# Isosceles Triangle

Written by Jerry Ratzlaff on . Posted in Plane Geometry

• A two-dimensional figure that 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.
• Radius of a circumcircle (outer) of a isosceles triangle if given legs and hypotrnuse ( R ).
• Radius of a inscribed circle (inner) of a isosceles triangle if given legs and hypotrnuse ( r ).
• Semiperimeter of a right triangle is one half of the perimeter.
• 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 (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
• Semi-perimeter:  $$s$$  -  One half of the perimeter
• Inradius of triangle:  $$r$$
• Outradius (circumcircle) of triangle:  $$R$$

### Area of an Isosceles Triangle formula

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

Where:

$$\large{ A }$$ = area

$$\large{ b }$$ = side

$$\large{ h }$$ = height

### Circumcircle of an Isosceles Triangle formula

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

Where:

$$\large{ R }$$ = outcircle

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

### Height of an Isosceles Triangle formula

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

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

Where:

$$\large{ h }$$ = height

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

$$\large{ A }$$ = area

### Inscribed Circle of an Isosceles Triangle formula

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

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

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

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

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

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

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

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

Where:

$$\large{ r }$$ = incircle

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

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

$$\large{ h }$$ = height

### Perimeter of an Isosceles Triangle formula

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

Where:

$$\large{ P }$$ = perimeter

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

### Semiperimeter of an Isosceles Triangle formula

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

Where:

$$\large{ s }$$ = semiperimeter

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

### Side of an Isosceles Triangle formula

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

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

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

Where:

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

$$\large{ h }$$ = height

$$\large{ P }$$ = perimeter

$$\large{ A }$$ = area