# What is the approximate area of the regular pentagon?

#### Understand the Problem

The question is asking for the approximate area of a regular pentagon. To solve this, we usually need to know either the length of the sides or the apothem (the distance from the center to the midpoint of a side) to calculate the area using the formula: Area = (5/2) * side_length * apothem or Area = (1/4) * √(5(5 + 2√5)) * side_length².

The approximate area of the regular pentagon is $43.0119$.

The area of a regular pentagon with a side length of 5 is approximately:

$$\text{Area} \approx 43.0119$$

#### Steps to Solve

1. Identify the given information

Determine the length of the side of the regular pentagon. If it is not provided, we will assume a length for demonstration.

1. Choose the formula

We can use either of the two area formulas provided. If we have the side length ($s$) of the pentagon, we can use:

$$\text{Area} = \frac{1}{4} \sqrt{5(5 + 2\sqrt{5})} s^2$$

1. Plug in the side length

Suppose our side length $s$ is 5 (as an example). Substitute this value into the area formula:

$$\text{Area} = \frac{1}{4} \sqrt{5(5 + 2\sqrt{5})} (5)^2$$

1. Calculate the inside part

First, calculate the expression inside the square root:

$$5(5 + 2\sqrt{5})$$

1. Take the square root

Calculate the square root of the result from the previous step.

1. Multiply and simplify

After obtaining the square root, multiply it by $\frac{1}{4}$ and then multiply the result by $25$ (since $5^2 = 25$) to find the area.

The area of a regular pentagon with a side length of 5 is approximately:

$$\text{Area} \approx 43.0119$$

The formula used to calculate the area of a regular pentagon derives from its geometric properties. Regular pentagons are often seen in architecture and art due to their symmetry.

#### Tips

• Forgetting to square the side length in the formula.
• Using the incorrect formula; make sure to use the one that matches the information you have (either side length or apothem).
• Miscalculating the square root.
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