## Volume of Cube Formula

V = S^{3}

- V is the volume enclosed by the cube
- S is the side length, it is also commonly represented by an A in other problem sets

This simple cube volume formula applies only to true cubes where all sides are an equal length. If all sides are not an equal length, but still parallel, please use the Length x Width x Height formula below.

## Volume of a Rectangular Prism

free Rectangular Prism Volume Calculator

V = L * W * H

- V is the volume enclosed by the rectangular prism
- L is the side length
- W is the side width
- H is the side height

For the purposes of the calculation it will not matter which sides are considered to be the Length, Width, or Height. However, all Length lines must run parallel to each other, as must all Width and Height lines to their respective partners.

## Surface Area of a Cube

free Cube Surface Area Calculator

A = 6S^{2}

- A is the surface area of the rectangular prism
- S is the side length

A cube is comprised of six equal squares. To find the total surface area first find the surface Area of a square by multiplying Side length by Side length, this is the same as saying Side^{2}. Then multiplying the area of one square (a face) by the total number of faces (six), we now know the total surface area of the cube.

## Face Diagonal Length

free Cube Face Diagonal Length Calculator

L = √2S

- L is the length of a diagonal line across a cube's face (through a square)
- S is the side length

This formula calculates the length of a diagonal line running across a single cube face from a vertex to another vertex that forms the opposite corner, this is the longest line that can be draw across a cube's face. Since the face of a cube is a square is formula also applies to diagonals across the face of a square and is essentially a two dimensional problem.

## Volume Diagonal Length

free Cube Volume Diagonal Length Calculator

L = √3S

- L is the maximum length of a diagonal through a cube
- S is the side length

This formula calculates the maximum length of a diagonal line running through cube volume from a vertex to another vertex that forms the opposite corner, this is the longest line that can be draw through a cube's volume.

## Inscribed Sphere Radius

free Inscribed Sphere Radius Calculator

R = S/2

- R is the maximum radius of a sphere inscribed in a cube
- S is the side length of the cube

A sphere that is inscribed in a cube is one that mathematically touches the face of the cube without penetrating any part of the cube. In mathematics the term tangent is used to refer to two entities that touch in one place without crossing. Therefore, the largest sphere that can be inscribed in a cube is tangential to the faces of the cube (it doesn't stick through the cube anywhere, it is totally inside the cube).

## Sphere Radius Tangent to Cube Edges

free Radius of a Sphere Tangent to Cube Edge Calculator

R = S/√2

- R is the radius of a sphere tangent to the edges of a cube
- S is the side length of the cube

A sphere that touches the edges of a cube only once on every side of the cube is said to be tangent to the edges of the cube. This sphere will poke through the faces of the cube but not be large enough to enclose the entire cube as the cube corners will poke out of the sphere.

## Circumscribed Sphere Radius

free Circumscribed Sphere Radius Calculator

R = S * √3/2

- R is the radius of a sphere that circumscribes a cube
- S is the side length of the cube

A sphere that completely contains a cube and touches each of its vertices is said to circumscribed the cube.

## Inscribed Sphere Volume

free Inscribed Sphere Radius Calculator

V = (4/3)π(S/2)^{3}

- V is the volume of a sphere inscribed in a cube
- S is the side length of the cube
- π (Pi) is a mathematical constant defined by the ratio of the circumference of a circle to the diameter of a circle

An inscribed sphere is completely contained by a cube touching only its faces. Previously we found the formula for the radius of an inscribed sphere, recall it was R = S/2. The volume of a sphere can be found using the formula V = (4/3)πR^{3}. To complete the formula substitute the radius R with our method for finding the radius of an inscribed sphere.

## Sphere Volume Tangent to Cube Edges

free Volume of a Sphere Tangent to Cube Edge Calculator

V = (4/3)π(S/√2)^{3}

- R is the radius of a sphere tangent to the edges of a cube
- S is the side length of the cube
- π (Pi) is a mathematical constant defined by the ratio of the circumference of a circle to the diameter of a circle

This formula substitutes the formula for finding the radius of a sphere tangent to the edge of a cube for the radius term in the formula for finding the volume of a sphere.

## Circumscribed Sphere Volume

free Circumscribed Sphere Volume Calculator

V = (4/3)π(S * √3/2)^{3}

- R is the radius of a sphere that circumscribes a cube
- S is the side length of the cube
- π (Pi) is a mathematical constant defined by the ratio of the circumference of a circle to the diameter of a circle

Substitute the formula for finding the radius of a sphere that circumscribes a cube for the radius term in the formula for the volume of a sphere.

## Cube Root

Y = ∛X

- X is the value that we are determining the cube root of
- Y is the cube root of X

This is the reverse of cubing a number. A number's cube root is another number that can be multiplied together three times to make the original number. For example 3 is the cube root of 27, let try it: 3 x 3 x 3, multiplying the first two numbers (3 x 3) results in 9, multiply by 3 again (9 x 3) equals 27, we have multiplied the number by itself three time.

## Cubing

Y = X^{3}

- X is the value to be cubed
- Y is the result of cubing X

To cube a number multiply is by itself three times. A shorthand notation for multiplying a number by itself is to use an exponent, also referred to as raising a number "to the power of...". In the case of cubing we are always raising a number to the power of three, written as X^{3}. Understanding the concept of cubing is key to understanding how to calculate the volume of a cube.