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Find the area of the regular 20​-gon with radius 9 mm.

User Mshnik
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2 Answers

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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.

User Graham Hannington
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6 votes
The radius is equal to the side of the polygon. The 20-gon has 20 sides.

let n = number of sides of the polygon
A = area of the polygon
s = measure of one side of the polygon
a = apothem

a = s / (2tan(180/n))
a = 9mm / (2tan(180/20))
a = 28.41 mm

P = s * n
P = 9mm * 20
P = 180 mm

A = a * P / 2
A = 28.41 mm * 180 mm / 2
A = 2556.9 mm^2
User Jaredready
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8.2k points

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