Arbitrary Triangle Calculator

Formulas for calculating an arbitrary triangle

Main theorems:

• Law of cosines: $$c^2=a^2+b^2-2\cdot a\cdot b\cdot \cos \left(\gamma \right)$$
• Law of sines: $$\frac{a}{\sin \left(\alpha \right)}=\frac{b}{\sin \left(\beta \right)}=\frac{c}{\sin \left(\gamma \right)}=2R$$
• Sum of angles of a triangle: $$\alpha +\beta +\gamma ={180}^{\circ }$$

Area of a triangle (S):

• Heron's formula: $$S=\sqrt{p\cdot \left(p-a\right)\cdot \left(p-b\right)\cdot \left(p-c\right)}$$ where: $$p=\frac{a+b+c}{2}$$ is the semiperimeter.
• By two sides and the angle between them: $$S=\frac{1}{2}\cdot a\cdot b\cdot \sin \left(\gamma \right)$$
• By side and height: $$S=\frac{1}{2}\cdot a\cdot h_a$$ where h_a is the height to side a.

Other parameters:

• Height (h_a): $$h_a=\frac{2\cdot S}{a}$$
• Median (m_a): $$m_a=\frac{1}{2}\cdot \sqrt{2\cdot b^2+2\cdot c^2-a^2}$$
• Angle bisector (l_a): $$l_a=\frac{2\cdot b\cdot c\cdot \cos \left(\frac{\alpha }{2}\right)}{b+c}$$
• Inradius (r): $$r=\frac{S}{p}$$
• Circumradius (R): $$R=\frac{a\cdot b\cdot c}{4\cdot S}$$

Note on the ambiguous case (SSA): If two sides and the angle opposite one of them are known, there may be two different triangles that satisfy these conditions (one with an acute angle, the other with an obtuse angle). In this case, the calculator will show both solutions.