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

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


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

For a triangle,


where Delta is the area of the triangle, a, b, and c are the side lengths, s is the semiperimeter, R is the circumradius, and A, B, and C are the angles opposite sides a, b, and c (Johnson 1929, p. 189). If two triangle side lengths a and b are known, together with the inradius r, then the length of the third side c can be found by solving (1) for c, 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 r:r:r. The ratio k of the exact trilinears to the homogeneous coordinates is given by


But since k=r in this case,



Other equations involving the inradius include


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

Let d be the distance between inradius r and circumradius R, d=rR^_. Then the Euler triangle formula states that


or equivalently


(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 r_d of the dual polyhedron can be expressed in terms of the circumradius R of the solid, midradius rho=rho_d, and edge length a as


and these radii obey


See also

Carnot's Theorem, Circumradius, Euler Triangle Formula, Japanese Theorem, Midradius

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Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., 1888.Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 10, 1967.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929.Mackay, J. S. "Historical Notes on a Geometrical Theorem and its Developments [18th Century]." Proc. Edinburgh Math. Soc. 5, 62-78, 1886-1887.Mackay, J. S. "Formulas Connected with the Radii of the Incircle and Excircles of a Triangle." Proc. Edinburgh Math. Soc. 12, 86-105, 1893.Mackay, J. S. "Formulas Connected with the Radii of the Incircle and Excircles of a Triangle." Proc. Edinburgh Math. Soc. 13, 103-104, 1894.

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Cite this as:

Weisstein, Eric W. "Inradius." From MathWorld--A Wolfram Web Resource.

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