Volume Calculator

Background.

The Quanta volume calculator handles the five most-asked 3-D solids in plane and solid geometry — the cube, the rectangular prism (a box), the sphere, the cylinder, and the cone — and returns both the volume enclosed by each shape and the surface area of its outer skin in a single keystroke. Volume is the measure of three-dimensional extent: the number of unit cubes that would fit inside a solid if you could pack it perfectly with no gaps. It is the answer to every "how much will fit?" or "how much will it weigh?" question in everyday life, engineering, manufacturing, construction, science, and trade. How many cubic metres of concrete do you need to pour a foundation? How many litres of water does that swimming pool, fish tank, or rainwater tank hold? How many cubic feet of soil go into the raised garden bed? How much helium does a spherical balloon contain? What is the displacement of an engine cylinder in cubic centimetres? How many cubic yards of gravel do you order for the driveway? Every one of those questions is a volume calculation, and every one of them depends on which 3-D shape best models the container you are measuring.

That is why each solid has its own formula: the cube is V = s³, the rectangular prism is V = l × w × h, the sphere is V = ⁴⁄₃πr³, the cylinder is V = πr²h, and the cone is V = ⅓πr²h. They are not interchangeable — the curvature of a sphere, the round cross-section of a cylinder, and the tapering point of a cone all change how much space the same linear dimension encloses, and the constants in each formula (the ⁴⁄₃, the π, the ⅓) reflect exactly that geometry. The reason volume is so much more sensitive to size than area is the cubic scaling: doubling a length doubles a one-dimensional measurement, quadruples a two-dimensional one, and multiplies a three-dimensional volume by eight. A storage box twice as wide, twice as long, and twice as tall holds eight times as much, not twice as much — a fact that catches almost everyone out the first time they meet it.

The formulas this page implements have a history as old as recorded mathematics. The Egyptian Moscow Mathematical Papyrus (~1850 BCE) records the correct volume of a truncated pyramid. Euclid devoted Book XI of the Elements (~300 BCE) to the foundations of solid geometry. Archimedes, in On the Sphere and Cylinder (~225 BCE), proved that the volume of a sphere is exactly two-thirds of the volume of the smallest cylinder that contains it — a result he asked to be engraved on his tombstone, and one Cicero later found preserved there in Syracuse. The factor of ⅓ in the cone formula was also proved by Euclid (Elements, Book XII, Proposition 10): every cone is exactly one-third of the cylinder that shares its base and height.

Use this calculator for homework, for engineering estimates, for renovation and landscaping projects, for shipping and packaging calculations, for cooking and baking conversions, or for any of the hundreds of everyday situations where you need to turn a few tape-measure readings into a useful cubic answer. Pick a shape from the dropdown, enter the dimensions it asks for, and the volume and surface area appear instantly. Everything below the widget is the full explainer: what volume really means, the six classical formulas and where each one comes from, a worked example with real numbers, and answers to the ten questions people most often ask when they reach for a volume calculator.

What is volume calculator?

Volume is the measure of how much three-dimensional space a solid occupies — the count of unit cubes that would fit inside it. The unit cube is the reference object: a cube one unit on every side encloses exactly one cubic unit of space (one cm³, one m³, one in³, one ft³, depending on which linear unit you choose). Every other solid's volume can be expressed as a multiple of that unit cube. Because the cube has three perpendicular dimensions and each contributes a factor of the side length, volume always scales as the cube of any linear dimension — a sphere with twice the radius has eight times the volume, a tank with all sides doubled holds eight times the water, a balloon at twice the diameter floats with eight times the buoyant force. The five shapes this calculator supports are the classical solids of elementary solid geometry: the cube (V = s³, SA = 6s²), the most symmetric box, where one side length determines everything. The rectangular prism (V = l × w × h, SA = 2(lw + lh + wh)), the general box with three potentially different side lengths — the model for rooms, shipping containers, bricks, books, and most man-made enclosures. The sphere (V = ⁴⁄₃πr³, SA = 4πr²), the perfectly round solid where every point on the surface is equidistant from the centre — the model for balls, bubbles, planets, water droplets at small scale, and any container shaped for minimum surface area at a given volume (this is why soap bubbles and stars are spherical: surface tension and gravity both minimise surface area per unit volume). The cylinder (V = πr²h, SA = 2πr(r + h)), a circular base extruded straight up — the model for cans, pipes, tanks, drinking glasses, log sections, and engine pistons. The cone (V = ⅓πr²h, SA = πr(r + √(r²+h²))), a circular base tapering to a single apex — the model for ice-cream cones, traffic cones, funnels, party hats, and the tapered end of a rocket nose. A sixth shape worth knowing — the pyramid — follows the same one-third rule as the cone: V = ⅓ × base_area × height for any pyramid whose base is a polygon. This calculator does not currently include the pyramid as a selectable shape, but the volume of a rectangular pyramid is simply ⅓ × l × w × h, and the volume of a triangular pyramid is ⅓ × ½ × base × triangle_height × pyramid_height. The Quanta surface-area calculator handles those polygonal pyramids if you need them. All five shapes share a deeper property: the volume formula always reduces to base area × height for prisms (cube and rectangular prism are prisms; cylinder is a circular prism) and to ⅓ × base area × height for tapering solids (cone and pyramid). The sphere is the odd one out — it is the only shape on the list with no flat face and no axis of extrusion, which is why its formula has the ⁴⁄₃ in it.

How to use this calculator.

1. 01Pick a shape from the dropdown. The five options are Cube, Rectangular Prism, Sphere, Cylinder, and Cone. The calculator will only use the dimension fields relevant to your chosen shape — entering a radius when you have selected Cube has no effect, and the field for Length is ignored when you have selected Sphere.
2. 02For a Cube, enter the side length. A cube has all twelve edges equal, so this single number gives you V = s³ and SA = 6s². Example: a 3 m × 3 m × 3 m storage container has a volume of 27 m³ and a surface area of 54 m².
3. 03For a Rectangular Prism, enter length, width, and height — in the same unit. The volume is the product of all three (V = lwh) and the surface area is 2(lw + lh + wh), which is the sum of the areas of the six rectangular faces. This is the formula for any box-shaped object: a room, a refrigerator, a shipping carton, a brick.
4. 04For a Sphere, enter only the radius (the distance from the centre to any point on the surface). Volume is ⁴⁄₃πr³ and surface area is 4πr². If you only know the diameter, divide it by two first. A ball of radius 10 cm has a volume of about 4189 cm³ (just over 4 litres) and a surface area of about 1257 cm².
5. 05For a Cylinder, enter the radius of the circular base and the height (the distance between the two flat ends). Volume is πr²h — the area of the circular base times the height — and surface area is 2πr(r + h), which combines the two circular ends (2πr²) with the curved side (2πrh).
6. 06For a Cone, enter the radius of the circular base and the perpendicular height from the base to the apex. Volume is ⅓πr²h — exactly one-third of the cylinder with the same base and height (Euclid, Elements XII.10) — and surface area is πr(r + √(r²+h²)), which combines the circular base with the slanted lateral surface.
7. 07Read the primary output, Volume, in the dark results panel. It is reported as "cubic units" — the cube of whatever unit you typed. Enter metres and the volume is in m³; enter centimetres and the volume is in cm³ (and 1 litre = 1000 cm³, so divide by 1000 if you want litres); enter feet and the volume is in ft³ (and 1 US gallon = 231 in³ = 0.1337 ft³, so multiply ft³ by 7.481 to get gallons).
8. 08Read the secondary output, Surface Area, in square units. Use it for paint coverage, heat-transfer estimates, fabric or sheet-metal cutting, or any calculation that depends on the outside skin rather than the enclosed space.

The formula.

Five shapes, two outputs each — and every formula traces back to one of two principles: prisms multiply base area by height, and tapering solids take one-third of that product. Cube. V = s³ and SA = 6s². A cube is a rectangular prism where length, width, and height are all the same number s, so the volume is s × s × s = s³ and the surface area is six identical square faces, each of area s², for a total of 6s². Rectangular Prism. V = l × w × h and SA = 2(lw + lh + wh). The volume is the area of the rectangular base (l × w) times the height h. The surface area is the sum of the areas of the six faces, paired into three opposing pairs: two faces of area lw (top and bottom), two of area lh (front and back), and two of area wh (left and right), giving 2(lw + lh + wh). Sphere. V = ⁴⁄₃πr³ and SA = 4πr². Archimedes proved both formulas in On the Sphere and Cylinder (~225 BCE) by the method of exhaustion — comparing the sphere to inscribed and circumscribed cylinders and cones. The modern proof uses calculus: integrate the area of horizontal circular cross-sections from -r to r, with each cross-section at height y having radius √(r² - y²) and area π(r² - y²), to get V = ∫_{-r}^{r} π(r² - y²) dy = π[r²y - y³/3]_{-r}^{r} = ⁴⁄₃πr³. The surface area is the derivative of the volume with respect to the radius: d/dr (⁴⁄₃πr³) = 4πr². Archimedes's most famous result on this shape was that the sphere's volume is exactly two-thirds of the smallest cylinder that contains it (a cylinder of radius r and height 2r, volume 2πr³); ⁴⁄₃πr³ ÷ 2πr³ = ⅔. Cylinder. V = πr²h and SA = 2πr(r + h). The volume is the area of the circular base (πr²) times the height h — exactly the prism formula applied to a circular base. The surface area is the two circular ends (2 × πr² = 2πr²) plus the lateral curved surface, which if you cut it vertically and unroll it is a rectangle of width 2πr (the circumference) and height h, area 2πrh — total SA = 2πr² + 2πrh = 2πr(r + h). Cone. V = ⅓πr²h and SA = πr(r + ℓ), where ℓ = √(r² + h²) is the slant height. The volume is one-third the volume of the cylinder with the same base and height — Euclid proved this in Elements Book XII, Proposition 10, by exhaustion, and it is the prototype for the more general fact that every pyramidal (tapering-to-a-point) solid has one-third the volume of the corresponding prism. The surface area combines the circular base (πr²) with the lateral surface, which when unrolled is a circular sector of radius ℓ and arc length 2πr, area πrℓ. The ℓ = √(r² + h²) follows from the Pythagorean theorem on the right triangle formed by the radius, the height, and the slant. Every formula in this calculator rejects negative inputs (a physical dimension cannot be negative) and returns exact zeros when any required dimension is zero (a degenerate solid with no extent). All arithmetic uses IEEE-754 double precision, which gives roughly 15–17 significant digits — well beyond any practical measurement precision.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Euclid, Elements, Book XI — the foundational treatise on solid geometry, defining solids, prisms, pyramids, cylinders, cones, and spheres and their properties (Heath translation, Cambridge University Press, 1908). Book XII, Proposition 10 contains the proof that every cone is exactly one-third of the cylinder with the same base and height.
2. [2]Archimedes, On the Sphere and Cylinder (~225 BCE) — the original proof that the volume of a sphere is 4πr³/3 and its surface area is 4πr², together with the celebrated theorem that the sphere occupies exactly two-thirds of the smallest cylinder that contains it. Translation by T. L. Heath, The Works of Archimedes (Cambridge University Press, 1897).
3. [3]NIST Digital Library of Mathematical Functions, §1.2 — formulas and constants for elementary solids including the cube, rectangular prism, sphere, cylinder, and cone.
4. [4]Zwillinger, Daniel (ed.). CRC Standard Mathematical Tables and Formulae, 33rd edition. CRC Press, 2018 — comprehensive reference of volume and surface-area formulas for all standard 3-D solids, including the five shapes in this calculator plus pyramids, frustums, ellipsoids, and tori.
5. [5]Coxeter, H. S. M. Introduction to Geometry, 2nd edition. Wiley, 1969 — classic graduate-level reference for Euclidean solid geometry including derivations of cube, prism, cylinder, cone, and sphere volume formulas.
6. [6]Weisstein, Eric W. "Sphere," "Cylinder," "Cone," "Cube," "Rectangular Parallelepiped." From MathWorld — A Wolfram Web Resource. Canonical formulas, derivations, and properties for each shape in this calculator.