Online calculator to calculate the third side \( c \) of a triangle given its two sides \( a \) and \( b \) and area \( A \).
The area \(A\) of a triangle with two sides \(a\) and \(b\) making an angle \(\alpha\) is given by:
\[ A = \frac12 ab \sin(\alpha) \]Use the cosine rule to express side \(c\) in terms of \(a\), \(b\), and \(\cos(\alpha)\):
\[ c^2 = a^2 + b^2 - 2ab\cos(\alpha) \]Solve for \(\cos(\alpha)\):
\[ \cos(\alpha)=\frac{a^2+b^2-c^2}{2ab} \]Using the identity \(\sin(\alpha)=\sqrt{1-\cos^2(\alpha)}\), rewrite the area formula as:
\[ A=\frac12 ab\sqrt{1-\left(\frac{a^2+b^2-c^2}{2ab}\right)^2} \]Simplifying gives:
\[ A=\frac14\sqrt{4a^2b^2-(a^2+b^2-c^2)^2} \]Square both sides and solve for the third side \(c\). This yields two possible solutions:
\[ c_1=\sqrt{a^2+b^2+\sqrt{4a^2b^2-16A^2}} \] \[ c_2=\sqrt{a^2+b^2-\sqrt{4a^2b^2-16A^2}} \]Note that the problem has:
1) Two solutions if \(4a^2b^2-16A^2 > 0\)
2) One solution (or two equal solutions) if \(4a^2b^2-16A^2 = 0\)
3) No solutions if \(4a^2b^2-16A^2 < 0\)
Enter the area, sides \(a\) and \(b\) and press "Calculate". The outputs are the discriminant, number of solutions, and the third sides \(c_1\) and \(c_2\) of the triangle if the problem has a solution. You can adjust the number of decimal places using the input field below.
Given area (A) and sides (a, b)
Area of Triangles
Cosine Law Problems
Online Geometry Calculators and Solvers