Volume Calculator for Common 3D Shapes

Volume formulas for six common three-dimensional shapes

Volume measures the amount of three-dimensional space enclosed within a solid object. It is expressed in cubic units: cubic centimetres, cubic metres, cubic feet, cubic inches, litres (which equal 1000 cubic centimetres), gallons, and so on. Volume calculations come up in science, engineering, cooking, construction, shipping logistics, and everyday tasks like estimating how much water fills a tank or how much concrete fills a form.

This calculator covers six common 3D shapes: cube, rectangular prism, cylinder, sphere, cone, and pyramid with a rectangular base. To use it, select the shape you need, enter the required measurements in the fields shown, and click Calculate volume. The result will display the volume in cubic units along with the exact formula used, so you can follow the logic step by step.

A cube has six equal square faces. Its volume is the side length cubed, written as s cubed. A rectangular prism (also called a cuboid or box) has volume equal to length times width times height. This is the most commonly needed volume formula in everyday life, covering everything from boxes and rooms to swimming pools and storage containers.

A cylinder has a circular cross-section and a straight height. Its volume is pi times the radius squared times the height. This formula comes from the fact that a cylinder is essentially a stack of circles (discs), and the area of each disc is pi times the radius squared. A sphere is a perfectly round 3D shape. Its volume is four thirds times pi times the radius cubed. The sphere formula is slightly less intuitive but is derived from integral calculus and is one of the fundamental results of classical geometry.

A cone has a circular base and tapers to a point. Its volume is one third of the volume of the cylinder with the same base and height: one third times pi times the radius squared times the height. A pyramid with a rectangular base has volume equal to one third of the base area times the height. For this calculator, you enter the base length and base width to compute the base area, then multiply by height and divide by three.

Why the one-third factor appears in cones and pyramids

Both cones and pyramids share the one-third factor in their volume formulas, and this is not a coincidence. It can be shown geometrically that exactly three congruent pyramids (or cones) can be packed together to fill the volume of one prism (or cylinder) with the same base and height. This result was known to ancient Greek mathematicians and is rigorously provable using Cavalieri's principle or integral calculus.

The practical implication is that a conical container holds exactly one third as much as a cylindrical container of the same base and height. Similarly, a square pyramid holds exactly one third as much as a box with the same base and height. This relationship is useful when designing containers, funnel systems, grain silos, and other structures where tapered shapes are involved.

Real-world uses for volume calculations

Volume calculations appear across many fields. In plumbing and engineering, calculating the volume of a cylindrical pipe or tank tells you how much fluid it holds. In construction, calculating the volume of a rectangular concrete pour tells you how many cubic metres of mix to order. In cooking and baking, converting between container sizes often requires knowing their volumes. In chemistry and pharmacy, solutions are measured by volume and prepared in cylindrical or conical vessels.

In shipping and packaging, the volume of a box determines dimensional weight, which affects freight costs. In agriculture, knowing the volume of a grain silo or water tank is essential for capacity planning. In home improvement, knowing the volume of a room allows you to calculate heating and cooling loads accurately.

The inputs required by this calculator are the standard geometric dimensions taught in school geometry. If you have measurements from a real object, make sure all dimensions are in the same unit before entering them. Mixing metres and centimetres, for example, will produce an incorrect result. Convert everything to a single unit first, then enter the numbers and read the result in the corresponding cubic unit.