Geometry Volume Calculator
Find the volume of a cube, cuboid, sphere, cylinder, cone, hemisphere, triangular prism or pyramid, with exact and decimal answers and steps.
Result
- Shape
- Sphere
- Dimensions
- Radius r = 5 cm
- Exact
- 500π/3 cm³
- Approximate
- ≈ 523.6 cm³
- Capacity
- ≈ 0.52 L (523.6 mL)
Volume of a sphere
V = (4/3)πr³
V = (4/3) × π × (5)³
V = 500π/3 cm³ ≈ 523.6 cm³
Step-by-step calculation
Step 1: Given
Radius: r = 5 cm
Step 2: Formula
Archimedes showed that a sphere fills exactly 2/3 of the cylinder that just encloses it: 2/3 × πr² × 2r = (4/3)πr³.
V = (4/3)πr³
Step 3: Substitute
V = (4/3) × π × (5)³
Step 4: Simplify
(5)³ = 125
V = (4 × 125 / 3) × π
V = 500π/3
Step 5: Result
Exact: V = 500π/3 cm³
Approximate: V ≈ 523.6 cm³
Volume formulas for 3D shapes
| Shape | Volume formula | Where |
|---|---|---|
| Cube | V = a³ | a = side |
| Cuboid / rectangular prism | V = lwh | l, w, h = length, width, height |
| Sphere | V = (4/3)πr³ | r = radius |
| Cylinder | V = πr²h | r = radius, h = height |
| Cone | V = (1/3)πr²h | h = perpendicular height |
| Hemisphere | V = (2/3)πr³ | half a sphere |
| Triangular prism | V = ½bhL | b, h = triangle base and height, L = length |
| Square pyramid | V = (1/3)a²h | a = base side, h = perpendicular height |
| Rectangular pyramid | V = (1/3)lwh | l × w base, h = perpendicular height |
Every prism and cylinder is base area × height; every pyramid and cone is one third of that. A sphere is (4/3)πr³ and a hemisphere half of it.
π is used at full precision and nothing is rounded until the final display; 1 L = 1,000 cm³ and 1 m³ = 1,000 L. For the area of the outside of a shape, use the Surface Area Calculator; for circles and triangles, the Circle Calculator and Triangle Calculator. Convert results with the Volume Converter, or size real projects with the Concrete Calculator and Pool Volume Calculator.
Volume is the amount of three-dimensional space a solid takes up. This geometry volume calculator finds the volume of a cube, cuboid (rectangular prism), sphere, cylinder, cone, hemisphere, triangular prism, square pyramid or rectangular pyramid from the dimensions you enter.
Pick a shape, and the diagram shows exactly which lengths to measure. Enter them in mm, cm, m, km, in, ft or yd, and the result appears in the matching cubic unit, such as cm³ or ft³, with its capacity in liters. When the answer involves π or a fraction, you get the exact value (for example 500π/3 cm³) as well as the decimal, followed by the formula, the substitution and every step of the calculation.
Worked Calculation Examples
| Scenario | Result | Calculation Step |
|---|---|---|
| Cube with side 5 cm | 125 cm³ | V = a³ = 5³ = 5 × 5 × 5 = 125 cm³, which is 0.125 liters. |
| Cylinder: radius 3 cm, height 8 cm | 72π ≈ 226.19 cm³ | V = πr²h = π × 3² × 8 = π × 9 × 8 = 72π ≈ 226.19 cm³ (about 0.23 liters). |
| Sphere with radius 6 m | 288π ≈ 904.78 m³ | V = (4/3)πr³ = (4/3) × π × 6³ = (4/3) × π × 216 = (4 × 216 / 3)π = 288π ≈ 904.78 m³. |
| Cone: radius 4 cm, height 9 cm | 48π ≈ 150.80 cm³ | V = (1/3)πr²h = (1/3) × π × 4² × 9 = (1/3) × π × 144 = 48π ≈ 150.80 cm³. A cylinder with the same base and height would hold 144π, three times as much. |
| Square pyramid: base 6 m, height 5 m | 60 m³ | V = (1/3)a²h = (1/3) × 6² × 5 = (1/3) × 36 × 5 = 180/3 = 60 m³. Use the perpendicular height, not the slant height of a face. |
| Triangular prism: base 6 in, triangle height 4 in, length 10 in | 120 in³ | Triangle area = ½ × 6 × 4 = 12 in². V = 12 × 10 = 120 in³. |
| Hemisphere with radius 3 cm | 18π ≈ 56.55 cm³ | V = (2/3)πr³ = (2/3) × π × 27 = 18π ≈ 56.55 cm³, exactly half of the 36π cm³ sphere with the same radius. |
What Is Volume?
Volume measures how much space a 3D object occupies: how many unit cubes it would take to fill it. A box 4 cm long, 3 cm wide and 2 cm tall holds 4 × 3 = 12 one-centimeter cubes in each layer, and there are 2 layers, so its volume is 24 cm³.
A length has one dimension and is measured in linear units (cm). An area has two, length × width, so it is measured in square units (cm²). Volume has three, length × width × height, so it is measured in cubic units: mm³, cm³, m³, in³, ft³ or yd³. A cubic meter is a cube 1 m on each side; a cubic centimeter, about the size of a sugar cube, is a cube 1 cm on each side.
Volume vs Surface Area
Volume measures the space inside a solid, in cubic units such as cm³, m³ or ft³. Surface area measures the total area covering its outside, in square units such as cm², m² or ft². Volume tells you how much a box holds; surface area tells you how much cardboard it takes to make it.
For a cube with 3 cm edges, the volume is 3³ = 27 cm³ and the surface area is 6 × 3² = 54 cm². Double the edges to 6 cm and the volume becomes 216 cm³ (8 times as much) while the surface area becomes 216 cm² (only 4 times as much): volume grows with the cube of the size, surface area with the square.
Volume vs Capacity
Volume is the amount of 3D space an object occupies; capacity is how much a container can hold, usually measured in liters or milliliters. For a thin-walled container they are practically the same number, just in different units. A thick-walled container has a capacity smaller than its outside volume.
The conversions are exact: 1 milliliter = 1 cm³, 1 liter = 1,000 cm³, and 1 cubic meter = 1,000 liters. So a tank 2 m × 1 m × 1.5 m has a volume of 3 m³ and holds 3,000 liters. In US units, 1 US gallon = 231 in³ ≈ 3.785 liters, and 1 ft³ ≈ 7.48 US gallons.
Real-World Uses of Volume
Construction: concrete, gravel, soil and fill are ordered by volume, so a slab or footing is measured as a rectangular prism and a column as a cylinder. Storage and shipping: carriers price by the volume (and dimensional weight) of boxes, and warehouse and container space is planned in cubic meters or cubic feet.
Tanks and pools: the capacity of a cylindrical water tank or rectangular pool is its volume in liters or gallons, which sets pump sizes and treatment doses. Packaging and manufacturing: designers compare shapes that hold the same volume, and the amount of material in a molded or cast part is its volume times density. Engineering and architecture: engines are rated by cylinder displacement, rooms by air volume for heating and ventilation, and roofs and domes by the space they enclose.
How to Use the Geometry Volume Calculator
- Choose the shape: cube, cuboid (rectangular prism), sphere, cylinder, cone, hemisphere, triangular prism, square pyramid or rectangular pyramid. The diagram shows which dimensions to measure.
- Enter the dimensions shown, all in the same unit. For cones and pyramids, use the perpendicular height from the apex straight down to the base, not the slant height.
- Pick the unit your measurements are in (mm, cm, m, km, in, ft or yd). The volume is given in the matching cubic unit, along with its capacity in liters.
- Read the volume: when the answer involves π or a fraction such as 1/3, the exact value (for example 500π/3 cm³) is shown next to the decimal approximation.
- Follow the step-by-step calculation to see the formula, the substituted values and each simplification, and choose how many decimal places to display.
Frequently Asked Questions
What is a volume calculator?
A volume calculator finds how much 3D space a shape occupies from its dimensions. This one handles cubes, cuboids, spheres, cylinders, cones, hemispheres, triangular prisms and square and rectangular pyramids, and shows the formula and steps for each.
How do you calculate volume?
Identify the shape, measure the dimensions its formula needs in one unit, and substitute them. For any prism or cylinder, multiply the base area by the height; for a cone or pyramid, take one third of that.
What is the formula for volume?
It depends on the shape: V = a³ for a cube, lwh for a cuboid, πr²h for a cylinder, (1/3)πr²h for a cone, (4/3)πr³ for a sphere, and (1/3) × base area × height for a pyramid.
What is the volume of a cube?
V = a³, the side length multiplied by itself three times. A cube with 5 cm sides has a volume of 5 × 5 × 5 = 125 cm³.
What is the volume of a cylinder?
V = πr²h: the area of the circular base times the height. For r = 3 cm and h = 8 cm, V = π × 9 × 8 = 72π ≈ 226.19 cm³.
What is the volume of a sphere?
V = (4/3)πr³. For r = 6 m, V = (4/3) × π × 216 = 288π ≈ 904.78 m³. If you know the diameter, halve it to get r first.
What is the volume of a cone?
V = (1/3)πr²h, one third of a cylinder with the same base and height. Use the perpendicular height, not the slant height. For r = 4 cm and h = 9 cm, V = 48π ≈ 150.80 cm³.
What is the difference between volume and surface area?
Volume is the space inside a solid, in cubic units such as cm³. Surface area is the area of its outside, in square units such as cm². A 3 cm cube has a volume of 27 cm³ and a surface area of 54 cm².
Why is volume measured in cubic units?
Volume multiplies three lengths (length × width × height), so the unit is multiplied three times: cm × cm × cm = cm³. One cubic unit is a cube with edges one unit long.
How do you convert cubic centimeters to liters?
Divide by 1,000, because 1 liter = 1,000 cm³ (and 1 cm³ = 1 mL). For example, 2,500 cm³ = 2.5 L. For cubic meters, multiply by 1,000: 1 m³ = 1,000 L.
Can this calculator calculate volume for different 3D shapes?
Yes: cube, cuboid (rectangular prism), sphere, cylinder, cone, hemisphere, triangular prism, square pyramid and rectangular pyramid, in mm, cm, m, km, in, ft or yd, with results in cubic units and liters.
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Last updated: September 27, 2026.