Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Circumradius"
(→‎Theorem)
 
(59 intermediate revisions by 31 users not shown)
Line 1: Line 1:
−
{{stub}}
+
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]].
 +
 
 +
==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 circumradius of the triangle is simply <math>R=\frac{abc}{4A}</math>. This can be rewritten as <math>A=\frac{abc}{4R}</math>.
 +
 
 +
== 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>
  
−
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==
+
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
−
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>
+
<cmath>\frac{BD}{BA} = \frac{BC}{BE},</cmath> or <cmath> \frac  {2R} c = \frac  ah.</cmath>
 +
However, remember that <math>[ABC] = \frac {bh} 2\implies h=\frac{2 \times [ABC]}b</math>. Substituting this in gives us
 +
<cmath> \frac  {2R} c = \frac  a{\frac{2 \times [ABC]}b},</cmath> and then simplifying to get
 +
<cmath> R=\frac{abc}{4\times [ABC]}\text{ or }[ABC]=\frac{abc}{4R}</cmath>
 +
and we are done.
 +
 
 +
==Formula for Circumradius==
 +
<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 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>
 +
==Circumradius, bisector and altitude==
 +
[[File:R H L angles.png|400px|right]]
 +
Circumradius and altitude are isogonals with respect bisector and vertex of triangle.
 +
 
 +
==Euler's Theorem for a Triangle==
 +
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 ==
 +
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 ==
 +
 
 +
<math>R=\frac{s}{\sqrt3}</math>
 +
 
 +
where <math>s</math> 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 ==
 +
 
 +
<math>R=\frac{abc}{\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}</math>
 +
 
 +
 
 +
Which follows from the Heron's Formula and <math>R=\frac{abc}{4A}</math>.
 +
 
 +
== If you know just one side and its opposite angle ==
 +
 
 +
<math>2R=\frac{a}{\sin{A}}</math> by the [[Law of Sines]].
 +
 
 +
(Extended Law of Sines)
  
 
==See also==
 
==See also==
−
* [[inradius]]
+
* [[Inradius]]
−
* [[semiperimeter]]
+
* [[Semiperimeter]]
 +
 
 +
[[Category:Geometry]]

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

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Circumradius&oldid=227441"