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Directions: Determine the area of the shaded region of each figure. The polygons in each problem are regular polygons. You must include your calculations.

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Problem 1

Imagine placing the nonpencil part of your compass at the center of both hexagons. Extend the compass so the pencil reaches the right-most point on the larger hexagon. Then sweep a full 360 degrees to form a circle. This surrounds the larger hexagon to form what is called the circumcircle. The circumradius is 6+6 = 12 feet.

If we know the circumradius r = 12 and the number of sides n = 6, then we can find the area of the larger regular hexagon using this formula

$$A = 0.5n*r^2*\sin(360/n)$$

Your calculator must be in degree mode. This formula only works for regular polygons.

Let's use that formula to calculate the area of the larger regular hexagon.

$$A = 0.5n*r^2*\sin(360/n)\\\\A = 0.5*6*12^2*\sin(360/6)\\\\A \approx 374.122974$$

Then do the same for the smaller regular hexagon that has circumradius of r = 6 this time. We still have n = 6 sides.

$$A = 0.5n*r^2*\sin(360/n)\\\\A = 0.5*6*6^2*\sin(360/6)\\\\A \approx 93.530744$$

Now subtract the areas

374.122974-93.530744 = 280.59223

The green region has an area of roughly 280.59223 square feet.

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Problem 2

The formula we use here is

$$A = 0.25n*s^2*\cot(180/n)$$

where

- n = number of sides = 5 in this case for both figures
- s = side length = will be different values depending on which figure you focus on
- "cot" refers to the cotangent function, which is the reciprocal of tangent. Your calculator should support cotangent. If not, then you'd have to do 1/tan of the value of 180/n. In this case, n = 5 so 180/n = 180/5 = 36. So you either calculate cot(36) or 1/tan(36)
- Your calculator must be in degree mode.
- This formula only works on regular polygons. These are polygons with the same side lengths and same angles.

Let's use that formula to calculate the area of the larger regular pentagon. It has side lengths of s = 8

$$A = 0.25n*s^2*\cot(180/n)\\\\A = 0.25*5*8^2*\cot(180/5)\\\\A = 0.25*5*8^2*\cot(36)\\\\A = 0.25*5*8^2*\frac{1}{\tan(36)}\\\\A \approx 110.110554\\\\$$

The larger regular pentagon has area of roughly 110.110554 square cm.

Repeat those steps for the smaller pentagon where each side is now s = 6. Keep n = 5 the same.

$$A = 0.25n*s^2*\cot(180/n)\\\\A = 0.25*5*6^2*\cot(180/5)\\\\A = 0.25*5*6^2*\cot(36)\\\\A = 0.25*5*6^2*\frac{1}{\tan(36)}\\\\A \approx 61.937186\\\\$$

Now subtract the two areas to get the area of the shaded green region.

110.110554 - 61.937186 = 48.173368

The green region has area of roughly 48.173368 square cm.
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