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)
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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)
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(3)
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(4)
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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)
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But since in this case,
(6)
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Other equations involving the inradius include
(7)
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(8)
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(9)
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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)
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or equivalently
(11)
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(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)
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(13)
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and these radii obey
(14)
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