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# Cube Formula

The cube formula helps us to find the surface area, diagonals, and volume of a cube. The cube of a number directly reflects the volume of a cube having an edge length equal to the given number. A cube is a 3-D solid object with six square faces whose all the sides are of the same length. Let us learn about the cube formula with a few solved examples in the end.

## What is the Cube Formula?

The cube is one of the five platonic solids and is also known as a regular hexahedron.

### Cube Formula

**Volume of Cube**

The volume of the cube can be calculated using different formulas based on the given parameters. It can be calculated using the side length as well as the measure of the cube's diagonal.

- The Volume of a Cube (based on side length) = a
^{3}cubic inches, where a is the length of the side of a cube - The volume of a Cube (based on diagonal) = (√3×d
^{3})/9 cubic inches, where d is the length of the diagonal of a cube

**Lateral Area of Cube**

The lateral area of a cube is the sum of areas of all side faces of the cube. There are 4 side faces so the sum of areas of all 4 side faces of a cube is its lateral area.

LSA of a Cube = 4a^{2}

where a is the side length.

**Total Area of Cube**

The total surface area of the cube will be the sum of the area of the base and the area of vertical surfaces of the cube. Since all the faces of the cube are made up of squares of the same dimensions then the total surface area of the cube will be the surface area of one face added five times to itself. Therefore, the formula to find the surface area of a cube is:

Total Surface Area (TSA) of a Cube = 6a^{2}

where a is the side length.

**Diagonal of Cube**

A cube has diagonals of two different lengths, the shorter ones lie on the square faces, and the longer ones pass through the center. The main diagonal of a cube is referred to as the one that cuts through the center which can be found by multiplying the length of one side by the square root of 3.

Diagonal of a cube = a√3

Let us understand the cube formulas better using a few solved examples.

## Examples Using Cube Formula

**Example 1:** Find the volume of a Rubik's Cube of length 4 in.

**Solution:**

To find the volume of a Rubik's cube:

The length of the side of the cube= 4 in. (given)

Using the cube formula,

volume = s × s × s = s^{3}

Put the values,

volume = 4 × 4 × 4 = 4^{3} = 64

**Answer: The volume of a Rubik's cube is 64 cubic inches.**

**Example 2: **The dimensions of a cube are 64 in. Find its diagonal using the cube formula.

**Solution: **

To find the diagonal of the cube:

The dimensions of the cube are: Length(l) = Width(w) = Height(h) = 64 in (given)

Using the cube formula,

diagonal = a√3

Put the values,

Diagonal = 64√3 = 110.848 inches

**Answer: The diagonal of the cube is **110.848 inches

**Example 3: **Find the total surface area of the cube if the length of the side of the cube is 25 in.

**Solution:**

Length of the side of the cube, a = 25 in

Using the formula for the area of the cube, which is: A = 6a^{2}

Put the values,

A = 6 × 25 × 25 = 3750 square inches

**Answer: The surface area of the cube is 3750 square inches.**

## FAQs on Cube Formula

### What Is the Cube Formula?

The cube formula helps us to find the surface area, diagonals, and volume of a cube. These are simple formulas, dependent mostly on one parameter that is the edge length or side of the cube.

### How to Calculate the Diagonal of a Cube Using Cube Formula?

The main diagonal of a cube that cuts through the center can be found by multiplying the length of one side by the square root of 3. Thus, the diagonal of a cube = a√3, where a is the edge of the cube.

### What is s in the Cube Formula?

In the cube formula, s refers to the edge of the cube. All the cube formulas - volume, surface area, and diagonals, are dependent on edge of the cube, either represented as s or a.

### How to Derive Cube Formula?

To derive the volume of a cube formula,

- Step 1: Consider any square-shaped sheet of paper.
- Step 2: Now, the area covered by this square sheet will be its surface area i.e. its length multiplied by its breadth. Both are the same in the case of a cube. Thus, the surface area will be “s
^{2}“. - Step 3: A cube is made by stacking multiple square sheets so that the height becomes equal to the length and breadth, i.e., “s” units. Thus, the height or thickness of the cube is “s”.

It can thus be concluded that the overall space covered by the cube, which is the volume, will be the area of the base multiplied by the height. Volume of a cube = s^{2 }× s = s^{3}

To derive the surface of a cube formula,

- Step 1: Consider any square-shaped sheet of paper.
- Step 2: In the case of a square, since the length and breadth are equal, the surface area will be “s
^{2}“.(its length multiplied by its breadth.) - Step 3: Since a cube has 6 faces thus, the total surface area of the cube equals 6 times the area of one face = 6s
^{2}

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