How to find the area of a decagon?
Understand the Problem
The question is asking for a method or formula to calculate the area of a decagon, which is a ten-sided polygon. To solve it, we would typically use a specific formula involving the length of the sides or the apothem.
Answer
The area of a regular decagon is given by $$ A = \frac{5s^2}{4 \tan\left(\frac{\pi}{10}\right)} $$ or $$ A = \frac{5}{2} s a $$.
Answer for screen readers
The area of a regular decagon can be calculated using the formula
$$ A = \frac{5}{2} s a $$
or
$$ A = \frac{5s^2}{4 \tan(\frac{\pi}{10})} $$
depending on which variables are provided.
Steps to Solve
- Identify the key variables
To calculate the area of a regular decagon, we need to know the length of one side, denoted as ( s ), or the apothem, denoted as ( a ).
- Use the area formula
For a regular decagon, the area can be calculated using the following formula:
$$ A = \frac{5}{2} s a $$
where ( A ) is the area, ( s ) is the length of one side, and ( a ) is the apothem.
- Simplifying the formula using side length
If only the side length ( s ) is known, the area can also be found using:
$$ A = \frac{5}{2} s \cdot \frac{s}{2 \tan(\frac{\pi}{10})} $$
This simplifies to:
$$ A = \frac{5s^2}{4 \tan(\frac{\pi}{10})} $$
This formula uses the tangent function to find the apothem in terms of the side length.
- Calculating the values
Plug the values of ( s ) or ( a ) into the chosen formula to compute the area ( A ).
The area of a regular decagon can be calculated using the formula
$$ A = \frac{5}{2} s a $$
or
$$ A = \frac{5s^2}{4 \tan(\frac{\pi}{10})} $$
depending on which variables are provided.
More Information
A decagon has ten equal sides and angles, and its area depends on either the side length or the apothem. The formulas can help find the area efficiently, and knowing the value of ( \tan(\frac{\pi}{10}) ) is useful for precise calculations.
Tips
- Mistaking the formula: Confusing the area formula for a decagon with that of other polygons.
- Incorrectly substituting values: Make sure to use consistent units when plugging in the side length or the apothem.
