What is an Obtuse Triangle?
An obtuse triangle is a triangle in which one interior angle is greater than 90°. Because the angles of any triangle sum to 180°, a triangle can have at most one obtuse angle. This calculator takes two sides and the angle between them (a side-angle-side, or SAS, configuration), solves the whole triangle, and tells you whether it is obtuse, right, or acute.
How to Use It
Enter the lengths of sides a and b, then the included angle C in degrees (the angle between sides a and b). The calculator returns the third side c, the area, the perimeter, the remaining angles A and B, and an obtuse/not-obtuse verdict based on the largest angle in the triangle.
The Formulas
The third side comes from the Law of Cosines: \(c^2 = a^2 + b^2 - 2ab\cos C\). The area uses the SAS formula \(\text{Area} = \tfrac{1}{2}\cdot a\cdot b\cdot\sin C\). The remaining angles are recovered with the Law of Cosines again, e.g. $$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$$ If the largest of A, B, C exceeds 90°, the triangle is obtuse.
Worked Example
For \(a = 8\), \(b = 5\), \(C = 120\degree\): $$c^2 = 64 + 25 - 2\cdot 8\cdot 5\cdot\cos(120\degree) = 89 - 80\cdot(-0.5) = 129$$ so \(c \approx 11.358\). $$\text{Area} = \tfrac{1}{2}\cdot 8\cdot 5\cdot\sin(120\degree) \approx 17.32$$ Since \(C = 120\degree > 90\degree\), the triangle is obtuse.
FAQ
What makes a triangle obtuse? Exactly one angle being larger than 90°.
Can a triangle be both obtuse and right? No. A right triangle has a 90° angle; an obtuse triangle has an angle strictly greater than 90°. A triangle can only have one of these.
What if I enter a 90° angle? The result is a right triangle and the verdict will read "not obtuse."
