# Isosceles Triangle

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.
• 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

## Area of an Isosceles Triangle formula

$$\large{ A_{area} = \frac {h\;b} {2} }$$
Symbol English Metric
$$\large{ A_{area} }$$ = area $$\large{ in^2 }$$ $$\large{ mm^2 }$$
$$\large{ a, b, c }$$ = side $$\large{ in }$$ $$\large{ mm }$$
$$\large{ h }$$ = height $$\large{ in }$$ $$\large{ mm }$$

## Circumcircle of an Isosceles Triangle formula

$$\large{ R = \frac { a^2 } { \sqrt { 4\; a^2 \;-\; b^2 } } }$$
Symbol English Metric
$$\large{ R }$$ = outcircle $$\large{ in^2 }$$ $$\large{ mm^2 }$$
$$\large{ a, b, c }$$ = side $$\large{ in }$$ $$\large{ mm }$$
$$\large{ h }$$ = height $$\large{ in }$$ $$\large{ mm }$$

## Height of an Isosceles Triangle formula

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

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

Symbol English Metric
$$\large{ h }$$ = height $$\large{ in }$$ $$\large{ mm }$$
$$\large{ A_{area} }$$ = area $$\large{ in^2 }$$ $$\large{ mm^2 }$$
$$\large{ a, b, c }$$ = side $$\large{ in }$$ $$\large{ mm }$$

## Inscribed Circle of an Isosceles Triangle formulas

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

Symbol English Metric
$$\large{ r }$$ = incircle $$\large{ in }$$ $$\large{ mm }$$
$$\large{ \alpha }$$  (Greek symbol alpha) = angle $$\large{ deg }$$ $$\large{ rad }$$
$$\large{ a, b, c }$$ = side $$\large{ in }$$ $$\large{ mm }$$
$$\large{ h }$$ = height $$\large{ in }$$ $$\large{ mm }$$

## Perimeter of an Isosceles Triangle formula

$$\large{ P = 2\;a + b }$$
Symbol English Metric
$$\large{ P }$$ = perimeter $$\large{ in }$$ $$\large{ mm }$$
$$\large{ a, b, c }$$ = side $$\large{ in }$$ $$\large{ mm }$$

## Semiperimeter of an Isosceles Triangle formula

$$\large{ s = \frac{ a \;+\; b \;+\; c }{ 2 } }$$
Symbol English Metric
$$\large{ s }$$ = semiperimeter $$\large{ in }$$ $$\large{ mm }$$
$$\large{ a, b, c }$$ = side $$\large{ in }$$ $$\large{ mm }$$

## Side of an Isosceles Triangle formulas

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

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

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

Symbol English Metric
$$\large{ a, b, c }$$ = side $$\large{ in }$$ $$\large{ mm }$$
$$\large{ A_{area} }$$ = area $$\large{ in^2 }$$ $$\large{ mm^2 }$$
$$\large{ h }$$ = height $$\large{ in }$$ $$\large{ mm }$$
$$\large{ P }$$ = perimeter $$\large{ in }$$ $$\large{ mm }$$

## Trig Functions

Find A
• given a c :  $$\; sin \; A= a \div c$$
• given b c :  $$\; cos \; A= b \div c$$
• given a b :  $$\; tan \; A= a \div b$$
Find B
• given a c :  $$\; sin \; B= a \div c$$
• given b c :  $$\; cos \; B= b \div c$$
• given a b :  $$\; tan \; B= b \div a$$
Find a
• given A c :  $$\; a= c*sin \; A$$
• given A b :  $$\; a= b*tan \; A$$
Find b
• given A c :  $$\; b= c*cos \; A$$
• given A a :  $$\; b= a \div tan \; A$$
Find c
• given A a :  $$\; c= a \div sin \; A$$
• given A b :  $$\; c= b \div cos \; A$$
• given a b :  $$\; c= \sqrt { a^2+b^2 }$$
Find Area
• given a b :  $$\; Area= a\;b \div 2$$

Tags: Triangle