Sphere Volume Calculator

What is a sphere?

A sphere is a perfectly round 3D shape. Every point on its surface is the same distance from its center. That distance is the radius. Balls, bubbles, marbles, ball bearings and planets are close to spheres. A sphere has no edges and no corners.

Parts of a sphere

• Center: the point in the middle, the same distance from every point on the surface.
• Radius (r): the distance from the center to the surface.
• Diameter (d): the distance across the sphere through the center. It equals 2r.
• Circumference (C): the distance around the sphere at its widest point. This circle is called a great circle.
• Surface area (A): the area of the curved outside of the sphere.
• Volume (V): the space inside the sphere. A sphere is the 3D version of a circle. A circle is flat and has an area. A sphere is solid and has a volume.

Equation of a sphere

In geometry with x, y and z axes, a sphere is every point at the distance r from its center (a, b, c). Its equation is:

For a sphere centered at the origin, this becomes x² + y² + z² = r². Students meet this sphere equation in algebra and calculus. The volume formula, 4/3 × π × r³, is a separate equation for the space inside.

Sphere volume formula

The formula for the volume of a sphere is:

• V is the volume.
• π (pi) is about 3.14159.
• r is the radius.
• r³ means r × r × r.

If you know the diameter instead, use this version of the sphere equation:

Both give the same answer, since r = d ÷ 2.

Volume of a sphere in other words

The volume of a sphere is about 4.19 × r³. Find the radius cubed and multiply by 4.19 for a quick estimate. For an exact answer, use 4/3 × π. A sphere with a radius of 1 inch holds 4.19 cubic inches. A sphere with a radius of 1 foot holds 4.19 cubic feet.

Sphere and circle formulas side by side

The sphere formulas build on the circle formulas you already know.

Notice the pattern: length uses r, area uses r², and volume uses r³.

How to find the volume of a sphere

To calculate the volume of a sphere by hand, follow these steps:

1. Find the radius. If you have the diameter, divide it by 2.
2. Cube the radius: multiply it by itself three times.
3. Multiply by π (3.14159).
4. Multiply by 4, then divide by 3.

The answer is in cubic units, such as cubic inches or cubic centimeters.

Sphere volume example with the radius

A ball has a radius of 5 inches.

1. r³ = 5 × 5 × 5 = 125.
2. 125 × 3.14159 = 392.7.
3. 392.7 × 4 ÷ 3 = 523.6 cubic inches.

That is about 2.27 gallons, or 8.58 liters.

Sphere volume example with the diameter

A gazing ball is 6 inches across.

V = 3.14159 × 6³ ÷ 6 = 3.14159 × 216 ÷ 6 = 113.1 cubic inches.

Sphere volume example in centimeters

A ball has a radius of 10 cm.

V = 4/3 × 3.14159 × 1,000 = 4,188.8 cm³, or about 4.19 liters.

Sphere volume example in feet

A round water tank has a radius of 2 feet.

V = 4/3 × 3.14159 × 2³ = 33.51 cubic feet. One cubic foot holds 7.48 gallons, so the tank holds about 250.7 gallons.

Sphere volume chart by diameter

Use this chart to check your answer or to pick a ball or tank size. Each row loads into the sphere volume calculator.

Notice that doubling the diameter from 12 to 24 inches makes the sphere hold 8 times as much: 3.9 gallons becomes 31.3 gallons.

Volume of a sphere from its circumference

You often cannot reach the center of a ball, but you can wrap a tape around it. Use the circumference to find the radius, then the volume.

Example: A size 7 basketball is about 29.5 inches around. r = 29.5 ÷ 6.2832 = 4.695 inches. V = 4/3 × 3.14159 × 4.695³ = 433.5 cubic inches, or about 1.88 gallons.

Volume of a sphere from its surface area

Find the radius from the surface area, then use the volume formula.

Example: A sphere has a surface area of 314.16 square inches. r = √(314.16 ÷ 12.566) = 5 inches. V = 314.16 × 5 ÷ 3 = 523.6 cubic inches.

Surface area of a sphere

The surface area of a sphere is the area of its curved outside. The formula is:

The surface area of a sphere is exactly four times the area of its great circle (π × r²). Picture four circles of the same radius. Together, they would cover the ball.

Example: A ball with a radius of 5 inches has a surface area of 4 × 3.14159 × 25 = 314.16 square inches.

How to find the surface area of a sphere

1. Find the radius.
2. Square it: r × r.
3. Multiply by π, then by 4.

Example: A softball is 12 inches around. Its radius is 12 ÷ 6.2832 = 1.91 inches. A = 4 × 3.14159 × 1.91² = 45.84 square inches.

Surface area of a sphere from its volume

Example: A sphere of 523.6 cubic inches has a surface area of ∛(36 × 3.14159 × 523.6²) = 314.16 square inches.

Surface area of a ball or globe

To paint, cover or wrap a ball, you need its surface area. A desk globe 12 inches across has a surface area of π × 12² = 452.4 square inches. Buy a little more paint or fabric than that, since seams and overlaps use some.

Sphere surface area to volume ratio

A small sphere has more surface for its size than a big one. That is why small drops of water evaporate fast and big animals lose heat more slowly than small ones. A sphere with a 5-inch radius has a ratio of 3 ÷ 5 = 0.6 per inch.

Circumference and radius of a sphere

Circumference of a sphere

The circumference of a sphere is the length around its widest part, the great circle.

A sphere does not have a perimeter in the usual sense, because it has no edges. When people ask for the "perimeter of a sphere", they mean this circumference.

Example: A ball with a 5-inch radius is 2 × 3.14159 × 5 = 31.42 inches around.

How to find the radius of a sphere

Pick the formula that matches what you know:

Example: A ball that holds exactly 1 gallon (231 cubic inches) has a radius of ∛(3 × 231 ÷ 12.566) = 3.81 inches.

How to find the diameter of a sphere

Double the radius, or use one of these formulas:

Example: A ball that holds 1 gallon (231 cubic inches) is ∛(6 × 231 ÷ 3.14159) = 7.61 inches across.

How the radius changes sphere volume

Because the radius is cubed, small changes make a big difference.

• Double the radius: the surface area grows 4 times, and the volume grows 8 times.
• Triple the radius: the surface area grows 9 times, and the volume grows 27 times.
• Halve the radius: the volume drops to one eighth.

At a radius of 3, the surface area and the volume have the same number, 113.1. Above 3, the volume grows faster.

Spherical cap and hemisphere volume

A spherical cap, or dome, is the part of a sphere cut off by one flat plane. A fish bowl, a dome roof and a contact lens are caps.

Here r is the radius of the sphere, h is the height of the cap, and a is the radius of the cap's flat base. Use the first formula when you know r. Use the second when you know a.

Example: A cap 3 inches tall cut from a sphere with a 5-inch radius has V = 3.14159 × 9 × (15 − 3) ÷ 3 = 113.1 cubic inches. That is 21.6% of the sphere.

Example: A round fish bowl is 7 inches tall and has an opening with a radius of 3.13 inches. V = 3.14159 × 7 × (3 × 3.13² + 49) ÷ 6 = 287.4 cubic inches, or about 1.24 gallons. The bowl comes from a sphere with a radius of 4.2 inches.

Spherical cap surface area

The curved surface of a cap depends only on the sphere's radius and the cap's height.

Example: The cap 3 inches tall from a 5-inch sphere has a curved surface of 2 × 3.14159 × 5 × 3 = 94.25 square inches. Its flat base adds π × 4.58² = 65.97 square inches.

Hemisphere volume

A hemisphere is half a sphere. It is a cap whose height equals the radius.

Example: A dome with a 5-inch radius holds 2/3 × 3.14159 × 125 = 261.8 cubic inches. Its curved surface is 2 × π × r² = 157.1 square inches. Add the flat base to get 235.6 square inches. For more on half spheres, use the hemisphere volume calculator.

Spherical segment volume

A spherical segment is a slice of a sphere between two parallel planes, like a band around a globe or the middle of a barrel-shaped bead.

a and b are the radii of the two flat faces, and h is the distance between them.

Example: A slice with face radii of 4 and 3 inches, 2 inches thick, holds 3.14159 × 2 × (48 + 27 + 4) ÷ 6 = 82.7 cubic inches.

Sphere vs cylinder, cone and cube

The sphere has neat links to other shapes with the same radius.

• Cylinder: a sphere fills exactly two thirds of the cylinder that just fits around it (height = 2r). For more cylinder shapes and liquid levels, use the cylinder volume calculator.
• Cone: a cone with the same base and height as that cylinder holds one third. So cone : sphere : cylinder = 1 : 2 : 3.
• Surface: the sphere's surface area equals the curved side of that cylinder: 4πr² = 2πr × 2r.
• Cube: a sphere fills about 52.4% of the smallest cube it fits in (π ÷ 6).

The Greek mathematician Archimedes proved the sphere and cylinder rules more than 2,000 years ago. He asked for a sphere inside a cylinder to be carved on his tomb.

Why a sphere holds the most for its surface

Of all shapes with the same volume, the sphere has the smallest surface area. To hold 1 gallon (231 cubic inches), a sphere needs 182.1 square inches of surface. A cube that holds the same gallon needs 225.9 square inches, about 24% more. That is why bubbles and drops are round, and why some tanks are spheres: they use the least material for the volume they hold.

Why the sphere volume formula works

Sphere volume with slices

Cut a sphere into thin, flat discs, like slicing an onion. Each disc is a short cylinder. Add up the volumes of all the discs, and you get the volume of the sphere. Thinner slices give a closer answer.

Sphere volume with calculus

A disc at height z has the radius √(r² − z²), so its area is π × (r² − z²). Add up the discs from the bottom (z = −r) to the top (z = r):

The surface area follows too. If you add a thin layer to a sphere, the volume grows by the surface area times the layer's thickness. So the surface area is the rate of change of the volume: dV/dr = 4πr².

Hollow sphere and spherical shell volume

A hollow sphere has an outer surface and an inner surface, like a ball, a globe ornament or a spherical tank. To find the volume of the wall, subtract the inner sphere from the outer sphere.

R is the outer radius, and r is the inner radius.

Example: A ball 10 inches across with a 0.5-inch wall has R = 5 and r = 4.5. The wall holds 4/3 × 3.14159 × (125 − 91.125) = 141.9 cubic inches of material. The space inside is 381.7 cubic inches, or 1.65 gallons.

To find how much liquid a spherical tank holds at a given level, use the tank volume calculator.

Volume of a ball: real examples

Sports balls are close to spheres, so the sphere volume formula works well for them. Their sizes are set by each sport's rules and vary a little between brands.

A basketball holds about 1.88 gallons of air at normal pressure. A soccer ball holds about 1.52 gallons.

Volume of a globe and the Earth as a sphere

The Earth is not a perfect sphere, but its mean radius is about 6,371 kilometers. Treated as a sphere, its volume is 4/3 × π × 6,371³ = about 1.083 trillion cubic kilometers. Its surface area is about 510 million square kilometers.

How many balls fit in a jar: sphere packing

Balls never fill a jar completely, because there are gaps between them. Equal balls poured into a jar fill about 64% of the space. The tightest possible stacking, like oranges in a pyramid, fills about 74%. To guess how many balls fit, multiply the jar's volume by 0.64 and divide by the volume of one ball.

Example: A 1-gallon jar holds 231 cubic inches. A gumball 1 inch across has a volume of 0.524 cubic inches. 231 × 0.64 ÷ 0.524 = about 282 gumballs. The real count depends on the jar's shape and how you shake it.

Sphere mass and density

To find how much a solid ball weighs, multiply its volume by the density of its material.

Example: A steel ball bearing with a 1-inch radius has a volume of 4.19 cubic inches, or 68.64 cm³. Steel weighs about 7.85 grams per cm³. So the ball weighs 68.64 × 7.85 = 538.8 grams, or 1.19 pounds. The same ball in aluminum (2.70 g/cm³) weighs 185.3 grams.

Real materials vary. For other materials, use the volume to weight calculator. To go the other way, pick "Find size from mass" in the calculator. Enter the mass and the material, and get the radius of the ball.

How to find the density of a sphere

Divide the mass by the volume. Use matching units, such as grams and cubic centimeters.

Example: A ball with a 1.5-inch radius weighs 1 pound (453.6 grams). Its volume is 14.14 cubic inches, or 231.7 cm³. Density = 453.6 ÷ 231.7 = 1.96 g/cm³. That is about twice as dense as water.

Sphere volume in gallons, liters and cubic feet

The sphere volume formula gives cubic units. Change them to gallons or liters to see how much a ball or round tank holds.

Example: A spherical tank with a 30-inch radius holds 4/3 × 3.14159 × 27,000 ÷ 231 = 489.6 gallons. For more unit changes, use the gallon converter or the cubic feet calculator.

How to measure a sphere

You cannot put a ruler through the center of a ball. Use one of these methods instead.

• Wrap a tape around it. Measure the circumference at the widest part. The calculator finds the radius from it. This works best for balls and globes.
• Use calipers. Close the jaws on the ball at its widest point to read the diameter directly. This works best for small balls, like marbles and bearings.
• Use two blocks. Place the ball between two flat blocks on a table, and measure the gap between the blocks. That gap is the diameter.
• Use water. Lower the ball into a straight-sided container of water and measure how far the water rises. The rise times the container's base area is the ball's volume.

Example of the water method: A round container has a 3-inch radius. The water rises 2.5 inches when the ball goes in. Volume = 3.14159 × 3² × 2.5 = 70.7 cubic inches. This trick, called water displacement, also works for shapes that are not perfect spheres.

Common sphere volume mistakes

• Using the diameter as the radius. This makes the volume 8 times too big. Halve the diameter first, or use π × d³ ÷ 6.
• Squaring instead of cubing. Volume uses r³. Surface area uses r².
• Forgetting the 4/3. π × r³ alone is only three quarters of the volume.
• Mixing units. Keep every size in the same unit. The answer comes out in that unit cubed.
• Measuring around the wrong line. For the circumference, measure around the widest part of the ball.
• Treating an egg or a rugby ball as a sphere. Those shapes are ellipsoids. Use the ellipsoid calculator.

Where sphere volume is used

• Sports: how much air a ball holds, and the size rules for balls.
• Tanks and storage: spherical water towers and gas tanks.
• Manufacturing: ball bearings, marbles, beads and shot.
• Science: drops, bubbles, cells and planets.
• Cooking: melon balls, meatballs and scoops of ice cream.
• Packaging: how many round items fit in a box or jar.

Related calculators

Sources

We checked the formulas and values on this page against these sources.

1. Sphere — Wolfram MathWorld. “Sphere volume, surface area and the great circle.”
2. Spherical Cap — Wolfram MathWorld. “Cap volume and curved surface area.”
3. Spherical Segment — Wolfram MathWorld. “Segment volume between two planes.”
4. Earth Fact Sheet — NASA Space Science Data Coordinated Archive. “The Earth's mean radius and volume.”