Volume Formulas for Common 3D Shapes
By the Calculator.hair team · Updated · 7 min read
Short answer
Volume is the space inside a 3D shape, in cubic units. The main formulas are: cube = s³, box = l × w × h, cylinder = π × r² × h, cone = ⅓ × π × r² × h, pyramid = ⅓ × base area × h, and sphere = 4/3 × π × r³. For example, a cylinder with a 3-foot radius and 10-foot height holds about 282.74 cubic feet.
Key takeaways
- A shape with straight sides has a volume of base area times height; one that tapers to a point has a third of that.
- A cone holds exactly one third of a cylinder with the same radius and height.
- Convert every measurement to one unit before multiplying.
- One cubic foot is about 7.48 US gallons or 28.32 liters, and one cubic yard is 27 cubic feet.
- Doubling every dimension multiplies volume by 8, because all three dimensions scale.
What are the volume formulas for common shapes?
The table gives the volume formula for each common solid, with a worked example. Volume means the amount of space a shape encloses, measured in cubic units such as cubic feet (ft³) or cubic meters (m³). Every result except the two hand-calculated rows at the end comes from our volume calculator, with all measurements in feet.
| Shape | Formula | Example inputs | Volume |
|---|---|---|---|
| Cube | V = s³ | s = 4 | 64 ft³ |
| Rectangular box | V = l × w × h | l = 4, w = 3, h = 2 | 24 ft³ |
| Cylinder | V = π × r² × h | r = 3, h = 10 | 282.743339 ft³ |
| Cone | V = ⅓ × π × r² × h | r = 3, h = 4 | 37.699112 ft³ |
| Square pyramid | V = ⅓ × a² × h | a = 6, h = 4 | 48 ft³ |
| Sphere | V = 4/3 × π × r³ | r = 3 | 113.097336 ft³ |
| Capsule | V = π × r² × h + 4/3 × π × r³ | r = 2, h = 5 | 96.342175 ft³ |
| Conical frustum | V = ⅓ × π × h × (r² + r × R + R²) | r = 2, R = 3, h = 6 | 119.380521 ft³ |
| Ellipsoid | V = 4/3 × π × a × b × c | a = 3, b = 2, c = 1 | 25.132741 ft³ |
| Tube | V = π × (d₁² − d₂²) × l ÷ 4 | d₁ = 4, d₂ = 3, l = 10 | 54.977871 ft³ |
| Rectangular pyramid (by hand) | V = ⅓ × l × w × h | l = 6, w = 4, h = 9 | 72 ft³ |
| Triangular prism (by hand) | V = ½ × b × t × l | b = 3, t = 4, l = 10 | 60 ft³ |
What is the cube volume formula?
The volume of a cube is the edge length cubed: V = s³. A cube with 4-foot edges has a volume of 4 × 4 × 4 = 64 cubic feet.
A rectangular box works the same way with three different measurements. A tank 4 feet long, 3 feet wide and 2 feet high holds 4 × 3 × 2 = 24 cubic feet. To work backward from a cube's volume to its edge, take the cube root: ∛64 = 4.
What is the cylinder volume formula?
The volume of a cylinder is the area of its circular base times its height: V = π × r² × h. With a radius of 3 feet and a height of 10 feet, the base covers π × 9 ≈ 28.27 square feet, and the volume is about 282.74 cubic feet.
The radius is half the diameter, so measure across the top and halve it. Our guide to area formulas covers the circle in more detail. Because the radius is squared, it matters more than the height. Doubling the height of a cylinder doubles its volume, while doubling the radius multiplies it by 4.
What is the cone volume formula?
The volume of a cone is one third of the base area times the height: V = ⅓ × π × r² × h. A cone with a 3-foot radius and a 4-foot height holds ⅓ × π × 9 × 4 ≈ 37.70 cubic feet.
A cylinder with that same radius and height holds about 113.10 cubic feet, exactly three times as much. The height in this formula is the vertical height, from the center of the base straight up to the tip. It is not the slant height, the distance along the sloping side. If you only know the slant height, use the Pythagorean theorem: h = √(l² − r²). A slant height of 5 and a radius of 3 give a vertical height of 4.
What is the pyramid volume formula?
The volume of any pyramid is one third of the base area times the height: V = ⅓ × B × h. For a square base with edge a, B = a², so a pyramid with a 6-foot base edge and a 4-foot height holds ⅓ × 36 × 4 = 48 cubic feet.
The base does not have to be square. A pyramid on a 6 × 4 rectangle with a height of 9 holds ⅓ × 24 × 9 = 72 cubic feet. Our calculator's pyramid form takes a square base, so that example is hand arithmetic. The same ⅓ applies to a triangular base once you have found its area.
What is the sphere volume formula?
The volume of a sphere is four thirds times pi times the radius cubed: V = 4/3 × π × r³. A sphere with a 3-foot radius holds 4/3 × π × 27 ≈ 113.10 cubic feet.
Because the radius is cubed, small changes have large effects. Doubling a sphere's radius from 3 to 6 feet multiplies its volume by 8. A capsule, a cylinder with a half sphere on each end, adds the cylinder and sphere formulas together. With a radius of 2 and a cylinder section 5 long, that is about 62.83 + 33.51 = 96.34 cubic feet.
How do you convert volume units?
Cube the length conversion factor, or multiply by a known conversion from the table. There are 3 feet in a yard, so one cubic yard is 3 × 3 × 3 = 27 cubic feet, not 3. Our calculator reports every result in cubic meters, liters, US gallons and cubic feet.
Worked example: the 4 × 3 × 2 foot tank holds 24 cubic feet. That is 24 × 28.316847 ≈ 679.60 liters, or 24 × 7.480519 ≈ 179.53 US gallons. If you measure one side in inches, convert first: a length of 48 inches is 4 feet. The calculator takes a separate unit for each measurement and returns the same 24 cubic feet.
Materials such as concrete and mulch are usually sold by the cubic yard, so divide cubic feet by 27: 24 ÷ 27 ≈ 0.89 cubic yards. See how much concrete you need for a full job. A gallon here is the US liquid gallon, defined as 231 cubic inches. The UK imperial gallon is larger, so check which one a source means.
| Unit | Equals |
|---|---|
| 1 cubic foot | 1,728 cubic inches |
| 1 cubic foot | 28.316847 liters |
| 1 cubic foot | 7.480519 US gallons |
| 1 cubic yard | 27 cubic feet |
| 1 cubic yard | 0.764555 cubic meters |
| 1 cubic meter | 1,000 liters |
| 1 cubic meter | 35.314667 cubic feet |
| 1 cubic meter | 264.172052 US gallons |
| 1 US gallon | 231 cubic inches |
| 1 liter | 1,000 cubic centimeters |
What is the difference between volume and surface area?
Volume measures the space inside a solid, while surface area measures the total area of its outer faces. Volume tells you how much a tank holds, and surface area tells you how much paint covers it. The surface area calculator handles the same shapes.
The table pairs both results for the examples above. For the cone and pyramid, the surface formula uses the slant height l, which is 5 feet in both examples. The two sets of numbers are in different units and can't be compared directly. A 3-foot-radius sphere shows why: its volume and surface area are both about 113.10, but one is cubic feet and the other is square feet.
| Shape | Surface area formula | Inputs | Surface area | Volume |
|---|---|---|---|---|
| Cube | 6 × s² | s = 4 | 96 ft² | 64 ft³ |
| Box | 2 × (lw + lh + wh) | 4, 3, 2 | 52 ft² | 24 ft³ |
| Cylinder | 2πr² + 2πrh | r = 3, h = 10 | 245.0442 ft² | 282.743339 ft³ |
| Cone | πr² + πrl | r = 3, h = 4 | 75.3982 ft² | 37.699112 ft³ |
| Square pyramid | a² + 2al | a = 6, h = 4 | 96 ft² | 48 ft³ |
| Sphere | 4πr² | r = 3 | 113.0973 ft² | 113.097336 ft³ |
What mistakes should you avoid when calculating volume?
Most errors come from units or from using the wrong length in the formula.
- Using the diameter where the formula calls for the radius. In the cylinder formula, this makes the answer 4 times too large.
- Using slant height instead of vertical height for a cone or pyramid.
- Forgetting the ⅓ for shapes that taper to a point.
- Mixing inches and feet in one multiplication.
- Using a linear conversion on a cubic quantity, such as dividing cubic feet by 3 to get cubic yards.
Calculators in this guide
Frequently asked questions
What is the formula for volume?
It depends on the shape, but most solids follow V = base area × height. Pointed shapes such as cones and pyramids take a third of that.
What is the volume of a cone with radius 3 and height 4?
V = ⅓ × π × 3² × 4 = 12π, or about 37.70 cubic units.
How many gallons are in a cubic foot?
About 7.48 US gallons. To get the volume of a tank in gallons, multiply its cubic feet by 7.480519.
How do you calculate the volume of a cylinder in gallons?
Find V = π × r² × h in cubic feet, then multiply by 7.480519. A cylinder with a 3-foot radius and a 10-foot height holds about 2,115 US gallons.
How do you find the volume of an irregular object?
Split it into simple solids and add their volumes. For a small solid object, you can also measure how much water it displaces when submerged.