Final answer:
The area of a regular 20-gon with a 9 mm radius can be found by first finding the apothem and the side length using trigonometric functions, then calculating the perimeter, and finally using the formula for the area of a regular polygon. The calculated area should be rounded according to the number of significant figures from the starting values.
Step-by-step explanation:
To find the area of a regular 20-gon (icosagon) with a radius (or circumradius) of 9 mm, we can use the formula for the area of a regular polygon:
A = \( \frac{1}{2} \times Perimeter \times Apothem \)
However, the perimeter of a regular 20-gon can be calculated as the sum of all its sides. Since all sides are of equal length, the perimeter (P) is:
P = 20 \times Side length
The apothem (a) of a regular polygon, which is the radius of the inscribed circle, can be found using the radius (R) of the circumscribed circle and the number of sides (n), using trigonometry:
a = R \times cos(\( \frac{\pi}{n} \))
For a regular 20-gon:
a = 9 mm \times cos(\( \frac{\pi}{20} \))and the side length of our regular 20-gon (s) can be found using:
s = 2 \times R \times sin(\( \frac{\pi}{n} \)) = 2 \times 9 mm \times sin(\( \frac{\pi}{20} \))
After finding the side length, we calculate the perimeter and then the area using the first formula. It's important to note that when using a calculator for \( \pi \), as it will give a long decimal, the significant figures of the starting values should determine the number of significant figures in your final answer, as demonstrated in the example:
A = \( \pi r^2 \) = (3.1415927...)\( \times \)(1.2 m)\( ^2 \) = 4.5238934 m\( ^2 \), rounded according to significant figures, gives us A=4.5 m\( ^2 \). Finally, remember to keep your units consistent throughout your calculations.