# Difference between revisions of "Circumradius"

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==Formula for Circumradius== | ==Formula for Circumradius== | ||

<math>R = \frac{abc}{4rs}</math> | <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>. | + | 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 and <math>s = (a+b+c)/2</math> is the semiperimeter. 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 13:55, 6 April 2016

*This article is a stub. Help us out by expanding it.*

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

## 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 circumradius of the triangle is simply . Also,

## Proof

We let , , , , and . We know that is a right angle because is the diameter. Also, because they both subtend arc . Therefore, by AA similarity, so we have or However, remember that area , so . Substituting this in gives us and then bash through algebra to get and we are done.

--Nosaj 19:39, 7 December 2014 (EST)

## Formula for Circumradius

Where is the Circumradius, is the inradius, and , , and are the respective sides of the triangle and is the semiperimeter. 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