A regular pentagonal pyramid has a side length of 4 mm, an apothem that measures 3 mm, and a height of 4.1 mm. What is the volume of the pyramid?

A regular pentagonal pyramid has a side length of 4 mm, an apothem that measures 3 mm, and a height of 4.1 mm. What is the volume of the pyramid?

To find the volume of a regular pentagonal pyramid, you can use the formula:

Volume = (1/3) * Base Area * Height

The base area of a regular pentagonal pyramid can be calculated using the formula:

Base Area = (5/4) * s^2 * cot(π/5)

Where:
- s is the side length of the base
- cot(π/5) represents the cotangent of the angle π/5 (36 degrees)

Let's plug in the values and calculate step by step:

1. Calculate the Base Area:
Base Area = (5/4) * s^2 * cot(π/5)
= (5/4) * 4^2 * cot(π/5)
= (5/4) * 16 * cot(π/5)
= 20 * cot(π/5)

2. Calculate the Volume:
Volume = (1/3) * Base Area * Height
= (1/3) * (20 * cot(π/5)) * 4.1

Now, let's calculate this using a calculator.

Note: Make sure your calculator is in degree mode when calculating the cotangent.

By substituting the values into the formula and performing the calculations, the volume of the regular pentagonal pyramid is obtained.