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.
Answer
The approximate area of the regular pentagon is $43.0119$.
Answer for screen readers
The area of a regular pentagon with a side length of 5 is approximately:
$$ \text{Area} \approx 43.0119 $$
Steps to Solve
- 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.
- 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:
$$ \text{Area} = \frac{1}{4} \sqrt{5(5 + 2\sqrt{5})} s^2 $$
- Plug in the side length
Suppose our side length $s$ is 5 (as an example). Substitute this value into the area formula:
$$ \text{Area} = \frac{1}{4} \sqrt{5(5 + 2\sqrt{5})} (5)^2 $$
- Calculate the inside part
First, calculate the expression inside the square root:
$$ 5(5 + 2\sqrt{5}) $$
- Take the square root
Calculate the square root of the result from the previous step.
- 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:
$$ \text{Area} \approx 43.0119 $$
More Information
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.