How to find the volume of a hexagonal pyramid?

Understand the Problem

The question is asking how to calculate the volume of a hexagonal pyramid. To find this volume, we can use the formula for the volume of a pyramid, which is V = (1/3) * base area * height. The base area for a hexagon can be calculated using the formula Area = (3√3/2) * side^2, where 'side' is the length of one of the sides of the hexagon.

Answer

V = \frac{\sqrt{3}}{2} \times a^2 \times h

Steps to Solve

Calculate the Base Area of the Hexagon

The first step is to calculate the area of the hexagonal base. The formula for the area of a regular hexagon is:

$$\text{Area} = \frac{3 \sqrt{3}}{2} \times \text{side}^2$$

Substitute the Side Length

Substitute the given length of one side of the hexagon into the formula. Suppose the side length is given as 'a'. Then the area of the hexagon becomes:

$$\text{Area} = \frac{3 \sqrt{3}}{2} \times a^2$$

Use the Volume Formula for a Pyramid

Next, we use the volume formula for a pyramid:

$$V = \frac{1}{3} \times \text{Base Area} \times \text{Height}$$

Substitute the Values

Substitute the calculated base area and the given height of the pyramid into the volume formula. Suppose the height is given as 'h'. Then the volume becomes:

$$V = \frac{1}{3} \times \left(\frac{3 \sqrt{3}}{2} \times a^2\right) \times h$$

Simplify the Expression

Finally, simplify the expression to find the volume:

$$V = \frac{3 \sqrt{3}}{6} \times a^2 \times h$$

Simplifying further, we get:

$$V = \frac{\sqrt{3}}{2} \times a^2 \times h$$

The final answer is V = \frac{\sqrt{3}}{2} \times a^2 \times h

More Information

This formula allows you to calculate the volume of any regular hexagonal pyramid given its side length and height.

Tips

A common mistake is to forget to apply the formula for the area of a hexagon correctly. Make sure to use the side length squared when calculating the base area.

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