d is the diameter. C is the circumference, A is the surface area and V is the volume. π is pi, about 3.14159. The circumference is the distance around the sphere's widest circle. The surface area is exactly four times the area of that circle. Archimedes, the ancient Greek mathematician, first proved this. The volume is two-thirds of the volume of the smallest can (cylinder) the sphere would fit snugly inside.
The last line turns the others around to get back to the radius. A square root (√) undoes the r² in the area. A cube root (∛) undoes the r³ in the volume. Everything else is then worked out from that radius, using every digit. The value you entered is kept exactly.
Volume depends on r³ (r × r × r), so small changes in size have large effects. A sphere with twice the radius has eight times the volume, and four times the surface area.
Worked example
A tape wrapped around the middle of a ball reads 75 cm. How big is the ball and how much air does it hold?
- Radius from the circumference: r = 75 ÷ 2π ≈ 11.9366 cm.
- Diameter: 2 × 11.9366 ≈ 23.8732 cm.
- Surface area: 4 × π × 11.9366² ≈ 1,790.4931 cm².
- Volume: (4 ÷ 3) × π × 11.9366³ ≈ 7,124.1457 cm³.
Result: The ball is about 23.87 cm across, and its surface is about 1,790.49 cm². It holds about 7,124.15 cm³, a little over 7.1 liters.
Understanding the result
Surface area is in square units, and volume is in cubic units, of the length you used. To turn cm³ into liters, divide by 1,000.
For a real object, a small error in the radius roughly triples in the volume, as a percentage. So measure carefully. A tape around the middle works better than a ruler across it.
Assumptions and limitations
- The object is a perfect sphere.
- Your input is exact. Results are rounded only for display.
- Real objects such as the Earth or a squashed ball are slightly flattened. Using an average radius gets you close.
- Half spheres (hemispheres), slices off the top (spherical caps) and hollow shells need adjusting. For a hemisphere, halve the values and add the flat face separately.
- It doesn't convert between units.
Frequently asked questions
How do I find the radius from the volume?
Turn V = (4 ÷ 3)πr³ around to r = ∛(3V ÷ 4π). For a volume of 1,000 cm³ (one liter), r³ ≈ 238.7324 and r ≈ 6.2035 cm. Choose volume as the measurement you know, and the calculator does this for you.
Why is a sphere's surface area 4πr²?
Archimedes showed that a sphere's surface has the same area as the curved side of the snug can (cylinder) around it. That cylinder has radius r and height 2r. Its side area is 2πr × 2r = 4πr². That's four times the area of the sphere's widest circle.
What is the volume of a hemisphere?
Half the sphere's volume: (2 ÷ 3)πr³. Its curved surface is half the sphere's surface, 2πr². If it's closed, add a flat round base of πr².
Can I use this for the mass of a ball?
Yes, in two steps. Find the volume here. Then multiply by the material's density (how heavy it is for its size), for example with the density calculator. A solid steel ball of radius 1 cm has a volume of about 4.19 cm³.
How this calculator is checked
Method: Standard sphere relations C = 2πr, A = 4πr² and V = (4/3)πr³. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.
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