Difference between revisions of "Circumradius"
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==Formula for a Triangle== | ==Formula for a Triangle== | ||
| − | Let <math>a, b</math> and <math>c</math> denote the triangle's three sides, and let <math>A</math> denote the area of the triangle. Then, the measure of the of the circumradius of the triangle is simply <math>\frac{abc}{4A}</math> | + | Let <math>a, b</math> and <math>c</math> denote the triangle's three sides, and let <math>A</math> denote the area of the triangle. Then, the measure of the of the circumradius of the triangle is simply <math>R=\frac{abc}{4A}</math> |
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| + | Also, <math>A=\frac{abc}{4R}</math> | ||
==Euler's Theorem for a Triangle== | ==Euler's Theorem for a Triangle== | ||
Revision as of 21:38, 17 October 2011
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The circumradius of a cyclic polygon is the radius of the cirumscribed circle of that polygon. For a triangle, it is the measure of the radius of the circle that circumscribes the triangle. Since every triangle is cyclic, every triangle has a circumscribed circle, or a circumcircle.
Formula for a Triangle
Let 
and 
denote the triangle's three sides, and let 
denote the area of the triangle. Then, the measure of the of the circumradius of the triangle is simply 
Also, 
Euler's Theorem for a Triangle
Let 
have circumradius 
and inradius 
. Let 
be the distance between the circumcenter and the incenter. Then we have ![\[d^2=R(R-2r)\]](http://latex.artofproblemsolving.com/7/e/2/7e2fcd01749b32bd1072381de2ac6589dd730703.png)
