# Isosceles Triangle

Written by Jerry Ratzlaff on . Posted in Plane Geometry

## Isosceles Triangle

• An isosceles triangle is when two sides are equal and two angles are equal.
• The total of angles equal $$\;x+y+z=180°$$.
• Edges $$\;A = C\;$$
• Angles $$\;x = y\;$$
• 3 edges
• 3 vertexs
• Sidess:  $$a$$,  $$b$$,  $$c$$
• Angles:  $$A$$,  $$B$$,  $$C$$
• Area:  $$K$$
• Perimeter:  $$P$$
• Height:  $$h_a$$,  $$h_b$$,  $$h_c$$
• Median:  $$m_a$$,  $$m_b$$,  $$m_c$$  -  A line segment from a vertex (coiner 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
• Semi-perimeter:  $$s$$  -  One half of the perimeter
• Inradius of triangle:  $$r$$
• Outradius (circumcircle) of triangle:  $$R$$

## Area of an Isosceles Triangle

### Area of an Isosceles Triangle formula

$$A= \frac {hb} {2}$$

Where:

$$A$$ = area

$$b$$ = side

$$h$$ = height

## Circumcircle of an Isosceles Triangle

The radius of a circumcircle (outer) of a an Isosceles triangle if given side $$( R )$$.

### Circumcircle of an Isosceles Triangle formula

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

Where:

$$R$$ = outcircle

$$a, b$$ = side

## Height of an Isosceles Triangle

### Height of an Isosceles Triangle formula

$$h = 2 \frac {A}{b}$$

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

Where:

$$h$$ = height

$$a, b$$ = side

$$A$$ = area

## Inscribed Circle of an Isosceles Triangle

### Inscribed Circle of an Isosceles Triangle formula

The radius of a inscribed circle (inner) of an Isosceles triangle if given side $$( r )$$.

$$r = \frac { b } { 2 } \sqrt { \frac { 2a - b } { 2a + b } }$$

The radius of a inscribed circle (inner) of an Isosceles triangle if given side and angle $$( r )$$.

$$r = a \frac { sine \; \alpha \;\;x\;\; cos \; \alpha } { 1 + cos \; \alpha } \;=\; \alpha \; cos \; \alpha \;\;x\;\; tan \frac { \alpha } { 2 }$$

$$r = \frac {b}{2} \;\;x\;\; \frac { sine \; \alpha } { 1 + cos \; \alpha } \;=\; \frac {b}{2} \;\;x\;\; tan \frac { \alpha } { 2 }$$

The radius of a inscribed circle (inner) of an Isosceles triangle if given side and height $$( r )$$.

$$r = \frac { bh } { b + \sqrt { 4h^2 + b^2 } }$$

$$r = \frac { h \sqrt { a^2 - h^2 } } { a + \sqrt { a^2 - h^2 } }$$

Where:

$$r$$ = incircle

$$a, b$$ = side

$$\alpha$$ (Greek symbol alpha) = angle

$$h$$ = height

## Perimeter of an Isosceles Triangle

### Perimeter of an Isosceles Triangle formula

$$P = 2a + b$$

Where:

$$P$$ = perimeter

$$a, b$$ = side

## Semiperimeter of an Isosceles Triangle

One half of the perimeter.

### Semiperimeter of an Isosceles Triangle formula

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

Where:

$$s$$ = semiperimeter

$$a, b, c$$ = side

## Side of an Isosceles Triangle

### Side of an Isosceles Triangle formula

$$a = \frac {P} {2} - \frac {b} {2}$$

$$b = P - 2a$$

$$b = 2 \frac {A} {h}$$

Where:

$$a, b$$ = side

$$h$$ = height

$$P$$ = perimeter

$$A$$ = area