The volume of shapes is the space a solid fills, measured in cubic units. Most formulas are base area times height: a cube is s x s x s, a prism or cylinder is base area x height, and a cone or pyramid is one third of that. A sphere is 4/3 x pi x r cubed, and 1 liter equals 1,000 cubic centimeters.
So whether you are filling a fish tank, ordering concrete or checking a homework answer, the same short list of formulas does the job. First, you measure a few lengths. Next, you plug them into the right formula. Finally, you convert the cubic answer into a unit you can actually buy or pour, such as gallons or liters.
This guide gives you one formula table for eight common solids, then works through a real example for each shape. After that, it shows how to turn cubic inches, feet and centimeters into liters and gallons. The conversion factors come from exact definitions published by the US National Institute of Standards and Technology (NIST) on its unit conversion page: for example, 1 inch is exactly 2.54 cm, so 1 cubic inch is exactly 16.387064 cubic centimeters.
Volume of Shapes Formula Table
Below are the eight formulas you will use most, so keep this table handy. In every row, s is a side, l is length, w is width, h is height, r is radius and B is the area of the base. Also, pi is about 3.14159.
| Shape | Formula | What to measure |
|---|---|---|
| Cube | V = s x s x s = s3 | One edge |
| Rectangular prism (box) | V = l x w x h | Length, width, height |
| Cylinder | V = pi x r2 x h | Radius of the circle, height |
| Cone | V = (1/3) x pi x r2 x h | Base radius, vertical height |
| Sphere | V = (4/3) x pi x r3 | Radius (half the diameter) |
| Hemisphere | V = (2/3) x pi x r3 | Radius |
| Pyramid (any base) | V = (1/3) x B x h | Base area, vertical height |
| Triangular prism | V = (1/2) x b x a x l | Triangle base b, triangle height a, prism length l |
Notice the pattern first. In other words, a prism or cylinder is simply its base area times its height. Meanwhile, a pointed solid with the same base and height, such as a cone or pyramid, holds exactly one third of that. As a result, you only need to remember a few ideas rather than eight separate rules.
Convert Your Volume of Shapes Answer to Liquid Units
After you have a volume in cubic centimeters, type it below as cc or mL (they are the same size). The converter then shows liters, US gallons, quarts, cups and fluid ounces, so you can see how much liquid a container really holds.
Similarly, for cubic inches, first multiply by 16.387 to get cubic centimeters. For cubic feet, multiply by 28.317 to get liters instead.
Volume of Shapes: Worked Examples
Each example below therefore uses round, realistic numbers. In addition, every answer was checked with a calculator and rounded to two decimals, so you can compare your own working step by step.
Cube
For example, a storage cube measures 4 in on each edge. So V = 4 x 4 x 4 = 64 cubic inches. In metric terms, that is 64 x 16.387 = about 1,048.77 cubic centimeters, or roughly 1.05 liters.
For more cube practice, including how to work backward from volume to edge length, see our guide to the volume of a cube.
Rectangular prism
For instance, an aquarium is 24 in long, 12 in wide and 16 in tall. Therefore V = 24 x 12 x 16 = 4,608 cubic inches. Because a US gallon is exactly 231 cubic inches, the tank holds 4,608 / 231 = about 19.95 gallons (about 75.51 liters) when filled to the brim.
Cylinder
For example, suppose a water tank has a radius of 2 ft and a height of 5 ft. So V = pi x 22 x 5 = about 62.83 cubic feet. Next, multiply by 7.4805 gallons per cubic foot to get about 470.01 gallons, or about 1,779.2 liters.
Cone
A cone has a base radius of 3 cm and a vertical height of 10 cm. First find the matching cylinder: pi x 9 x 10 = 282.74 cubic centimeters. Then take one third, which gives about 94.25 cubic centimeters (94.25 mL).
Sphere
An official size 7 basketball is about 29.5 in around. Dividing by 2 x pi gives a radius of about 4.70 in. As a result, V = (4/3) x pi x 4.703 = about 433.5 cubic inches, or roughly 7.10 liters of air space.
If you only know the diameter or circumference of a ball, our page on the volume of a sphere walks through each starting point.
Hemisphere
A round mixing bowl is half a sphere with a 5 cm inside radius. Therefore V = (2/3) x pi x 53 = about 261.80 cubic centimeters. That is exactly half of the full 5 cm sphere, which is 523.60 cubic centimeters.
Pyramid
Likewise, a garden pyramid has a square base 6 m on each side and stands 4 m tall. First, the base area is 6 x 6 = 36 square meters. So V = (1/3) x 36 x 4 = 48 cubic meters.
Triangular prism
A tent-shaped block has a triangular end with a 4 in base and a 3 in height, and it is 10 in long. First, the triangle area is (1/2) x 4 x 3 = 6 square inches. Then multiply by the length: 6 x 10 = 60 cubic inches.
Volume of Shapes in Different Units
In general, formulas give you cubic units of whatever length you measured. Consequently, inches give cubic inches and meters give cubic meters. Use this table to switch between them; the values follow from the exact inch and gallon definitions.
| From | Equals |
|---|---|
| 1 cubic inch (in³) | 16.387064 cm³ (exact) |
| 1 cubic foot (ft³) | 1,728 in³ = 28.317 L = 7.4805 US gal |
| 1 cubic yard (yd³) | 27 ft³ = 0.7646 m³ |
| 1 US gallon | 231 in³ = 3.785411784 L (exact) |
| 1 liter (L) | 1,000 cm³ = 61.024 in³ |
| 1 milliliter (mL) | 1 cm³ = 1 cc |
| 1 cubic meter (m³) | 1,000 L = 35.315 ft³ = 264.17 US gal |
Also, concrete and mulch are sold by the cubic yard. So a 10 ft by 10 ft slab that is 4 in thick needs 10 x 10 x (4/12) = 33.33 cubic feet, which is 33.33 / 27 = about 1.23 cubic yards before you add a waste allowance.
How to Find the Volume of Shapes Step by Step
- Identify the solid. Decide whether it is a prism, a cylinder, a pointed solid or a round one, then pick its formula from the table.
- Measure in one unit. Take every length in inches, or every length in centimeters, but never mix them.
- Halve any diameter. Formulas use the radius, so divide the measured width of a circle by 2.
- Work out the base area. Use a square, rectangle, circle or triangle area first, because most formulas build on it.
- Multiply by height and apply any fraction. Use one third for cones and pyramids, and 4/3 or 2/3 for spheres and hemispheres.
- Convert and round at the end. Change cubic units into liters or gallons with the table above, or enter the result in the converter.
Recommended Tools for Measuring the Volume of Shapes
However, good measurements matter more than the formula. For example, a digital caliper reads small parts to a hundredth of a millimeter, and a set of wooden solids makes the formulas easy to see. Meanwhile, graduated cylinders let you check any answer by water displacement, and a scientific calculator handles pi and powers.
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Volume vs Surface Area vs Capacity
At first glance these three words look alike, yet they answer different questions.
| Term | What it measures | Typical units |
|---|---|---|
| Volume | Space the solid itself fills | cm³, in³, ft³, m³ |
| Capacity | How much a hollow container holds inside | mL, L, fl oz, gal |
| Surface area | Total area of the outside faces | cm², in², ft² |
For instance, a thick ceramic mug has a larger outer volume than its capacity, because the walls take up space. Similarly, paint is bought by surface area, while water and soil are bought by volume.
Common Mistakes With the Volume of Shapes
Do
- Use the radius, not the diameter.
- Keep every length in one unit.
- Measure inside walls for capacity.
- Round only at the very end.
- Check a result with water displacement.
Don’t
- Forget the one third for cones and pyramids.
- Use slant height as the height.
- Convert a volume with a length factor.
- Round pi to 3 for real projects.
- Square the radius when the sphere needs it cubed.
Irregular and Composite Solids
Of course, real objects are rarely perfect shapes. However, you can usually split them into simple parts. A silo, for example, is a cylinder with a hemisphere on top, so you add the two volumes. Similarly, a pipe is the outer cylinder minus the inner one.
For a rock, a toy or any lumpy object, use water displacement instead. First, half fill a graduated cylinder and note the level. Next, lower the object in until it is fully covered. Then the rise in milliliters equals the object’s volume in cubic centimeters. Also, this trick, linked to Archimedes, works for any solid that sinks and does not soak up water.
Honest Limits of These Formulas
First, the formulas are exact for ideal shapes with flat faces and perfect curves. Real containers, though, have rounded corners, thick walls and slightly tapered sides. Therefore a measured tank often holds a little less than the math suggests. In addition, your answer is only as good as your tape: a 1 percent error in the radius becomes about a 3 percent error in a sphere’s volume, because the radius is cubed. In short, treat the result as a close estimate and leave a margin when you order materials.
Volume of Shapes FAQs
What is the volume of shapes in simple words?
In short, it is the amount of 3D space a solid takes up. You find it with a formula based on the shape’s lengths, and the answer comes out in cubic units such as cubic inches or cubic centimeters.
What is the formula for the volume of a cylinder?
V = pi x r squared x h. First find the area of the circular base, then multiply it by the height.
Why is a cone one third of a cylinder?
Because of its taper, a cone with the same base and height as a cylinder holds exactly one third as much. Calculus proves it, and you can check it by filling the cone three times to fill the cylinder.
How do I find the volume of a sphere from its diameter?
Halve the diameter to get the radius, then use V = (4/3) x pi x r cubed. A 10 cm ball has a 5 cm radius and a volume of about 523.6 cubic centimeters.
What is the volume of a pyramid?
V = (1/3) x base area x height. A square pyramid with a 6 m base and a 4 m height holds 48 cubic meters.
How many cubic centimeters are in a liter?
There are exactly 1,000 cubic centimeters in 1 liter, and also 1 cubic centimeter equals 1 milliliter.
How many gallons are in a cubic foot?
One cubic foot holds about 7.48 US gallons, because a cubic foot is 1,728 cubic inches and a US gallon is 231 cubic inches.
Which units should I use for the volume of shapes?
Use the cube of whatever length unit you measured. Then convert to liters, gallons or cubic yards depending on what you need to buy or fill.
What is the difference between volume and capacity?
Volume is the space a solid fills. Capacity, on the other hand, is how much a hollow container can hold inside its walls, so it is usually a little smaller than the outer volume.
How do I find the volume of an irregular object?
Split it into simple shapes and add their volumes, or sink it in a graduated cylinder and read how many milliliters the water rises.
The Bottom Line on the Volume of Shapes
To sum up, most solids follow one idea: base area times height, with one third for pointed shapes and 4/3 pi r cubed for spheres. So measure carefully in a single unit, use the radius, and convert only at the end. Finally, use the converter above to turn your answer into liters, gallons or cups.