Circumradius: Difference between revisions
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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>R=\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>. Also, <math>A=\frac{abc}{4R}</math> | ||
Also, <math>A=\frac{abc}{4R}</math> | |||
==Formula for Circumradius== | ==Formula for Circumradius== | ||
Revision as of 19:34, 7 December 2014
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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, 
Formula for Circumradius
Where 
is the Circumradius, 
is the inradius, and 
, 
, and are the respective sides of the triangle. Note that this is similar to the previously mentioned formula; the reason being that 
.
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)
