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.
Read more about this topic: Incircle And Excircles Of A Triangle
Famous quotes containing the words relation to, relation and/or area:
“Any relation to the land, the habit of tilling it, or mining it, or even hunting on it, generates the feeling of patriotism. He who keeps shop on it, or he who merely uses it as a support to his desk and ledger, or to his manufactory, values it less.”
—Ralph Waldo Emerson (1803–1882)
“Art should exhilarate, and throw down the walls of circumstance on every side, awakening in the beholder the same sense of universal relation and power which the work evinced in the artist, and its highest effect is to make new artists.”
—Ralph Waldo Emerson (1803–1882)
“Whatever an artist’s personal feelings are, as soon as an artist fills a certain area on the canvas or circumscribes it, he becomes historical. He acts from or upon other artists.”
—Willem De Kooning (b. 1904)