How to find the area of a nonagon?
Understand the Problem
The question is asking how to calculate the area of a nonagon, which is a nine-sided polygon. To solve this, we need to understand the formula for the area of a nonagon, which generally involves knowing the length of its sides and using specific geometric formulas.
Answer
The area of the nonagon is approximately \( 56.68 \) square units.
Answer for screen readers
The area of the nonagon is approximately ( A \approx 56.68 ) square units.
Steps to Solve
- Identify the formula for the area of a nonagon
The area ( A ) of a regular nonagon (where all sides and angles are equal) can be calculated using the formula:
$$ A = \frac{9}{4} s^2 \cot\left(\frac{\pi}{9}\right) $$
where ( s ) is the length of a side.
- Substitute the side length into the formula
If we know the length of a side ( s ), we can substitute that value into the formula. For example, if ( s = 5 ):
$$ A = \frac{9}{4} \times 5^2 \cot\left(\frac{\pi}{9}\right) $$
- Calculate the numeric value
Now we will calculate ( A ) by first finding ( 5^2 ) and then multiplying by ( \frac{9}{4} ) and evaluating ( \cot\left(\frac{\pi}{9}\right) ):
$$ 5^2 = 25 $$
Then calculate:
$$ A = \frac{9}{4} \times 25 \times \cot\left(\frac{\pi}{9}\right) $$
- Use a calculator for the cotangent value and final multiplication
Using a scientific calculator, find ( \cot\left(\frac{\pi}{9}\right) ) which is approximately ( 3.07768 ).
So we can compute:
$$ A \approx \frac{9}{4} \times 25 \times 3.07768 $$
- Final calculation
Now we perform the final multiplication to find the area:
$$ A \approx 56.6826 $$
The area of the nonagon is approximately ( A \approx 56.68 ) square units.
More Information
The area formula for a regular nonagon can also be derived from the area of triangles formed from the center of the nonagon to its vertices. The cotangent in the formula helps connect the side length to the angles formed by the triangles.
Tips
- Not using the correct formula for a regular nonagon; remember it only applies if all sides and angles are equal.
- Forgetting to properly calculate the cotangent; always double-check your calculator input.
