Volume Calculator

Calculate the volume of a cube, rectangular prism, cylinder, sphere, cone, square pyramid, or triangular prism — just pick a shape and the formula does the rest. Also computes surface area for cylinders, spheres, and cones.

Volume formulas by shape

Shape Formula
Cube Volume = side × side × side
Rectangular prism Volume = length × width × height
Cylinder Volume = π × radius2 × height, Surface area = 2π×radius2 + 2π×radius×height
Sphere Volume = (4/3) × π × radius3, Surface area = 4π×radius2
Cone Volume = (1/3) × π × radius2 × height, Surface area = π×radius2 + π×radius×slant height
Square pyramid Volume = (1/3) × side2 × height
Triangular prism Volume = base area (triangle) × prism length

The volume's unit is the cube of whatever length unit you entered (e.g. enter values in cm and the result is in cm³). Surface area is in the square of that unit.

Tips

  • While the Area Calculator handles 2D shapes, this tool is its 3D counterpart. Prisms and pyramids both use the shared idea of "base area × height" (or × 1/3), so understanding the area formulas makes the volume formulas click faster.
  • When estimating how much lumber or concrete you need for a DIY project, calculating the volume of a rectangular prism or cylinder first helps you avoid ordering too much or too little material.
  • To find the capacity of a fish tank or water tank, pick a cylinder or rectangular prism, enter the interior dimensions in centimeters, and divide the resulting volume (cm³) by 1,000 to convert to liters.
  • The slant height a cone needs for its surface area (the distance from the apex to a point on the base circle) is computed automatically from the radius and height, so there's no need to work out the Pythagorean theorem by hand.
  • A pyramid or cone's volume is always exactly one third of a prism or cylinder with the same base and height. Keeping that 1/3 ratio in mind alongside the formulas makes them much easier to remember for homework.

FAQ

A cube is a special rectangular prism where all edges are equal in length. A rectangular prism can have a different length, width, and height, while a cube only needs one side length entered to compute its volume.

The slant height is the straight-line distance from the cone's apex to a point on the edge of its base. Given the base radius and height, it can be computed automatically using the Pythagorean theorem (slant2 = radius2 + height2) — this tool uses exactly that formula internally to get the surface area.

Ancient Greek mathematicians such as Eudoxus and Democritus demonstrated this relationship using an early technique called the method of exhaustion. Intuitively, a triangular prism can be split into three congruent tetrahedra (pyramids), and you can also confirm it experimentally: filling a prism-shaped container by pouring in water from a pyramid-shaped mold of the same base and height takes exactly three pourings.

This tool doesn't assume a specific length unit, so the volume's unit is simply the cube of whatever unit you entered. For example, if you enter lengths in centimeters, read the volume as cm³ (cubic centimeters). Surface area comes out in the square of that unit (e.g. cm²).

This relationship comes from building up a sphere's volume as a stack of infinitely thin spherical shells. In fact, differentiating the sphere's volume formula with respect to the radius r gives exactly the surface area formula, 4πr² — because the extra volume gained from a tiny increase in radius is approximately the sphere's surface area at that moment times the thickness of the increase.
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Side Note — the cylinder-and-sphere relationship Archimedes had carved on his tombstone

The ancient Greek mathematician Archimedes discovered that a sphere and the cylinder that exactly circumscribes it (a cylinder with the same height and diameter as the sphere) always have a volume ratio of 2:3. The cylinder's volume is πr²×2r = 2πr³, and the sphere's volume is (4/3)πr³, giving exactly 2πr³ : (4/3)πr³ = 3 : 2. Archimedes was said to be so proud of this discovery that he requested a diagram of a cylinder and an inscribed sphere be carved on his tombstone.

The fact that a pyramid or cone's volume equals one third of a prism or cylinder with the same base and height is credited to the mathematician Eudoxus, working around the 4th century BCE, and was later recorded in Euclid's "Elements". That geometers arrived at this kind of near-limit reasoning (the method of exhaustion) more than two millennia before calculus was formalized is one of the notable milestones in the history of geometry.

Calculating the volume of everyday solids still matters a great deal today, from sizing water tanks and estimating 3D-printer material usage to computing box capacity for packaging design. The formulas themselves have barely changed since antiquity, but the range of their applications keeps expanding.

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