A pentagon inscribed in a circle

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Question from Elaine, a student:

My question as written on my homework is: Given a pentagon inscribed in a circle of radius r, determine a) the angle between any two sides of the pentagaon b) the perimeter of the pentagaon c) the area of the pentagon. I know this kind of counts as three questions, so if you can only answer one, that's okay. Any help will be much appreciated. Thanks!

Hi Elaine.

a) The total of the angles of a polygon go up by 180 degrees for every extra side you have. A triangle's total is 180 degrees, a quadrilateral's total is 360 degrees, so a five-sided polygon has 540 degrees in all. In a regular polygon, all angles are the same, so you just divide the number of angles (same as the number of sides) into the total angle.

b and c) Take a look at Tracy's question and our responses here:
http://mathcentral.uregina.ca/QQ/database/QQ.09.06/s/tracy1.html

Since all pentagons are the same shape (just the size differs), then all side ratios are the same (remember similar triangles do this). So the ratio between the side length and the radius in Tracy's question is the same as the corresponding ratio in your question.

Hope this helps,
Stephen La Rocque.

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