Difference between revisions of "Circumradius"

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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 circumradius of the triangle is simply $R=\frac{abc}{4A}$ . Also, $A=\frac{abc}{4R}$

Proof

$[asy] pair O, A, B, C, D; O=(0,0); A=(-5,1); B=(1,5); C=(5,1); dot(O); dot (A); dot (B); dot (C); draw(circle(O, sqrt(26))); draw(A--B--C--cycle); D=-B; dot (D); draw(B--D--A); label("$A$", A, W); label("$B$", B, N); label("$C$", C, E); label("$D$", D, S); label("$O$", O, W); pair E; E=foot(B,A,C); draw(B--E); dot(E); label("$E$", E, S); draw(rightanglemark(B,A,D,20)); draw(rightanglemark(B,E,C,20)); [/asy]$

We let $AB=c$ , $BC=a$ , $AC=b$ , $BE=h$ , and $BO=R$ . We know that $\angle BAD$ is a right angle because $BD$ is the diameter. Also, $\angle ADB = \angle BCA$ because they both subtend arc $AB$ . Therefore, $\triangle BAD \sim \triangle BEC$ by AA similarity, so we have $\frac{BD}{BA} = \frac{BC}{BE},$ or $\frac {2R} c = \frac ah.$ However, remember that area $\triangle ABC = \frac {bh} 2$ , so $h=\frac{2 \times \text{Area}}b$ . Substituting this in gives us $\frac {2R} c = \frac a{\frac{2 \times \text{Area}}b},$ and then bash through algebra to get $R=\frac{abc}{4\times \text{Area}},$ and we are done.

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

Formula for Circumradius

$R = \frac{abc}{4rs}$ Where $R$ is the Circumradius, $r$ is the inradius, and $a$ , $b$ , and $c$ are the respective sides of the triangle and $s = (a+b+c)/2$ is the semiperimeter. Note that this is similar to the previously mentioned formula; the reason being that $A = rs$ .

Euler's Theorem for a Triangle

Let $\triangle ABC$ have circumcenter $O$ and incenter $I$ .Then $OI^2=R(R-2r) \implies R \geq 2r$

Proof

Right triangles

The hypotenuse of the triangle is the diameter of its circumcircle, and the circumcenter is its midpoint, so the circumradius is equal to half of the hypotenuse of the right triangle.

$[asy] pair A,B,C,I; A=(0,0); B=(0,3); C=(4,0); draw(A--B--C--cycle); I=circumcenter(A,B,C); draw(circumcircle(A,B,C)); label("$C$",I,N); dot(I); draw(rightanglemark(B,A,C,10)); [/asy]$

Equilateral triangles

$R=\frac{s}{\sqrt3}$

where $s$ is the length of a side of the triangle.

$[asy] pair A,B,C,I; A=(0,0); B=(1,0); C=intersectionpoint(arc(A,1,0,90),arc(B,1,90,180)); draw(A--B--C--cycle); I=circumcenter(A,B,C); draw(circumcircle(A,B,C)); label("$C$",I,E); dot(I); label("$s$",A--B,S); label("$s$",A--C,N); label("$s$",B--C,N); [/asy]$

If all three sides are known

$R=\frac{abc}{\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}$

And this formula comes from the area of Heron and $R=\frac{abc}{4A}$ .

If you know just one side and its opposite angle

$2R=\frac{a}{\sin{A}}$

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Difference between revisions of "Circumradius"

Revision as of 19:02, 1 August 2017

Contents

Formula for a Triangle

Proof

Formula for Circumradius

Euler's Theorem for a Triangle

Proof

Right triangles

Equilateral triangles

If all three sides are known

If you know just one side and its opposite angle

See also

@@ Line 61: / Line 61: @@
 I=circumcenter(A,B,C);
 draw(circumcircle(A,B,C));
-label("$I$",I,N);
+label("$C$",I,N);
 dot(I);
 draw(rightanglemark(B,A,C,10));