Difference between revisions of "Circumradius"
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Also, <math>A=\frac{abc}{4R}</math> | Also, <math>A=\frac{abc}{4R}</math> | ||
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| + | ==Formula for Circumradius== | ||
| + | <math>R = \frac{abc}{4rs}</math> | ||
| + | Where <math>R</math> is the Circumradius, <math>r</math> is the inradius, and <math>a</math>, <math>b</math>, and <math>c</math> are the respective sides of the triangle. Note that this is similar to the previously mentioned formula; the reason being that <math>A = rs</math>. | ||
==Euler's Theorem for a Triangle== | ==Euler's Theorem for a Triangle== | ||
Revision as of 18:28, 8 August 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.
Contents
[hide]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)
