# Pyramidal Frustum Calculator

## Pyramidal frustum definition

A **pyramidal frustum** is a three-dimensional geometric shape that resembles a **truncated pyramid**. It is created by cutting off the top portion of a pyramid, leaving the bottom base and sides intact.

## Parts of a pyramidal frustum

There are eight parts of a pyramidal frustum mentioned in this tool: number of sides, bottom base length (a), top base length (b), height (h), top area (A1), bottom area (A2), area (A) and volume (V).

### Number of sides (n)

A pyramidal frustum has the same number of sides as the original pyramid. For example, if the original pyramid is a square pyramid, the frustum will also have four sides. If the original pyramid is a triangular pyramid, the frustum will have three sides.

### Bottom base length (a)

The **bottom base length**, or "a", refers to the length of the bottom base of the pyramidal frustum. This is the length of one of the sides of the bottom base.

We can calculate the bottom base length with the following formula:

`a = √(A2 * 4 * tan(180/n) / n)`

### Top base length (b)

The **top base length**, or "b", refers to the length of the top base of the pyramidal frustum. This is the length of one of the sides of the top base.

We can find the top base length by using a similar formula, only replacing the bottom area (A2) with the top area (A1):

`b = √(A1 * 4 * tan(180/n) / n)`

### Height (h)

The **height**, or "h", refers to the distance between the top base and bottom base of the pyramidal frustum. It can be calculated with the following formula:

`h = 3 * V / (A1 + A2 + √(A1*A2))`

### Top area (A1)

The **top area**, or "A1", refers to the surface area of the top base of the pyramidal frustum. The formula for calculating the top area of a pyramidal frustum is:

`A1 = b² * n / (4 * tan(180/n))`

### Bottom area (A2)

The **bottom area**, or "A2", refers to the surface area of the bottom base of the pyramidal frustum. The formula for calculating the bottom area of a pyramidal frustum is:

`A2 = a² * n / (4 * tan(180/n))`

### Area (A)

The **area**, or "A", refers to the total surface area of the pyramidal frustum.

To calculate the total area we first calculate the lateral area (A3) of the pyramidal frustum:

`A3 = n * (a + b) / 2 * √((h² + (a / (2 * sin(180/n)) - b / (2 * sin(180/n)))²) - ((a - b) / 2)²)`

then, we add all three areas together:

`A = A1 + A2 + A3`

### Volume (V)

The **volume**, or "V", refers to the amount of space inside the pyramidal frustum. The formula for calculating the volume of a pyramidal frustum is:

`V = h / 3 * (A1 + A2 + √(A1*A2))`