SOLUTION: What is the area of a regular pentagon with perimeter 10x and apothem 1/2y?

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Question 739070: What is the area of a regular pentagon with perimeter 10x and apothem 1/2y?
Answer by fcabanski(1391) About Me  (Show Source):
You can put this solution on YOUR website!
Additional info is needed for a numeric answer. Without knowing x or y or some other relation between them, the answer will contain at least one variable.


A = 1/2 a *P where a is apothem and P is perimeter.


1/2 * 1/2 y * 10x = 5xy/2: Answer 1.


You can go further by cutting the pentagon into 5 triangles, and cutting each of those into two right triangles with height 1/2 y and base x. The triangles thus formed are 36 (opposite the side length x), 54 (opposite the side length 1/2 y) 90 triangles.


Then tan(36) = x/(1/2y) so x = 1/2 y * tan(36)


Substitute that back into the 5xy/2 gives 5*(1/2 y *tan(36))*y /2 = 5%2Ay%5E2%2Atan%2836%29%2F4 Answer 2.


tan(54) = y/2x and 2x*tan(54) = y


Then 5xy/2 = 5x*2x*tan(54)/2 = 5x%5E2%2Atan%2854%29 Answer 3.


Those three answers are equivalent.

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