Rectangular Prism Volume Calculator
Volume, surface area and diagonal from three measurements.
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- Volume in litres—
- Volume in cubic feet—
- Surface area—
- Lateral surface area—
- Space diagonal—
- Total edge length—
How to use this calculator
- Measure the three edges that meet at any corner: length, width and height. On a rectangular prism it does not matter which you call which, because multiplying them together gives the same answer in any order.
- Enter all three in the same unit and pick that unit from the dropdown.
- Read the volume, plus the surface area, the diagonal and the total edge length, which are the other figures people usually need next.
If you are working out capacity rather than bulk, measure the inside of the box. The difference matters as soon as the walls are thick.
What is a rectangular prism?
A rectangular prism is a solid with six rectangular faces meeting at right angles. It is also called a cuboid, or simply a box, and it is the most common three dimensional shape in everyday life: rooms, shipping cartons, bricks, aquariums, concrete pours, phone screens in profile.
It has six faces, twelve edges and eight vertices. Opposite faces are identical, and every angle in the solid is ninety degrees. When all twelve edges are the same length, it is a cube, which is a rectangular prism rather than a different shape.
The rectangular prism volume formula
The logic behind it is worth seeing once. The base is a rectangle of area l × w. Stack that base up to a height of h and you have filled the solid, so the volume is the base area multiplied by the height. That principle covers every prism, not just this one; only the shape of the base changes.
Because multiplication does not care about order, labelling an edge length rather than width changes nothing. This is the only common volume formula where you genuinely cannot mix up the inputs.
The other formulas on this page
| Quantity | Formula | What it tells you |
|---|---|---|
| Volume | V = lwh | How much it holds |
| Surface area | A = 2(lw + lh + wh) | Material to cover all six faces |
| Lateral surface area | A = 2h(l + w) | The four sides, excluding top and base |
| Space diagonal | d = √(l² + w² + h²) | Longest item that fits inside |
| Total edge length | E = 4(l + w + h) | Framing or trim needed |
The diagonal is the sleeper useful one. It is the Pythagorean theorem applied twice, and it answers the practical question of whether a long object will fit in a box at an angle when it will not fit along any edge.
A worked example
Take a shipping carton measuring 30 cm by 20 cm by 15 cm.
- Volume. 30 × 20 × 15 = 9,000 cm³, which is 9 litres.
- Surface area. 2 × (30×20 + 30×15 + 20×15) = 2 × (600 + 450 + 300) = 2,700 cm² of cardboard.
- Diagonal. √(900 + 400 + 225) = √1,525 = 39.05 cm, so a 38 cm ruler fits corner to corner, though a 40 cm one will not.
Those three numbers answer three different practical questions about the same box: what it holds, what it costs to make, and what will fit inside it.
Volume or surface area?
People reach for the wrong one surprisingly often, and the units are the giveaway. Volume is measured in cubic units and describes the space inside. Surface area is measured in square units and describes the skin around it.
- Filling it — water, soil, concrete, grain — is volume.
- Covering it — paint, wrapping, cardboard, tiles — is surface area.
- Shipping it is usually volume, since couriers charge on volumetric weight.
- Heating or cooling it involves both: volume sets the air to move, surface area sets the heat lost.
A detail worth knowing: double every edge of a box and the surface area becomes four times larger while the volume becomes eight times larger. Big containers are more efficient per litre than small ones, which is why bulk packaging uses less material per unit than single servings.
Scaling a prism up or down
| Multiply every edge by | Surface area changes by | Volume changes by |
|---|---|---|
| 2 | 4× | 8× |
| 3 | 9× | 27× |
| 10 | 100× | 1,000× |
| 0.5 | 0.25× | 0.125× |
Area scales with the square of the change and volume with the cube. This is the square-cube law, and it explains a great deal beyond boxes, from why large animals need proportionally thicker legs to why bigger server racks are harder to cool.
Where this comes up
- Shipping. Couriers price parcels on volumetric weight, which is the box volume divided by a fixed divisor, so the carton size can cost more than the contents weigh.
- Aquariums. Tank volume in litres sets the stocking level, the heater wattage and the filter rating.
- Concrete and soil. Both are ordered by the cubic metre, and a trench or slab is a rectangular prism.
- Room air. Air conditioning and ventilation are sized from room volume, not floor area.
- 3D printing. The build volume of a printer is a rectangular prism, and the diagonal tells you the longest part you can print at an angle.
Frequently asked questions
What is the formula for the volume of a rectangular prism?
Volume equals length times width times height, written V = lwh. Measure all three edges in the same unit and the answer comes out in cubic units of that same unit. For a box 30 by 20 by 15 cm, the volume is 9,000 cubic centimetres.
Is a rectangular prism the same as a cuboid?
Yes. Cuboid is the more common term in British schools and rectangular prism in American ones, but they describe the same solid: six rectangular faces meeting at right angles. A cube is the special case where all twelve edges are equal.
Does it matter which side I call the length?
No. Multiplication gives the same result in any order, so labelling the edges differently cannot change the volume. This makes the rectangular prism unusually forgiving compared with formulas that distinguish radius from height.
How do I convert the volume to litres?
If you measured in centimetres, divide the cubic centimetres by 1,000. If you measured in metres, multiply the cubic metres by 1,000. A box of 9,000 cubic centimetres holds 9 litres.
What is the difference between volume and surface area?
Volume is the space inside, in cubic units, and tells you what the box holds. Surface area is the total of the six faces, in square units, and tells you how much material covers it. Use volume for filling and surface area for wrapping or painting.
How do I find the diagonal of a rectangular prism?
Take the square root of the sum of the three squared edges: d equals the square root of l squared plus w squared plus h squared. For a 30 by 20 by 15 box that is the square root of 1,525, or about 39.05 units. This is the longest straight object that fits inside.
What happens to the volume if I double every side?
The volume becomes eight times larger, while the surface area only becomes four times larger. Volume scales with the cube of the change and area with the square, which is why larger containers use less material per litre held.
How do I calculate the volume of an irregular box-like shape?
Split it into rectangular sections, calculate each separately and add the results. An L-shaped room is two rectangular prisms. For a shape with a hollow section, work out the outer volume and subtract the inner one.