Difference between revisions of "Incircle"

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*The radius of an incircle of a triangle (the inradius) with sides <math>a,b,c</math> and area <math>A</math>  is <math>\frac{2A}{a+b+c}</math>
 
*The radius of an incircle of a triangle (the inradius) with sides <math>a,b,c</math> and area <math>A</math>  is <math>\frac{2A}{a+b+c}</math>
 
*The radius of an incircle of a right triangle (the inradius) with legs <math>a,b</math> and hypotenuse <math>c</math> is <math>r=\frac{ab}{a+b+c}=\frac{a+b-c}{2}</math>.
 
*The radius of an incircle of a right triangle (the inradius) with legs <math>a,b</math> and hypotenuse <math>c</math> is <math>r=\frac{ab}{a+b+c}=\frac{a+b-c}{2}</math>.
*For any polygon with an incircle, <math>A=sr</math>, where <math>K</math> is the area, <math>s</math> is the semiperimeter, and <math>r</math> is the inradius.
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*For any polygon with an incircle, <math>A=sr</math>, where <math>A</math> is the area, <math>s</math> is the semiperimeter, and <math>r</math> is the inradius.
  
 
*The formula for the semiperimeter is <math>s=\frac{a+b+c}{2}</math>.
 
*The formula for the semiperimeter is <math>s=\frac{a+b+c}{2}</math>.
  
 
*And area of the triangle by Heron is <math>A^2=s(s-a)(s-b)(s-c)</math>.
 
*And area of the triangle by Heron is <math>A^2=s(s-a)(s-b)(s-c)</math>.
 
  
 
==See also==
 
==See also==

Revision as of 10:32, 6 August 2017

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An incircle of a convex polygon is a circle which is inside the figure and tangent to each side. Every triangle and regular polygon has a unique incircle, but in general polygons with 4 or more sides (such as non- square rectangles) do not have an incircle. A quadrilateral that does have an incircle is called a Tangential Quadrilateral.

Formulas

  • The radius of an incircle of a triangle (the inradius) with sides $a,b,c$ and area $A$ is $\frac{2A}{a+b+c}$
  • The radius of an incircle of a right triangle (the inradius) with legs $a,b$ and hypotenuse $c$ is $r=\frac{ab}{a+b+c}=\frac{a+b-c}{2}$.
  • For any polygon with an incircle, $A=sr$, where $A$ is the area, $s$ is the semiperimeter, and $r$ is the inradius.
  • The formula for the semiperimeter is $s=\frac{a+b+c}{2}$.
  • And area of the triangle by Heron is $A^2=s(s-a)(s-b)(s-c)$.

See also

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