Formulas for the volume of figures

The volume of a figure is a quantitative characteristic of the space occupied by a body or substance. The volume of a body or the capacity of a vessel is determined by its shape and linear dimensions.

volume of a cube
volume of a prism
volume of a parallelepiped
volume of a rectangular parallelepiped
volume of a pyramid
volume of a cylinder
volume of a cone
volume of a sphere

Volume of a cube

A cube is a regular polyhedron, each face of which is a square.

The volume of a cube equals the cube of the length of its edge.

V = a^{3}

where V is the volume of the cube,
a is the length of the edge of the cube.

Volume of a prism

A prism is a polyhedron, two faces of which are congruent (equal) polygons lying in parallel planes, and the remaining faces are parallelograms having common sides with these polygons.

The volume of a prism equals the product of the area of the base of the prism and the height.

If the base is:

a triangle, find the area of the triangle,
a square, then the area of the square,
an arbitrary figure, find the area of that arbitrary figure.

V = S_{о} \cdot h

where V is the volume of the prism,
S_о is the area of the base of the prism,
h is the height of the prism — the distance between its bases. For a right prism, in which all edges are perpendicular to the bases, this is any of the edges.

Volume of a parallelepiped

A parallelepiped is a polyhedron with six faces, each of which is a parallelogram.

The volume of a parallelepiped equals the product of the area of the base of the parallelepiped and the height.

V = S_{о} \cdot h

where V is the volume of the parallelepiped,
S_о is the area of the base of the parallelepiped,
h is the height of the parallelepiped — the distance between its bases.

Volume of a rectangular parallelepiped

A rectangular parallelepiped is a parallelepiped in which all faces are rectangles.

The volume of a rectangular parallelepiped equals the product of its length, width, and height.

V = a \cdot b \cdot h

where V is the volume of the rectangular parallelepiped,
a is the length,
b is the width,
h is the height.

Volume of a pyramid

A pyramid is a polyhedron whose base is an arbitrary polygon, and all faces are triangles with a common vertex, which is the apex of the pyramid.

Volume of a pyramid with an arbitrary base

The volume of a pyramid equals one third of the product of the area of the base and the height.

V = \frac{1}{3} \cdot S \cdot h

where V is the volume of the pyramid,
S is the area of the base of the pyramid,
h is the height of the pyramid.

Volume of a truncated pyramid

A truncated pyramid is the part of a pyramid between its base and this section. A section parallel to the base of the pyramid divides the pyramid into two parts.

The volume of a truncated pyramid equals one third of the product of the height h and the sum of the areas of the top base S1, the bottom base of the truncated pyramid S2, and the geometric mean between them.

V = \frac{1}{3} \cdot h \cdot (S1 + \sqrt{S1 \cdot S2} + S2)

where V is the volume of the truncated pyramid,
S1 is the area of the top base of the truncated pyramid,
S2 is the area of the bottom base of the truncated pyramid,
h is the height of the truncated pyramid.

Volume of a regular pyramid

A regular pyramid is a pyramid whose base is a regular polygon and whose height passes through the center of the circle inscribed in the base.

V = \frac{n \cdot a^{2} \cdot h}{12 \cdot tg \frac{180°}{n}}

where V is the volume of the pyramid,
a is the side of the base of the pyramid,
n is the number of sides of the polygon at the base,
h is the height of the regular pyramid.

Volume of a regular triangular pyramid

A regular triangular pyramid is a pyramid whose base is an equilateral triangle and whose faces are equal isosceles triangles.

V = \frac{h \cdot a^{2}}{4 \cdot \sqrt{3}}

where V is the volume of the regular triangular pyramid,
a is the side of the base of the pyramid,
h is the height of the regular triangular pyramid.

Volume of a regular quadrangular pyramid

A regular quadrangular pyramid is a pyramid whose base is a square and whose faces are equal isosceles triangles.

The volume of a regular quadrangular pyramid equals one third of the product of the height of the pyramid and the cube of the side of the base of the pyramid.

V = \frac{h \cdot a^{2}}{3}

where V is the volume of the regular quadrangular pyramid,
a is the side of the base of the pyramid,
h is the height of the regular quadrangular pyramid.

Volume of a tetrahedron

A tetrahedron is a pyramid in which all faces are equilateral triangles.

The volume of a tetrahedron equals one twelfth of the product of the cube of the length of a face of the tetrahedron and the square root of two.

V = \frac{\sqrt{2}}{12} \cdot a^{3}

where V is the volume of the tetrahedron,
a is any of the faces of the tetrahedron.

Volume of a cylinder

A cylinder is a geometric body bounded by a cylindrical surface and two parallel planes intersecting it.

The volume of a cylinder equals the product of the area of its base and the height.

V = π \cdot R^{2} \cdot h = S \cdot h

where V is the volume of the cylinder,
S is the area of the base of the cylinder,
R is the radius of the cylinder,
π is pi ≈ 3.14159265.

Volume of a cone

A cone is a body obtained by combining all rays emanating from a single point (the apex of the cone) and passing through a flat surface.

The volume of a cone equals one third of the product of the area of its base and the height.

V = \frac{1}{3} \cdot π \cdot R^{2} \cdot h = \frac{1}{3} \cdot S \cdot h

where V is the volume of the cone,
S is the area of the base of the cone,
R is the radius of the base of the cone,
π is pi ≈ 3.14159265.

Volume of a sphere

A sphere is the set of all points in space that are located at a distance from the center no greater than a given value.

The volume of a sphere equals four thirds of the product of pi and the cube of the radius of the sphere.

V = \frac{4}{3} \cdot π \cdot R^{3}

where V is the volume of the sphere,
R is the radius of the sphere,
π is pi ≈ 3.14159265.

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