# Oblique Triangle (Acute and Obtuse)

on . Posted in Plane Geometry

• Oblique 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

## area of an Oblique triangle formulas

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

$$\large{ A_{area} = a\;b\; \frac {\sin y} {2} }$$

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

## Circumcircle of an Oblique triangle formulas

$$\large{ R = \sqrt { \frac { a^2 \; b^2 \; c^2 } { \left( a \;+\; b \;+\;l \;c \right) \; \left( - a \;+\; b \;+\; c \right) \; \left( a \;-\; b \;+\; c \right) \; \left( a \;+\; b \;-\; c \right) } } }$$

$$\large{ R = \frac { a \; b \; c } { 4 \; \sqrt { s \; \left( s \;-\; a \right) \; \left( s \;-\; b \right) \; \left( s \;-\; c \right) } } }$$

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

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

$$\large{ r = \sqrt { \frac { \left( s \;-\; a \right) \; \left( s \;-\; b \right) \; \left( s \;-\; c \right) } { s } } }$$
Symbol English Metric
$$\large{ r }$$ = incircle $$\large{ in }$$   $$\large{ mm }$$
$$\large{ a, b, c }$$ = edge $$\large{ in }$$ $$\large{ mm }$$

## perimeter of an Oblique triangle formula

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

## semiperimeter of an Oblique triangle formula

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

$$\large{ a = P - b - c }$$

$$\large{ a = 2\; \frac {A_{area}} {b\;\sin y} }$$

$$\large{ b = P - a - c }$$

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

$$\large{ c = P - a - b }$$

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 :  $$\; s= (a+b+c) \div 2$$
Find d
• given a b c s :  $$\; d= (b^2+c^2-a^2) \div 2b$$
Find e
• given a b c s :  $$\; e= (a^2+b^2-c^2) \div 2b$$
Find h
• given c A :  $$\; h= c*sin \;A$$
• given a C :  $$\; h= a*sin \;C$$
Find Area
• given a b c :  $$\; A_{area}=\frac {1}{2}(bh)$$
• given a b c :  $$\; A_{area}= \sqrt { s(s-a)(s-b)(s-c) }$$
• given C a b :  $$\; A_{area}= \frac{1}{2} ab* \sin \; c$$
Find A
• given a b c s :  $$\; \sin \frac{1}{2} A= \sqrt {(s-b)(s-c) \div bc }$$
• given a b c s :  $$\; \cos \frac{1}{2} A= \sqrt {s(s-a) \div bc }$$
• given a b c s :  $$\; \tan \frac{1}{2} A= \sqrt {(s-b)(s-c) \div s(s-a) }$$
• given c h :  $$\;\ sin \; A= \frac {h}{c}$$
• given B a b :  $$\;\ sin \; A= a* sin\; B \div b$$
• given B a c :  $$\; A= \frac{1}{2}(A+C)+\frac{1}{2}(A-C)$$
• given C a b :  $$\; A= \frac{1}{2}(A+B)+\frac{1}{2}(A-B)$$
• given C a c :  $$\;\ sin \; A= a* sin \; C \div c$$
Find B
• given a b c s :  $$\; \sin \frac{1}{2} B= \sqrt {(s-a)(s-c) \div ac}$$
• given a b c s :  $$\; \cos \frac{1}{2} B= \sqrt {s(s-b) \div ac}$$
• given a b c s :  $$\; \tan \frac{1}{2} B= \sqrt {(s-a)(s-c) \div s(s-b)}$$
• given a h :  $$\;\ sin \; B= \frac {h}{a}$$
• given A a b :  $$\;\ sin B= b* sin \; A \div a$$
• given A b c :  $$\;\ B= \frac{1}{2}(B+C)+\frac{1}{2}(B-C)$$
• given C a b :  $$\; B= \frac{1}{2}(A+B)-\frac{1}{2}(A-B)$$
• given C a c :  $$\;\ sin \; B= b* sin \; C \div c$$
Find C
• given a b c s :  $$\; \sin \frac{1}{2} C= \sqrt {(s-a)(s-b) \div ab}$$
• given a b c s :  $$\; \cos \frac{1}{2} C= \sqrt {s(s-c) \div ab}$$
• given a b c s :  $$\; \tan \frac{1}{2} C= \sqrt {(s-a)(s-b) \div s(s-c) }$$
• given A a c :  $$\;\ sin \; C= c* sin \; A \div a$$
• given A b c :  $$\; C= \frac{1}{2}(B+C)-\frac{1}{2}(B-C)$$
• given B a c :  $$\; C= \frac{1}{2}(A+C)-\frac{1}{2}(A-C)$$
• given B b c :  $$\;\ sin \; C= c* sin \; B \div c$$
Find a
• given A B b :  $$\; a= b* \sin \; A \div \sin \; B$$
• given A B c :  $$\; a= c* \sin \; A \div \sin (A+B)$$
• given A C b :  $$\; a= b* \sin \; A \div \sin (A+C)$$
• given A C c :  $$\; a= c* \sin \; A \div \sin \; C$$
• given B C b :  $$\; a= b* \sin (A+C) \div \sin \; B$$
• given B C c :  $$\; a= c* \sin (A+C) \div \sin \; C$$
• given A b c :  $$\; a= \sqrt {b^2+c^2-2bc * \cos \; A}$$
Find b
• given A B a :  $$\; b= a* \sin \; B \div \sin \; A$$
• given A B c :  $$\; b= c* \sin \; B \div \sin (A+B)$$
• given A C a :  $$\; b= a* \sin (A+C) \div \sin \; A$$
• given A C c :  $$\; b= c* \sin (A+C) \div \sin \; C$$
• given B C a :  $$\; b= a* \sin \; B \div \sin (B+C)$$
• given B C c :  $$\; b= c* \sin \; B \div \sin \; C$$
• given B a c :  $$\; b= \sqrt {a^2+c^2-2ac * \cos B}$$
Find c
• given A B a :  $$\; c= a* \sin (A+B) \div \sin \; A$$
• given A B b :  $$\; c= b* \sin (A+B) \div \sin \; B$$
• given A C a :  $$\; c= a* \sin \; C \div \sin \; A$$
• given A C b :  $$\; c= b* \sin \; C \div \sin (A+C)$$
• given B C a :  $$\; c= a* \sin \; C \div \sin (B+C)$$
• given B C b :  $$\; c= b* \sin \; C \div \sin \; B$$
• given C a b :  $$\; c= \sqrt {a^2+b^2-2ab * \cos \; C}$$

Tags: Triangle