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:
“Much poetry seems to be aware of its situation in time and of its relation to the metronome, the clock, and the calendar. ... The season or month is there to be felt; the day is there to be seized. Poems beginning “When” are much more numerous than those beginning “Where” of “If.” As the meter is running, the recurrent message tapped out by the passing of measured time is mortality.”
—William Harmon (b. 1938)
“To be a good enough parent one must be able to feel secure in one’s parenthood, and one’s relation to one’s child...The security of the parent about being a parent will eventually become the source of the child’s feeling secure about himself.”
—Bruno Bettelheim (20th century)
“Self-esteem is the real magic wand that can form a child’s future. A child’s self-esteem affects every area of her existence, from friends she chooses, to how well she does academically in school, to what kind of job she gets, to even the person she chooses to marry.”
—Stephanie Martson (20th century)