Answer:
To find the apothem of a regular polygon, we can use the formula:
A = (ns^2)/(4tan(180°/n))
Where A is the area, n is the number of sides, s is the length of each side, and tan(180°/n) is the tangent of half of the central angle.
First, we need to find the number of sides of the polygon. To do this, we can use the formula for the sum of the interior angles of a polygon:
(n - 2)180° = 1080°
Solving for n:
n = 6
So the polygon has 6 sides.
Next, we can plug in the values we have into the formula for the apothem:
A = (6s^2)/(4tan(180°/6)) = 160
Expanding the right side:
A = (6s^2)/(4tan(30°)) = 160
Using the tangent identity:
A = (6s^2)/(4(√3/3)) = 160
Multiplying both sides by 4(√3/3) to isolate s^2:
4(√3/3)A = 6s^2
Expanding the left side:
4(√3)A/3 = 6s^2
Dividing both sides by 6:
2(√3)A/3 = s^2
Taking the square root of both sides:
s = √(2(√3)A/3)
Plugging in the value for A:
s = √(2(√3) * 160/3)
Calculating:
s = √(106.667) = 3.25
So the length of each side is 3.25 inches.
Finally, we can find the apothem using the formula:
a = s/2tan(180°/n)
Expanding the right side:
a = 3.25/2tan(30°)
Using the tangent identity:
a = 3.25/2(√3/3)
Dividing both sides by √3/3:
a = 3.25√3
Calculating:
a = 5.5
So the length of the apothem is 5.5 inches.