Difference between revisions of "Incircle"
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| − | An '''incircle''' of a [[convex]] [[polygon]] is a [[circle]] which is inside the figure and [[tangent line | 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 (geometry) | square]] [[rectangle]]s) 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]]. | + | An '''incircle''' of a [[convex]] [[polygon]] is a [[circle]] which is inside the figure and [[tangent line | 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 (geometry) | square]] [[rectangle]]s) 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== | ==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 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 [[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 [[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>A</math> is the area, <math>s</math> is the semi perimeter, and <math>r</math> is the inradius. | *For any polygon with an incircle, <math>A=sr</math>, where <math>A</math> is the area, <math>s</math> is the semi perimeter, and <math>r</math> is the inradius. | ||
*The coordinates of the incenter (center of incircle) are <math>(\dfrac{aA_x+bB_x+cC_x}{a+b+c}, \dfrac{aA_y+bB_y+cC_y}{a+b+c})</math>, if the coordinates of each vertex are <math>A(A_x, A_y)</math>, <math>B(B_x, B_y)</math>, and <math>C(C_x, C_y)</math>, the side opposite of <math>A</math> has length <math>a</math>, the side opposite of <math>B</math> has length <math>b</math>, and the side opposite of <math>C</math> has length <math>c</math>. | *The coordinates of the incenter (center of incircle) are <math>(\dfrac{aA_x+bB_x+cC_x}{a+b+c}, \dfrac{aA_y+bB_y+cC_y}{a+b+c})</math>, if the coordinates of each vertex are <math>A(A_x, A_y)</math>, <math>B(B_x, B_y)</math>, and <math>C(C_x, C_y)</math>, the side opposite of <math>A</math> has length <math>a</math>, the side opposite of <math>B</math> has length <math>b</math>, and the side opposite of <math>C</math> has length <math>c</math>. | ||
| − | *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=\sqrt{s(s-a)(s-b)(s-c)}</math>. |
==See also== | ==See also== | ||
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[[Category:Geometry]] | [[Category:Geometry]] | ||
| − | Click here to learn about the | + | Click here to learn about the orthocenter, and Line's Tangent |
Latest revision as of 17:38, 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
