How to find the area of a nonagon?

Steps to Solve

1. 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.

2. 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 ):

$$ A = \frac{9}{4} \times 5^2 \cot\left(\frac{\pi}{9}\right) $$

3. 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:

$$ A = \frac{9}{4} \times 25 \times \cot\left(\frac{\pi}{9}\right) $$

4. 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:

$$ A \approx \frac{9}{4} \times 25 \times 3.07768 $$

5. Final calculation

Now we perform the final multiplication to find the area:

$$ A \approx 56.6826 $$