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Oblique Triangle (Acute and Obtuse)

  • acute triangle 1obtuse triangle 2Oblique triangle (a two-dimensional figure) is tilted at an angle, not horizontal or vertical.
  • Acute oblique triangle (a two-dimensional figure) has all three angles less than 90°.
  • Circumcircle is a circle that passes through all the vertices of a two-dimensional figure.
  • Inscribed circle is the largest circle possible that can fit on the inside of a two-dimensional figure.
  • Obtuse oblique triangle (a two-dimensional figure) has one of the three angles more than 90°.
  • Semiperimeter is one half of the perimeter.
  • x + y + z = 180°
  • 3 edges
  • 3 vertexs
  • 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

 

scalene triangle 5hobtuse triangle 4h

obtuse triangle angle bisector 1

obtuse triangle circ 1

 

 

 

 

 

 

scalene triangle 5mscalene triangle 5tobtuse triangle 4t

scalene triangle 4

 

 

 

 

 

 

 

area of an Oblique triangle formulas

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

\( A_{area} \;=\;   a \cdot b \cdot \dfrac{ \sin( y) }{ 2 } \) 
Symbol English Metric
\( A_{area} \) = area \(\large{ in^2 }\)   \(\large{ mm^2 }\)
\( a, b, c \) = edge \(\large{ in }\) \(\large{ mm }\)

 

Circumcircle of an Oblique triangle formulas

\( R \;=\;   \sqrt {   \dfrac{ a^2 \cdot b^2 \cdot c^2 }{  \left( a + b + l \cdot c  \right)   \cdot   \left( - a + b + c  \right)   \cdot   \left( a - b + c  \right)    \cdot    \left( a + b - c  \right)   }   }  \) 

\( R \;=\;   \dfrac{ a \cdot b \cdot c }{   4 \cdot  \sqrt  {  s \cdot \left( s - a  \right)   \cdot   \left( s - b  \right)    \cdot    \left( s - c  \right)  }     }  \) 
Symbol English Metric
\(\large{ R }\) = outcircle \(\large{ in }\)   \(\large{ mm }\)
\(\large{ a, b, c }\) = edge \(\large{ in }\) \(\large{ mm }\)
\(\large{ s }\) = semiperimeter \(\large{ in }\) \(\large{ mm }\)

 

height of an Oblique triangle formula
\( h \;=\;   2 \cdot \dfrac{A_{area} }{ b } \) 
Symbol English Metric
\(\large{ h }\) = height \(\large{ in }\) \(\large{ mm }\)
\(\large{ A_{area} }\) = area \(\large{ in^2 }\)   \(\large{ mm^2 }\)
\(\large{ a, b, c }\) = edge \(\large{ in }\) \(\large{ mm }\)

 

inscribed circle of an Oblique triangle formula
\( r \;=\;     \sqrt{   \dfrac{  \left( s - a  \right)  \cdot \left( s - b  \right)  \cdot \left( s - c  \right)  }{ s }   }  \) 
Symbol English Metric
\( r \) = incircle \(\large{ in }\)   \(\large{ mm }\)
\( a, b, c \) = edge \(\large{ in }\) \(\large{ mm }\)

 

perimeter of an Oblique triangle formula
\(P \;=\;   a + b + c \) 
Symbol English Metric
\( P \) = perimeter \(\large{ in }\)   \(\large{ mm }\)
\( a, b, c \) = edge \(\large{ in }\) \(\large{ mm }\)

 

semiperimeter of an Oblique triangle formula
\( s \;=\;   \dfrac{ a + b + c }{ 2  }   \) 
Symbol English Metric
\(\large{ s }\) = semiperimeter \(\large{ in }\)   \(\large{ mm }\)
\(\large{ a, b, c }\) = edge \(\large{ in }\) \(\large{ mm }\)

 

Side of an Oblique triangle formulas

\( a \;=\;   P - b - c   \) 

\( a \;=\;   2\cdot \dfrac{ A_{area} }{ b\cdot \sin( y) } \) 

\( b \;=\;   P - a - c   \) 

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

\( c \;=\;   P - a - b   \)
Symbol English Metric
\(\large{ a, b, c }\) = edge \(\large{ in }\) \(\large{ mm }\)
\(\large{ A_{area} }\) = area \(\large{ in^2 }\)   \(\large{ mm^2 }\)
\(\large{ P }\) = perimeter \(\large{ in }\) \(\large{ mm }\)

  

Trig Functions
Find p
  • given a b c :  \(\; p \;=\; a + b + c \)
Find s
  • given a b c :  \(\; s \;=\;  \dfrac{ a + b + c  }{  2 }\)
Find d
  • given a b c s :  \(\; d \;=\;   \dfrac{ b^2 + c^2 - a^2 }{  2 \cdot b }\)
Find e
  • given a b c s :  \(\; e \;=\;  \dfrac{ a^2 + b^2 - c^2 }{  2 \cdot b }\)
Find h
  • given c A :  \(\; h \;=\; c \cdot   sin(A)\)
  • given a C :  \(\; h \;=\; a \cdot sin(C) \)
Find Area
  • given a b c :  \(\; A_{area} \;=\; \dfrac{1}{2} \cdot (b \cdot h) \)
  • given a b c :  \(\; A_{area} \;=\; \sqrt{ s \cdot (s - a)  \cdot  (s - b) \cdot  (s - c)    }\)
  • given C a b :  \(\; A_{area} \;=\; \dfrac{1}{2} \cdot ( a \cdot b) \cdot  \sin( c) \)
Find A
  • given a b c s :  \(\; sin \dfrac{1}{2} (A) \;=\; \sqrt{ \dfrac{ (s - b) \cdot (s - c) }{ b \cdot c } }\)
  • given a b c s :  \(\; cos \dfrac{1}{2} (A) \;=\; \sqrt{  \dfrac{ s \cdot (s - a)  }{ b \cdot c } }\)
  • given a b c s :  \(\; tan \dfrac{1}{2} (A) \;=\; \sqrt{  \dfrac{ (s - b) \cdot (s - c)  }{  s \cdot (s - a) } } \)
  • given c h :  \(\; sin(A) \;=\; \dfrac{ h }{ c }\)
  • given B a b :  \(\; sin( A) \;=\; \dfrac{ a \cdot sin(B) }{  b }\)
  • given B a c :  \(\; A \;=\; \dfrac{1}{2} \cdot (A + C) + \dfrac{1}{2} \cdot (A - C) \)
  • given C a b :  \(\; A \;=\; \dfrac{1}{2} \cdot  (A + B) + \dfrac{1}{2} \cdot (A - B) \)
  • given C a c :  \(\; sin(A) \;=\;  \dfrac{ a \cdot sin(C) }{ c }\)
Find B
  • given a b c s :  \(\; sin \dfrac{1}{2} (B) \;=\; \sqrt{ \dfrac{ (s - a) \cdot (s - c) }{ a \cdot c} }\)
  • given a b c s :  \(\; cos \dfrac{1}{2} (B) \;=\; \sqrt{ \dfrac{ s \cdot (s - b) }{  a \cdot c} }\)
  • given a b c s :  \(\; tan \dfrac{1}{2} (B) \;=\; \sqrt{ \dfrac{ (s - a) \cdot ( s - c)   }{  s \cdot (s - b) }  }\)
  • given a h :  \(\; sin( B) \;=\; \dfrac{ h }{ a } \)
  • given A a b :  \(\; sin( B) \;=\; \dfrac{ b \cdot sin( A) }{ a }\)
  • given A b c :  \(\; B \;=\; \dfrac{1}{2} \cdot (B + C) + \dfrac{1}{2} \cdot (B - C) \)
  • given C a b :  \(\; B \;=\; \dfrac{1}{2} \cdot (A+B) - \dfrac{1}{2} \cdot (A-B)  \)
  • given C a c :  \(\; sin( B) \;=\; \dfrac{ b \cdot sin( C)  }{  c }\)
Find C
  • given a b c s :  \(\; sin \dfrac{1}{2} (C) \;=\; \sqrt{ \dfrac{ (s - a) \cdot (s - b)  }{ a \cdot b} }\)
  • given a b c s :  \(\; cos \dfrac{1}{2} (C) \;=\; \sqrt{ \dfrac{ s \cdot (s - c) }{  a \cdot b} }\)
  • given a b c s :  \(\; tan \dfrac{1}{2} (C) \;=\; \sqrt{ \dfrac{ (s - a) \cdot (s - b)  }{ s \cdot (s - c) } }\)
  • given A a c :  \(\; sin( C) \;=\;  \dfrac{ c \cdot sin( A)  }{  a }\)
  • given A b c :  \(\; C \;=\; \dfrac{1}{2} \cdot (B + C) - \dfrac{1}{2} \cdot (B - C) \)
  • given B a c :  \(\; C \;=\; \dfrac{1}{2} \cdot (A + C) - \dfrac{1}{2} \cdot (A - C) \)
  • given B b c :  \(\; sin( C) \;=\; \dfrac{ c \cdot sin( B)  }{ c }\)
Find a
  • given A B b :  \(\; a \;=\;  \dfrac{ b \cdot sin( A) }{ sin( B) }\)
  • given A B c :  \(\; a \;=\;  \dfrac{ c cdot sin( A) }{ sin (A+B) }\)
  • given A C b :  \(\; a \;=\;  \dfrac{ b \cdot sin( A) }{ sin (A+C)  }\)
  • given A C c :  \(\; a \;=\;  \dfrac{ c \cdot sin( A) }{ sin( C) }\)
  • given B C b :  \(\; a \;=\;  \dfrac{ b \cdot sin (A + C) }{ sin( B) }\)
  • given B C c :  \(\; a \;=\;  \dfrac{ c \cdot sin (A + C) }{ sin( C) }\)
  • given A b c :  \(\; a \;=\;  \sqrt{ b^2 + c^2 - (2 \cdot b \cdot c)  cdot  \cos( A)  } \)
Find b
  • given A B a :  \(\; b \;=\;  \dfrac{ a \cdot sin( B) }{ sin( A) }\)
  • given A B c :  \(\; b \;=\;  \dfrac{ c \cdot sin( B) }{ sin (A + B) }\)
  • given A C a :  \(\; b \;=\;  \dfrac{ a \cdot sin (A + C) }{ sin( A) }\)
  • given A C c :  \(\; b \;=\;  \dfrac{ c \cdot sin (A + C) }{ sin( C) }\)
  • given B C a :  \(\; b \;=\; \dfrac{  a \cdot sin( B) }{ sin (B + C) }\)
  • given B C c :  \(\; b \;=\;  \dfrac{ c \cdot sin( B) }{ sin( C) }\)
  • given B a c :  \(\; b \;=\;  \sqrt{ a^2 + c^2 - (2 \cdot a \cdot c) \cdot cos( B) } \)
Find c
  • given A B a :  \(\; c \;=\;  \dfrac{ a \cdot sin (A + B) }{ sin( A) }\)
  • given A B b :  \(\; c \;=\;  \dfrac{ b \cdot sin (A + B) }{ sin( B) }\)
  • given A C a :  \(\; c \;=\;  \dfrac{ a \cdot sin( C) }{ sin( A) }\)
  • given A C b :  \(\; c \;=\;  \dfrac{ b \cdot sin( C) }{ sin (A + C) }\)
  • given B C a :  \(\; c \;=\;  \dfrac{ a \cdot sin( C) }{ sin (B + C) }\)
  • given B C b :  \(\; c \;=\;  \dfrac{ b \cdot sin( C) }{ sin( B) }\)
  • given C a b :  \(\; c \;=\;  \sqrt{ a^2 + b^2 - (2 \cdot a \cdot b)  \cdot cos( C) } \)

 

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