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We draw the pentagon first. Let its side be S.

We have to draw the incircle. that incircle touches the sides tangentially at one point and that point is the center of the side. If we draw a perpendicular from the center of the polygon on to the side, it meets the side at the center of the side.
That perpendicular is called apothem.   The APOTHEM is the radius of incircle.

So we have a triangle (Isosceles) with two radii and drawn at two ends of the side, and the side itself.
Now with the apothem, half of side and one radius we have a right angle triangle.

Radius = apothem = a =  1/2  S  cot π/5  =  0.6882  S

From the center of pentagon, draw a circle of radius 0.6882 * Side. That is the INCIRCLE.

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