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.
The area of the purple color is the area of the dodecagon minus the area of the circle. Therefore, we must find the area of the dodecagon. By drawing diagonals between opposite vertices, we can divide the dodecagon into 12 congruent isosceles triangles with a vertex angle that is 360^(∘) divided by 12. 360^(∘)/12=30^(∘) If we draw the height from the vertex angle of one triangle, it will bisect this angle and the base of the triangle.
Substitute values
Substitute values
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.