# Isosceles Triangle

Written by Jerry Ratzlaff on . Posted in Plane Geometry

• Isosceles triangle (a two-dimensional figure) has two sides that are the same length or at least two congruent sides.
• Angle bisector of a right isosceles triangle is a line that splits an angle into two equal angles.
• Circumcircle is a circle that passes through all the vertices of a two-dimensional figure.
• Congruent is all sides having the same lengths and angles measure the same.
• Inscribed circle is the largest circle possible that can fit on the inside of a two-dimensional figure.
• Semiperimeter is one half of the perimeter.
• a = c
• x = y
• x + y + z = 180°
• Height:  $$h_a$$, $$h_b$$, $$h_c$$
• Median:  $$m_a$$, $$m_b$$, $$m_c$$  -  A line segment from a vertex (corner point) to the midpoint of the opposite side
• Angle bisectors:  $$t_a$$, $$t_b$$, $$t_c$$  -  A line that splits an angle into two equal angles
• 3 edges
• 3 vertexs

## Formulas that use Area of an Isosceles Triangle

 $$\large{ A_{area} = \frac {h\;b} {2} }$$

### Where:

$$\large{ A_{area} }$$ = area

$$\large{ b }$$ = side

$$\large{ h }$$ = height

## Formulas that use Circumcircle of an Isosceles Triangle

 $$\large{ R = \frac { a^2 } { \sqrt { 4\; a^2 \;-\; b^2 } } }$$

### Where:

$$\large{ R }$$ = outcircle

$$\large{ a, b }$$ = side

## Formulas that use Height of an Isosceles Triangle

 $$\large{ h = 2 \frac {A_{area}}{b} }$$ $$\large{ h = \sqrt { a^2 - \frac {b^2}{4} } }$$

### Where:

$$\large{ h }$$ = height

$$\large{ a, b }$$ = side

$$\large{ A_{area} }$$ = area

## Formulas that use Inscribed Circle of an Isosceles Triangle

 $$\large{ r = \frac { b } { 2 } \; \sqrt { \frac { 2\;a \;-\; b } { 2\;a \;+\; b } } }$$ The radius of a inscribed circle (inner) of an Isosceles triangle if given side $$( r )$$. $$\large{ r = a \; \frac { sine \; \alpha \;x\; cos \; \alpha } { 1 \;+\; cos \; \alpha } = \alpha \; cos \; \alpha \;\;x\;\; tan \frac { \alpha } { 2 } }$$ The radius of a inscribed circle (inner) of an Isosceles triangle if given side and angle $$( r )$$. $$\large{ r = \frac {b}{2} \;x\; \frac { sine \; \alpha } { 1 \;+\; cos \; \alpha } = \frac {b}{2} \;x\; tan \frac { \alpha } { 2 } }$$ The radius of a inscribed circle (inner) of an Isosceles triangle if given side and angle $$( r )$$. $$\large{ r = \frac { b\;h } { b \;+\; \sqrt { 4\;h^2 \;+\; b^2 } } }$$ The radius of a inscribed circle (inner) of an Isosceles triangle if given side and height $$( r )$$. $$\large{ r = \frac { h\; \sqrt { a^2 \;-\; h^2 } } { a \;+\; \sqrt { a^2 \;-\; h^2 } } }$$ The radius of a inscribed circle (inner) of an Isosceles triangle if given side and height $$( r )$$.

### Where:

$$\large{ r }$$ = incircle

$$\large{ a, b }$$ = side

$$\large{ \alpha }$$  (Greek symbol alpha) = angle

$$\large{ h }$$ = height

## Formulas that use Perimeter of an Isosceles Triangle

 $$\large{ P = 2\;a + b }$$

### Where:

$$\large{ P }$$ = perimeter

$$\large{ a, b }$$ = side

## Formulas that use Semiperimeter of an Isosceles Triangle

 $$\large{ s = \frac { a \;+\; b \;+\; c } { 2 } }$$

### Where:

$$\large{ s }$$ = semiperimeter

$$\large{ a, b, c }$$ = side

## Formulas that use Side of an Isosceles Triangle

 $$\large{ a = \frac {P} {2} - \frac {b} {2} }$$ $$\large{ b = P - 2\;a }$$ $$\large{ b = 2\; \frac {A_{area}} {h} }$$

### Where:

$$\large{ a, b }$$ = side

$$\large{ h }$$ = height

$$\large{ P }$$ = perimeter

$$\large{ A_{area} }$$ = area