Incircle and Excircles of A Triangle - Relation To Area of The Triangle

Relation To Area of The Triangle

The radii of the in- and excircles are closely related to the area of the triangle. Let K be the triangle's area and let a, b and c, be the lengths of its sides. By Heron's formula, the area of the triangle is

where is the semiperimeter and P = 2s is the perimeter.

The radius of the incircle (also known as the inradius, r ) is

Thus, the area K of a triangle may be found by multiplying the inradius by the semiperimeter:

The radii in the excircles are called the exradii. The excircle at side a has radius

Similarly the radii of the excircles at sides b and c are respectively

and

From these formulas one can see that the excircles are always larger than the incircle and that the largest excircle is the one tangent to the longest side and the smallest excircle is tangent to the shortest side. Further, combining these formulas with Heron's area formula yields the result that

The ratio of the area of the incircle to the area of the triangle is less than or equal to, with equality holding only for equilateral triangles.