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Difference between revisions of "Circumradius"

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The '''circumradius''' of a [[cyclic]] [[polygon]] is the radius of the circumscribed circle of that polygon. For a triangle, it is the measure of the [[radius]] of the [[circle]] that [[circumscribe]]s the triangle. Since every triangle is [[cyclic]], every triangle has a circumscribed circle, or a [[circumcircle]].
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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 [[circumscribe]]s the triangle. Since every triangle is [[cyclic]], every triangle has a circumscribed circle, or a [[circumcircle]].
 
  
 
==Formula for a Triangle==
 
==Formula for a Triangle==
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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>R=\frac{abc}{4A}</math>. Also, <math>A=\frac{abc}{4R}</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 circumradius of the triangle is simply <math>R=\frac{abc}{4A}</math>. This can be rewritten as <math>A=\frac{abc}{4R}</math>.
  
 
== Proof ==
 
== Proof ==
Line 31: Line 29:
 
draw(rightanglemark(B,E,C,20));
 
draw(rightanglemark(B,E,C,20));
 
</asy>
 
</asy>
 +
  
 
We let <math>AB=c</math>, <math>BC=a</math>, <math>AC=b</math>, <math>BE=h</math>, and <math>BO=R</math>. We know that <math>\angle BAD</math> is a right angle because <math>BD</math> is the diameter. Also, <math>\angle ADB = \angle BCA</math> because they both subtend arc <math>AB</math>. Therefore, <math>\triangle BAD \sim \triangle BEC</math> by AA similarity, so we have
 
We let <math>AB=c</math>, <math>BC=a</math>, <math>AC=b</math>, <math>BE=h</math>, and <math>BO=R</math>. We know that <math>\angle BAD</math> is a right angle because <math>BD</math> is the diameter. Also, <math>\angle ADB = \angle BCA</math> because they both subtend arc <math>AB</math>. Therefore, <math>\triangle BAD \sim \triangle BEC</math> by AA similarity, so we have
 
<cmath>\frac{BD}{BA} = \frac{BC}{BE},</cmath> or <cmath> \frac  {2R} c = \frac  ah.</cmath>
 
<cmath>\frac{BD}{BA} = \frac{BC}{BE},</cmath> or <cmath> \frac  {2R} c = \frac  ah.</cmath>
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However, remember that area <math>\triangle ABC = \frac {bh} 2</math>, so <math>h=\frac{2 \times \text{Area}}b</math>. Substituting this in gives us
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However, remember that <math>[ABC] = \frac {bh} 2\implies h=\frac{2 \times [ABC]}b</math>. Substituting this in gives us
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<cmath> \frac  {2R} c = \frac  a{\frac{2 \times \text{Area}}b},</cmath> and then bash through algebra to get
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<cmath> \frac  {2R} c = \frac  a{\frac{2 \times [ABC]}b},</cmath> and then simplifying to get
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<cmath> R=\frac{abc}{4\times \text{Area}},</cmath>
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<cmath> R=\frac{abc}{4\times [ABC]}\text{ or }[ABC]=\frac{abc}{4R}</cmath>
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and we are donel
+
and we are done.
  
 
==Formula for Circumradius==
 
==Formula for Circumradius==
 
<math>R = \frac{abc}{4rs}</math>
 
<math>R = \frac{abc}{4rs}</math>
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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>.
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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>.
 +
 
 +
But, if you don't know the inradius, you can find the area of the triangle by [[Heron's Formula|Heron’s Formula]]:
 +
 
 +
<math>A=\sqrt{s(s-a)(s-b)(s-c)}</math>
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==Circumradius, bisector and altitude==
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[[File:R H L angles.png|400px|right]]
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Circumradius and altitude are isogonals with respect bisector and vertex of triangle.
  
 
==Euler's Theorem for a Triangle==
 
==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>
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Let <math>\triangle ABC</math> have circumcenter <math>O</math> and incenter <math>I</math>.Then <cmath>OI^2=R(R-2r) \implies R \geq 2r</cmath>
 +
 
 +
==Proof==
 +
 
 +
See https://www.cut-the-knot.org/triangle/EulerIO.shtml
 +
 
 +
== Right triangles ==
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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;
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A=(0,0);
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B=(0,3);
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C=(4,0);
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draw(A--B--C--cycle);
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I=circumcenter(A,B,C);
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draw(I--A,gray);
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label("$r$",(I+A)/2,NW,gray);
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draw(circumcircle(A,B,C));
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label("$C$",I,N);
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dot(I);
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draw(rightanglemark(B,A,C,10));
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</asy>
 +
 
 +
This results in a well-known theorem:
 +
===Theorem===
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The midpoint of the hypotenuse is equidistant from the vertices of the right triangle.
 +
 
 +
== Equilateral triangles ==
 +
 
 +
<math>R=\frac{s}{\sqrt3}</math>
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 +
where <math>s</math> is the length of a side of the triangle.
 +
 
 +
<asy>
 +
pair A,B,C,I;
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A=(0,0);
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B=(1,0);
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C=intersectionpoint(arc(A,1,0,90),arc(B,1,90,180));
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draw(A--B--C--cycle);
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I=circumcenter(A,B,C);
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draw(circumcircle(A,B,C));
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label("$C$",I,E);
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dot(I);
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label("$s$",A--B,S);
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label("$s$",A--C,N);
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label("$s$",B--C,N);
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</asy>
 +
 
 +
== If all three sides are known ==
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 +
<math>R=\frac{abc}{\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}</math>
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 +
 
 +
Which follows from the Heron's Formula and <math>R=\frac{abc}{4A}</math>.
 +
 
 +
== If you know just one side and its opposite angle ==
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<math>2R=\frac{a}{\sin{A}}</math> by the [[Law of Sines]].
 +
 
 +
(Extended Law of Sines)
  
 
==See also==
 
==See also==

Latest revision as of 18:27, 17 September 2024

The circumradius of a cyclic polygon is the radius of the circumscribed 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 $R=\frac{abc}{4A}$. This can be rewritten as $A=\frac{abc}{4R}$.

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 $[ABC] = \frac {bh} 2\implies h=\frac{2 \times [ABC]}b$. Substituting this in gives us \[\frac {2R} c = \frac a{\frac{2 \times [ABC]}b},\] and then simplifying to get \[R=\frac{abc}{4\times [ABC]}\text{ or }[ABC]=\frac{abc}{4R}\] and we are done.

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 $A = rs$.

But, if you don't know the inradius, you can find the area of the triangle by Heron’s Formula:

$A=\sqrt{s(s-a)(s-b)(s-c)}$

Circumradius, bisector and altitude

R H L angles.png

Circumradius and altitude are isogonals with respect bisector and vertex of triangle.

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

See https://www.cut-the-knot.org/triangle/EulerIO.shtml

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(I--A,gray); label("$r$",(I+A)/2,NW,gray); draw(circumcircle(A,B,C)); label("$C$",I,N); dot(I); draw(rightanglemark(B,A,C,10)); [/asy]

This results in a well-known theorem:

Theorem

The midpoint of the hypotenuse is equidistant from the vertices of the right triangle.

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

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


Which follows from the Heron's Formula and $R=\frac{abc}{4A}$.

If you know just one side and its opposite angle

$2R=\frac{a}{\sin{A}}$ by the Law of Sines.

(Extended Law of Sines)

See also

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