Difference between revisions of "Incircle"
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==Formulas== | ==Formulas== | ||
*The radius of an incircle of a triangle (the inradius) with sides <math>a,b,c</math> and area <math>A</math> is <math>r =</math> <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>r =</math> <math>\frac{2A}{a+b+c}.</math> | ||
| − | *The [[area]] of any [[triangle]] is <math>r | + | *The [[area]] of any [[triangle]] is <math>r \times s,</math> where <math>s</math> is the [[Semiperimeter]] of the [[triangle]]. |
*The formula above can be simplified with Heron's Formula, yielding <math>r = \sqrt{\frac{(s-a)(s-b)(s-c)}{s}}.</math> | *The formula above can be simplified with Heron's Formula, yielding <math>r = \sqrt{\frac{(s-a)(s-b)(s-c)}{s}}.</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>. | ||
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*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>. | ||
| − | *The [[area]] of the [[triangle]] by [[Heron's Formula]] is <math>A | + | *The [[area]] of the [[triangle]] by [[Heron's Formula]] is <math>A=\sqrt{s(s-a)(s-b)(s-c)}</math>. |
==See also== | ==See also== | ||
Latest revision as of 18:38, 9 March 2023
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. For a triangle, the center of the incircle is the Incenter, where the incircle is the largest circle that can be inscribed in the polygon. The Incenter can be constructed by drawing the intersection of angle bisectors.
Formulas
- The radius of an incircle of a triangle (the inradius) with sides
and area 
is 
- The area of any triangle is
where 
is the Semiperimeter of the triangle. - The formula above can be simplified with Heron's Formula, yielding
- The radius of an incircle of a right triangle (the inradius) with legs
and hypotenuse 
is 
. - For any polygon with an incircle,
, where is the area, is the semi perimeter, and 
is the inradius. - The coordinates of the incenter (center of incircle) are
, if the coordinates of each vertex are 
, 
, and 
, the side opposite of has length 
, the side opposite of 
has length 
, and the side opposite of 
has length .
and area 
is 
where 
is the 
and hypotenuse 
is 
.
, where 
is the inradius.
, if the coordinates of each vertex are 
, 
, and 
, the side opposite of 
, the side opposite of 
has length 
, and the side opposite of 
has length - The formula for the semiperimeter is
.
.- The area of the triangle by Heron's Formula is
.
.See also
Click here to learn about the orthocenter, and Line's Tangent
