Volume formulas, and turning volume into litres
The volume and surface area formulas for a box, cylinder, cone, sphere, hemisphere and pyramid, how to turn cubic units into litres and gallons, and worked examples for tanks, pools and concrete.
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In short
- Volume is the space inside a solid, in cubic units. A tank 100 cm by 50 cm by 60 cm holds 300,000 cm³, which is exactly 300 litres.
- One litre is 1,000 cm³, one cubic metre is 1,000 litres, one cubic foot is 28.3168 litres and one US gallon is 3.7854 litres.
- A cone is a third of the cylinder that fits around it, and a sphere is two thirds of its cylinder: radius 10 gives 4,188.79 for the sphere and 2,094.40 for the hemisphere.
- Use the radius, not the diameter, in a cylinder, cone or sphere formula; using the diameter makes the volume four times too large for a cylinder and eight times for a sphere.
- For a cone or pyramid, use the vertical height, not the slanted side.
Volume is how much space a solid takes up, and for a container it is how much the container can hold. It is the number you need to size a water tank, fill a pool, order concrete or compare two packaging boxes. The formulas are short, but three things routinely go wrong: using a diameter where the formula wants a radius, using a slanted side where it wants a vertical height, and converting cubic units to litres with the wrong factor. This page covers all three.
The formulas
| Solid | Volume | Surface area |
|---|---|---|
| Box (cuboid) | V = length × width × height | S = 2 × (l×w + l×h + w×h) |
| Cube | V = side³ | S = 6 × side² |
| Cylinder | V = π × r² × h | S = 2 × π × r × (r + h) |
| Cone | V = ⅓ × π × r² × h | S = π × r × (r + slant), with slant = √(r² + h²) |
| Sphere | V = 4⁄3 × π × r³ | S = 4 × π × r² |
| Hemisphere | V = ⅔ × π × r³ | S = 3 × π × r² |
| Square pyramid | V = ⅓ × base² × h | S = base² + 2 × base × slant, with slant = √((base ÷ 2)² + h²) |
The relationships among them are the easiest way to remember them. A cone is exactly one third of the cylinder with the same base and height. A pyramid is exactly one third of the box with the same base and height. A sphere has two thirds of the volume of the cylinder that just contains it, which is 4⁄3 × π × r³ against 2 × π × r³.
Worked examples
A box or rectangular tank. A tank 100 cm long, 50 cm wide and 60 cm high has a volume of 100 × 50 × 60 = 300,000 cm³. As 1,000 cm³ is one litre, that is 300 litres. Its surface area, the amount of material to make it, is 2 × (5,000 + 6,000 + 3,000) = 28,000 cm².
A cylinder. A drum with a radius of 15 cm and a height of 40 cm has a volume of π × 15² × 40 = 3.14159 × 225 × 40 = 28,274.33 cm³, or 28.27 litres, which is 7.47 US gallons. Its surface area is 2 × π × 15 × (15 + 40) = 5,183.63 cm².
A cone. A funnel with a base radius of 6 cm and a vertical height of 12 cm holds ⅓ × π × 36 × 12 = 452.39 cm³. Its slanted side is √(36 + 144) = 13.416 cm, which is longer than the height, and the surface area is π × 6 × (6 + 13.416) = 365.99 cm².
A sphere. A ball of radius 10 cm has a volume of 4⁄3 × π × 1,000 = 4,188.79 cm³, and a surface area of 4 × π × 100 = 1,256.64 cm². A hemisphere, such as a dome or a bowl, has half the volume, 2,094.40 cm³.
A square pyramid. A pyramid with a 10 cm square base and a vertical height of 12 cm has a volume of ⅓ × 100 × 12 = 400 cm³. The slanted face height is √(25 + 144) = 13 cm, so the surface area is 100 + 2 × 10 × 13 = 360 cm².
Radius, diameter and the four-times mistake
Cylinders, cones and spheres are described by a radius in the formulas, but they are measured with a tape across the widest point, which gives the diameter. The radius is half of it. Forgetting to halve makes the error larger than it feels, because the radius is squared in a cylinder or cone and cubed in a sphere.
If a drum is 30 cm across and 40 cm tall, the radius is 15, and the volume is 28,274.33 cm³. Putting 30 into the formula as the radius gives π × 900 × 40 = 113,097 cm³, exactly four times too much. For a sphere the same slip gives eight times too much, because the radius is cubed.
The second error is the height of a cone or pyramid. The formula wants the vertical height, measured straight from the centre of the base up to the tip. The slanted side, measured along the surface, is always longer. Using it overstates the volume. The calculator asks only for the vertical height and works out the slant itself where it is needed for the surface area.
Volume to litres, gallons and weight
The factors to learn are these:
| Quantity | In litres |
|---|---|
| 1 cubic centimetre (cm³ or ml) | 0.001 |
| 1,000 cm³ | 1 |
| 1 cubic metre (m³) | 1,000 |
| 1 cubic foot | 28.3168 |
| 1 cubic inch | 0.016387 |
| 1 US gallon | 3.78541 |
| 1 imperial (UK) gallon | 4.54609 |
So a volume in cubic centimetres divided by 1,000 is litres, and a volume in cubic metres multiplied by 1,000 is litres. A cubic foot, a unit still used for room and freight volumes, is 28.3168 litres.
For water the conversion goes one step further, because a litre of water weighs about a kilogram. A full 300 litre tank holds roughly 300 kg of water, which matters when you are deciding what a shelf or a roof can carry. Other liquids and materials are heavier or lighter than water, so the weight conversion applies to water only.
Practical uses
A water tank. A household needs a tank of 1,000 litres, which is exactly 1 m³. If the tank is a cylinder with a radius of 0.5 m, its height must be 1 ÷ (π × 0.25) = 1.273 m. If it is a box 1 m by 1 m, it is 1 m high.
A swimming pool. A rectangular pool 8 m long, 4 m wide and filled to an average depth of 1.5 m holds 8 × 4 × 1.5 = 48 m³, which is 48,000 litres. Pools with a sloping floor need an average depth, found by adding the shallow and deep depths and halving.
Concrete. A slab 10 ft by 10 ft and 4 inches thick has a volume of 10 × 10 × (4 ÷ 12) = 33.33 cubic feet. In cubic metres that is 0.944 m³, and in cubic yards, the unit used in many countries to order it, 1.235, because a cubic yard is 27 cubic feet. Order about 10 percent more, since the ground is never perfectly flat and some is lost in mixing: 36.7 cubic feet here.
Shipping and storage. Cartons are measured by volume to price a shipment and to see how many fit in a container; the CBM calculator handles that, including volumetric weight.
Cooking and containers. A cylinder of 8 cm radius and 12 cm height is 2,412.7 cm³, about 2.4 litres. Knowing the volume of a pot is the easiest way to scale a recipe.
Measuring a shape that is not a standard solid
Split the object into solids that are standard and add the volumes. A silo is a cylinder with a cone on top. A grain hopper is a box with a pyramid below it. A bowl is a hemisphere, or a hemisphere with a cylinder collar. For something hollow, such as a pipe, subtract the inner cylinder from the outer one: a pipe with an outer radius of 5 cm, an inner radius of 4 cm and a length of 100 cm holds π × 16 × 100 = 5,026.5 cm³ of fluid, and has π × (25 − 16) × 100 = 2,827.4 cm³ of material.
For truly irregular objects, the practical answer is displacement: lower the object into a container of water and measure how much the level rises. The water displaced is the volume of the object.
When a calculation matters, such as a tank that must hold a particular amount, measure inside dimensions rather than outside, because the wall thickness is not part of the capacity.
What this calculator assumes
- All lengths are in the one unit you choose, and the volume comes out in that unit cubed.
- Shapes are exact: a cylinder has a flat circular base and top, and a cone comes to a single point.
- The pyramid has a square base, and its tip is directly above the centre of the base.
- Litres and gallons are for liquids that fill the shape completely. A partly filled tank holds less.
- The hemisphere is a solid half-sphere, and its surface area includes the flat circular face.