Inradius

The radius of a Triangle's Incircle or of a Polyhedron's Insphere, denoted $r$. For a Triangle,

$$r = {1\over 2}{\sqrt{(b+c-a)(c+a-b)(a+b-c)\over a+b+c}} = {\Delta\over s}$$ (1)

$$= 4R\sin({\textstyle{1\over 2}}\alpha_1)\sin({\textstyle{1\over 2}}\alpha_2)\sin({\textstyle{1\over 2}}\alpha_3),$$ (2)

where $\Delta$ is the Area of the Triangle, $a$, $b$, and $c$ are the side lengths, $s$ is the Semiperimeter, and $R$ is the Circumradius (Johnson 1929, p. 189).

Equation (1) 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

$$k={2\Delta\over a+b+c}={\Delta\over s}.$$ (3)

But since $k=r$ in this case,

$$r=k={\Delta\over s},$$ (4)

Q. E. D.

Other equations involving the inradius include

$$Rr={abc\over 4s}$$ (5)

$$\Delta^2=rr_1r_2r_3$$ (6)

$$\cos A+\cos B+\cos C=1+{r\over R}$$ (7)

$$r=2R\cos A\cos B\cos\ C$$ (8)

$$a^2+b^2+c^2=4rR+8R^2,$$ (9)

where $r_i$ are the Exradii (Johnson 1929, pp. 189-191).

As shown in Right Triangle, the inradius of a Right Triangle of integral side lengths $x$, $y$, and $z$ is also integral, and is given by

$$r={xy\over x+y+z},$$ (10)

where $z$ is the Hypotenuse. Let $d$ be the distance between inradius $r$ and Circumradius $R$, $d=\overline{rR}$. Then

$$R^2-d^2=2Rr$$ (11)

$${1\over R-d}+{1\over R+d}={1\over r}$$ (12)

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

Expressing the Midradius $\rho$ and Circumradius $R$ in terms of the midradius gives

$$r = {\rho^2\over\sqrt{\rho^2+{\textstyle{1\over 4}}a^2}}$$ (13)

$$r = {R^2-{\textstyle{1\over 4}}a^2\over R}$$ (14)

for an Archimedean Solid.

See also Carnot's Theorem, Circumradius, Midradius

References

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.

Mackay, J. S. ``Formulas Connected with the Radii of the Incircle and Excircles of a Triangle.'' Proc. Edinburgh Math. Soc. 13, 103-104.