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 14:55, 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