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Volume Formula
Volume is the measurement of three-dimensional space. Knowing how to calculate volume is crucial for shipping, construction, gardening, and daily tasks like cooking or filling pools. For an instant calculation, use the volume calculator to solve for cubic yards, gallons, or liters across all major 3D shapes.
8 min read
What are the volume formulas?
Unlike area, which only requires multiplying two dimensions, volume requires multiplying three dimensions (or base area and height). Because different 3D shapes have unique geometry, they each rely on a different volume formula.
Standard volume formulas
Here are the standard formulas for the five most common 3D geometric shapes:
1. Cube
V = s³
Where s = side length
2. Rectangular Prism (Box)
V = l × w × h
Where l = length, w = width, h = height
3. Cylinder
V = π × r² × h
Where r = radius of base, h = height, π ≈ 3.14159
4. Sphere
V = (4/3) × π × r³
Where r = radius, π ≈ 3.14159
5. Cone
V = (1/3) × π × r² × h
Where r = radius of base, h = height, π ≈ 3.14159
Worked examples: Calculating volume
Let's calculate the volume for each of the five shapes using real numbers:
- Example 1: Cube
Find the volume of a box with equal side lengths of 4 cm.
V = s³ = 4³ = 4 × 4 × 4 = 64 cm³. - Example 2: Rectangular Prism (Box)
Find the volume of a package that is 10 inches long, 5 inches wide, and 6 inches high.
V = l × w × h = 10 × 5 × 6 = 300 in³. - Example 3: Cylinder
Find the volume of a soda can with a base radius of 3 cm and a height of 12 cm.
V = πr²h = π × 3² × 12 = π × 9 × 12 = 108π ≈ 339.29 cm³. - Example 4: Sphere
Find the volume of a toy ball with a radius of 6 inches.
V = (4/3)πr³ = (4/3) × π × 6³ = (4/3) × π × 216 = 288π ≈ 904.78 in³. - Example 5: Cone
Find the volume of a funnel with a base radius of 3 inches and a height of 8 inches.
V = (1/3)πr²h = (1/3) × π × 3² × 8 = (1/3) × π × 9 × 8 = 24π ≈ 75.40 in³.
Common volume mistakes
- Forgetting to convert units. If your height is in feet but your width is in inches, you will calculate an incorrect volume. Make sure all dimensions are converted to the same unit before multiplying.
- Squaring instead of cubing. Volume is three-dimensional, meaning the output unit must be cubic (e.g. cm³, yd³), not square (e.g. cm², yd²).
- Using diameter instead of radius. For cylinders, spheres, and cones, the formulas require the radius (r). If you are given a diameter, remember to divide it by 2 first.
Cone Volume calculations
When working specifically with conical containers, calculating the volume involves an extra step if you are given the slant height instead of the vertical height. In these cases, you must use the Pythagorean theorem to find the vertical height before solving:
height = √(slant_height² − radius²)
To skip the manual geometry, the cone volume calculator resolves the height and volume directly.
Run the numbers
Explore volume and dimension calculators to check your work:
Frequently asked questions
Volume is the measure of the three-dimensional space enclosed by a closed boundary or occupied by an object. It is measured in cubic units (e.g., cubic inches, cubic centimeters, cubic yards).
Volume is the amount of space an object occupies. Capacity is the ability of a container to hold fluid or other materials. While mathematically identical, capacity is typically expressed in units like gallons, liters, or cups.
Because a yard has 3 feet, a cubic yard has 3 × 3 × 3 = 27 cubic feet. Divide the volume in cubic feet by 27 to find the equivalent volume in cubic yards.
Cones and pyramids have exactly one-third the volume of cylinders and prisms with the same base area and height. This geometric constant is proven using calculus.
The volume calculator supports 3D shapes including cubes, cylinders, and spheres, providing calculations and unit conversions. The cone volume calculator offers specialized calculations for conical shapes.