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>\frac{abc}{4A}</math>
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==Euler's Theorem for a Triangle==
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Let <math>\triangle ABC</math> have circumradius <math>R</math> and inradius <math>r</math>. Let <math>d</math> be the distance between the circumcenter and the incenter. Then we have <cmath>d^2=R(R-2r)</cmath>
  
 
==See also==
 
==See also==

Revision as of 20:45, 6 November 2007

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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 $a, b$ and $c$ denote the triangle's three sides, and let $A$ denote the area of the triangle. Then, the measure of the of the circumradius of the triangle is simply $\frac{abc}{4A}$

Euler's Theorem for a Triangle

Let $\triangle ABC$ have circumradius $R$ and inradius $r$. Let $d$ be the distance between the circumcenter and the incenter. Then we have \[d^2=R(R-2r)\]

See also

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