Volume Calculator

What is Volume?

Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains. Volume is often quantified numerically using the SI derived unit, the cubic meter. Each volume calculator provided below includes the formula, step by step calculation and, solved example problem to understand the calculation concept.

The formula for calculating the volume depends on the shape of the object. Here are the general formulas for some common shapes:

• Cube: V = s³
• Cylinder: V = πr²h
• Sphere: V = 4/3πr³
• Cone: V = 1/3πr²h
• Rectangular Prism: V = lwh
• Triangular Prism: V = 0.5 * b * h * l
• Pyramid: V = 1/3Bh
• Ellipsoid: V = 4/3πabc
• Torus: V = (πr²)(2πR)
• Capsule: V = πr²h + 4/3πr³
• Spherical Cap: V = 1/6πh(3a²+h²)
• Conical Frustum: V = 1/3πh(r1²+r2²+r1*r2)
• Square Pyramid: V = 1/3b²h
• Tube (Hollow cylinder): V = πh(R²-r²)

Cube

A cube is a three-dimensional shape with six equal square faces. The volume of a cube is calculated with the formula: V = s³

Formula: The formula to calculate the volume of a cube is: V = s³

Example: If a cube has sides of 4 units, its volume would be V = 4³ = 64 cubic units.

Cylinder

A cylinder is a three-dimensional solid object with two parallel circular bases and a curved surface connecting the bases.

Formula: The formula to calculate the volume of a cylinder is: V = πr²h

Example: If the radius (r) is 2 cm and height (h) is 5 cm, then V = π*2²*5 = 62.83 cm³.

Sphere

A sphere is a perfectly symmetrical three-dimensional solid object, where every point on its surface is equidistant from its center.

Formula: The formula to calculate the volume of a sphere is: V = 4/3*πr³

Example: If the radius (r) is 3 cm, then V = 4/3*π*3³ = 113.1 cm³.

Cone

A cone is a three-dimensional solid object with a circular base and a single vertex, connected by a curved surface.

Formula: The formula to calculate the volume of a cone is: V = 1/3*πr²h

Example: If the radius (r) is 3 cm and height (h) is 5 cm, then V = 1/3*π*3²*5 = 47.12 cm³.

Rectangular Prism

A rectangular prism, also known as a cuboid, is a three-dimensional solid object with six rectangular faces.

Formula: The formula to calculate the volume of a rectangular prism is: V = lwh

Example: If the length (l) is 2 cm, width (w) is 3 cm and height (h) is 4 cm, then V = 2*3*4 = 24 cm³.

Triangular Prism

A triangular prism is a three-dimensional solid object with two identical triangular bases and three rectangular faces.

Formula: The formula to calculate the volume of a triangular prism is: V = 1/2*bhl

Example: If the base (b) is 3 cm, height (h) of the triangle is 4 cm, and length (l) of the prism is 5 cm, then V = 1/2*3*4*5 = 30 cm³.

Pyramid

A pyramid is a three-dimensional solid object with a polygonal base and triangular faces that meet at a single point (vertex).

Formula: The formula to calculate the volume of a pyramid is: V = 1/3*Bh

Example: If the area of the base (B) is 9 cm² and the height (h) is 6 cm, then V = 1/3*9*6 = 18 cm³.

Ellipsoid

An ellipsoid is a three-dimensional solid object, symmetrical about three perpendicular planes, akin to a sphere stretched along its axes.

Formula: The formula to calculate the volume of an ellipsoid is: V = 4/3*πabc

Example: If the axes (a, b, c) are 2 cm, 3 cm, and 4 cm respectively, then V = 4/3*π*2*3*4 = 100.53 cm³.

Torus

A torus is a three-dimensional solid object shaped like a doughnut, symmetrical about an axis, with a circular cross-section.

Formula: The formula to calculate the volume of a torus is: V = 2*π²*Rr²

Example: If the major radius (R) is 5 cm and the minor radius (r) is 2 cm, then V = 2*π²*5*2² = 251.33 cm³.

Capsule

A capsule is a three-dimensional solid object comprising of a cylinder with hemispherical ends, like a typical pill shape.

Formula: The formula to calculate the volume of a capsule is: V = πr²h + 4/3*πr³

Example: If the radius (r) is 2 cm and the cylinder height (h) is 4 cm, then V = π*2²*4 + 4/3*π*2³ = 50.27 cm³.

Spherical Cap

A spherical cap is the portion of a sphere cut off by a plane, resulting in a curved surface and a circular base.

Formula: The formula to calculate the volume of a spherical cap is: V = 1/6*πh(3a² + h²)

Example: If the height (h) is 3 cm and the radius of the base (a) is 2 cm, then V = 1/6*π*3*(3*2² + 3²) = 37.7 cm³.

Conical Frustum

A conical frustum is a portion of a cone that remains after its top is cut off with a plane parallel to its base.

Formula: The formula to calculate the volume of a conical frustum is: V = 1/3*πh(r₁² + r₂² + r₁r₂)

Example: If the height (h) is 4 cm, the radius of the smaller base (r₁) is 2 cm, and the radius of the larger base (r₂) is 3 cm, then V = 1/3*π*4*(2² + 3² + 2*3) = 60.32 cm³.

Square Pyramid

A square pyramid is a three-dimensional solid object with a square base and four triangular faces meeting at a single vertex.

Formula: The formula to calculate the volume of a square pyramid is: V = 1/3*Bh

Example: If the area of the base (B) is 9 cm² and the height (h) is 4 cm, then V = 1/3*9*4 = 12 cm³.

Tube

A tube is a three-dimensional solid object that is hollow, with circular cross-sections, akin to a cylinder but empty in the middle.

Formula: The formula to calculate the volume of a tube is: V = πh(R² - r²)

Example: If the height (h) is 5 cm, the outer radius (R) is 3 cm, and the inner radius (r) is 2 cm, then V = π*5*(3² - 2²) = 47.12 cm³.