Chapter Closure
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The area of the purple color is the area of the dodecagon minus the area of the circle.
The area of the purple region is greater.
Let's illustrate the dimensions of the circle and dodecagon.
Since we know the radius of the circle, r= 9, we can directly find the green area by calculating the area of the circle.
We can find the length of the leg of the right triangle by using the cosine ratio and sine ratio.
Substitute values
The longer leg is 14cos 15^(∘) cm long. Let's also calculate the length of b.
Substitute values
The shorter leg is 14sin 15^(∘) cm long, which means the base of the isosceles triangle must be twice this length.
Now we can calculate the area of the triangle and finally the dodecagon by multiplying this number by 12. (1/2(14cos 15^(∘))(28cos 15^(∘)))12=588 cm^2 Finally, we can find the purple area by subtracting the circle's area from the dodecagon's area. 588-81π ≈ 333.5 cm^2 Since 333.5>254.5, the purple area is greater.