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Isosceles Triangle

  • 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

 

Area of an Isosceles Triangle formula
\( A_{area} \;=\; \dfrac{ h \cdot b }{ 2 } \) 
Symbol English Metric
\( A_{area} \) = area \( in^2 \) \( mm^2 \)
\( a, b, c \) = side \( in \) \( mm \)
\( h \) = height \( in \) \( mm \)

 

Circumcircle of an Isosceles Triangle formula
\( R \;=\;  \dfrac{ a^2 }{ \sqrt{  4 \cdot a^2 - b^2 } }  \) 
Symbol English Metric
\( R \) = outcircle \( in^2 \) \( mm^2 \)
\( a, b, c \) = side \( in \) \( mm \)
\( h \) = height \( in \) \( mm \)

 

Height of an Isosceles Triangle formula

\( h \;=\;  2 \cdot \dfrac {A_{area} }{ b } \) 

\( h \;=\; \sqrt{   a^2 - \dfrac{ b^2 }{ 4 }  } \)
Symbol English Metric
\( h \) = height \( in \) \( mm \)
\( A_{area} \) = area \( in^2 \) \( mm^2 \)
\( a, b, c \) = side \( in \) \( mm \)

 

Inscribed Circle of an Isosceles Triangle formulas

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

\( r \;=\;   \dfrac{ b }{ 2 } \cdot \sqrt{ \dfrac{ 2 \cdot a - b  }{  2 \cdot a + b }   } \)

\( r \;=\;   a \cdot  \dfrac{ sine( \alpha) \cdot  cos( \alpha)  }{ 1 + cos( \alpha)  }    \;=\;   \alpha \cdot cos( \alpha)  \cdot  tan\left( \dfrac{ \alpha }{ 2 } \right)  \)

\( r \;=\; \dfrac{ b}{2}  \cdot  \dfrac{ sine( \alpha)  }{ 1  +  cos( \alpha)  }    \;=\;   \dfrac {b}{2}  \cdot  tan\left( \dfrac{ \alpha  }{ 2 } \right)  \)

\( r \;=\;   \dfrac{ b \cdot h  }{  b +  \sqrt{ 4 \cdot h^2 + b^2  }   } \)

\( r \;=\;   \dfrac{  h \cdot  \sqrt{ a^2 - h^2  }  }{  a +  \sqrt{ a^2 - h^2  }   } \)
Symbol English Metric
\( r \) = incircle \( in \) \( mm \)
\( \alpha \)  (Greek symbol alpha) = angle \( deg \) \( rad \)
\( a, b, c \) = side \( in \) \( mm \)
\( h \) = height \( in \) \( mm \)

 

Perimeter of an Isosceles Triangle formula
\( P \;=\; 2 \cdot a + b \) 
Symbol English Metric
\( P \) = perimeter \( in \) \( mm \)
\( a, b, c \) = side \( in \) \( mm \)

 

Semiperimeter of an Isosceles Triangle formula
\( s \;=\;   \dfrac{ a + b + c }{ 2  }   \) 
Symbol English Metric
\( s \) = semiperimeter \( in \) \( mm \)
\( a, b, c \) = side \( in \) \( mm \)

 

Side of an Isosceles Triangle formulas

\( a \;=\; \dfrac{P}{2} - \dfrac{b}{2} \) 

\( b \;=\; P - 2 \cdot a   \) 

\( b \;=\;  2 \cdot \dfrac{A_{area} }{ h } \) 
Symbol English Metric
\( a, b, c \) = side \( in \) \( mm \)
\( A_{area} \) = area \( in^2 \) \( mm^2 \)
\( h \) = height \( in \) \( mm \)
\( P \) = perimeter \( in \) \( mm \)

 

Trig Functions
Find A
  • given a c :  \(\; sin( A) \;=\; \dfrac{ a }{ c }\)
  • given b c :  \(\; cos( A) \;=\; \dfrac{ b }{ c }\)
  • given a b :  \(\; tan( A) \;=\; \dfrac{ a }{ b }\)
Find B
  • given a c :  \(\; sin( B) \;=\; \dfrac{ a }{ c }\)
  • given b c :  \(\; cos( B) \;=\; \dfrac{ b }{ c }\)
  • given a b :  \(\; tan( B) \;=\; \dfrac{ b }{ a }\)
 Find a
  • given A c :  \(\; a \;=\; c \cdot sin( A) \)
  • given A b :  \(\; a \;=\; b \cdot tan( A) \)
Find b
  • given A c :  \(\; b \;=\; c \cdot  cos( A) \)
  • given A a :  \(\; b \;=\;  \dfrac{ a }{ tan( A) }\)
Find c
  • given A a :  \(\; c \;=\; \dfrac{ a }{ sin( A) }\)
  • given A b :  \(\; c \;=\; \dfrac{ b }{ cos( A) }\)
  • given a b :  \(\; c \;=\; \sqrt{ a^2 + b^2 } \)
Find Area
  • given a b :  \(\; Area \;=\; \dfrac{ a \cdot b }{ 2 }\)

 

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