Question 1208232
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what is the {{{highlight(cross(largest))}}} diameter of the circle that will fit inside a regular pentagon with 2-inch sides
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    I crossed above,  since there is  NO  the largest diameter.
    All diameters of a circle have the same length.



<pre>
Consider one of 5 isosceles triangles that make this pentagon.


The base of this triangle is 2 inches, hence, half of the base is 1 inch long.


The angle of the triangle, opposite to the base, is  360/5 = 72 degrees;
half of this angle is 36 degrees.


Let r be the radius of the circle, inscribed in this pentagon (in inches).


Then  {{{1_inch/r}}} = tan(36).


Hence,  r = {{{1/tan(36)}}} = {{{1/0.726542528}}} = 1.3763819 inches.


For tan(36), there is special formula  tan(36) = {{{sqrt(5-2*sqrt(5))}}}.


For deriving this formula, see, for example, this source

    https://www.cuemath.com/trigonometry/tan-36-degrees/

or many others (it is a classic in Trigonometry).



So, the answer to your question for the diameter is  

    D = {{{2/tan(36)}}} = {{{2/sqrt(5-2*sqrt(5))}}} =  2.752763841 inches.
</pre>

Solved.