Isosceles Triangle

on . Posted in Plane Geometry

  • isosceles triangle 2isosceles triangle 1Isosceles 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

Isosceles Triangle Index

 

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 :  \(\; Area= a\;b \div 2 \)

 

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Tags: Triangle