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What is the length of a diagonal of a regular pentagon of unit side length?

 May 28, 2020
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Sketch a regular pentagon and one of its diagonals

--  that diagonal, along with two sides forms an isosceles triangle

 

You know two of the sides of that triangle; they are each 1

 

You can find the size of the vertex angle of the pentagon by using this formula:  (n - 2)·180o / n

--  since  n = 5:  each vertex angle  =  (5 - 2)·180o / 5  =  108o.

 

To find the length of the diagonal (the base of the isosceles triangle), use the Law of Cosines:

--  c2  =  a2 + b2 - 2·a·b·cos(C)

--  c2  =  12 + 12 - 2·1·1·cos(108o)

 

The value of c will be the length of a diagonal of a regular pentagon of unit side length ...

 May 28, 2020

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