Inradius

The radius of a polygon's incircle or of a polyhedron's insphere, denoted or sometimes (Johnson 1929). A polygon possessing an incircle is same to be inscriptable or tangential.

The inradius of a regular polygon with sides and side length is given by

(1)

The following table summarizes the inradii from some nonregular inscriptable polygons.

polygon	inradius
3, 4, 5 triangle
30-60-90 triangle
bicentric quadrilateral
diamond
golden triangle
isosceles right triangle
isosceles triangle
lozenge
rhombus
right triangle
tangential quadrilateral

For a triangle,

			(2)
			(3)
			(4)

where is the area of the triangle, , , and are the side lengths, is the semiperimeter, is the circumradius, and , , and are the angles opposite sides , , and (Johnson 1929, p. 189). If two triangle side lengths and are known, together with the inradius , then the length of the third side can be found by solving (1) for , resulting in a cubic equation.

Equation (◇) can be derived easily using trilinear coordinates. Since the incenter is equally spaced from all three sides, its trilinear coordinates are 1:1:1, and its exact trilinear coordinates are . The ratio of the exact trilinears to the homogeneous coordinates is given by

(5)

But since in this case,

(6)

Q.E.D.

Other equations involving the inradius include

			(7)
			(8)
			(9)

where is the semiperimeter, is the circumradius, and are the exradii of the reference triangle (Johnson 1929, pp. 189-191).

Let be the distance between inradius and circumradius , . Then the Euler triangle formula states that

(10)

or equivalently

(11)

(Mackay 1886-87; Casey 1888, pp. 74-75). These and many other identities are given in Johnson (1929, pp. 186-190).

For a Platonic or Archimedean solid, the inradius of the dual polyhedron can be expressed in terms of the circumradius of the solid, midradius , and edge length as

			(12)
			(13)

and these radii obey

(14)

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