Geometry

Volume

Volume formulas for 3D solids: cubes, prisms, cylinders, pyramids, cones, spheres, and ellipsoids.

Volume measures the 3D extent of a solid in cubic units (cubic metres, cubic inches, and so on). Prismatic solids such as cubes, rectangular prisms, and cylinders follow the pattern base area × height. Pyramids and cones give one-third of the corresponding prism. Spheres and ellipsoids have their own forms.

Volume measures 3D space in cubic units.

Cube

Edge length a. Volume is the side cubed; doubling the edge multiplies volume by eight:

Volume=a^{3}

Rectangular Prism

Edges a, b, c (length, width, height in any order). Volume is the product of all three:

Volume=a\times b\times c

Irregular Prism

B is the base polygon (any shape) and h the perpendicular height between the two parallel bases. Volume is the base area times that height, generalising the rectangular-prism rule:

Volume=\left(Area\text{ }of\text{ }B\right)\times h

Cylinder

Base radius r and height h. The circular base has area πr², so volume is the disc area times the height: the prism rule applied to a circular base:

Volume=\pi r^{2}h

Pyramid

b is the base area (for a square pyramid, b = side²) and h the perpendicular height from base to apex. Volume is one-third the volume of the prism with the same base and height:

Volume=\left(\frac{1}{3}\right)bh

whare b = area of square

Cone

Base radius r and perpendicular height h from base to apex. The circular case of the pyramid rule: one-third the base area (πr²) times the height:

Volume=\left(\frac{1}{3}\right)\pi r^{2}h

Sphere

Radius r. Volume scales with the cube of r. The constant 4π/3 is the same factor that appears when integrating a hemisphere of revolution:

Volume=\left(\frac{4}{3}\right)\pi r^{3}

Ellipsoid

Three semi-axes r1, r2, r3 (half the lengths along each principal direction). Volume is 4π/3 times their product. When all three are equal this reduces to the sphere formula:

Volume=\left(\frac{4}{3}\right)\pi\left(r1\times r2\times r3\right)