What This Calculator Does
This tool finds an unknown angle of a triangle using the Law of Sines. If you know two sides and an angle opposite one of them (the SSA configuration), you can solve for the angle opposite the other known side. It applies to any triangle — right, acute, or obtuse — and works in degrees.
The Formula
The Law of Sines states that \( a / \sin A = b / \sin B \). Rearranging to find angle A gives:
$$A = \arcsin\!\left(\frac{\text{Side }a \cdot \sin\!\left(\text{Angle }B\right)}{\text{Side }b}\right)$$
Here a is the side opposite the unknown angle A, while b and B are a known side–angle pair that lie opposite each other.
How to Use It
Enter side a (opposite the angle you want), side b, and the known angle B in degrees. The calculator computes \( \sin(A) = a\cdot\sin(B)/b \) and then takes the inverse sine to return angle A. If the ratio exceeds 1, no triangle exists with those measurements, so the value is clamped to 90°.
Worked Example
Suppose a = 7, b = 10, and B = 40°. Then $$\sin(A) = 7 \cdot \sin(40°) / 10 = 7 \cdot 0.642788 / 10 = 0.449951.$$ Taking arcsin gives A ≈ 26.74°. That is the missing angle.
FAQ
Why might there be two answers? In the ambiguous SSA case, both A and 180° − A can be valid. This calculator returns the acute solution from arcsin; check whether the obtuse alternative also fits your triangle.
What if sin(A) is greater than 1? No triangle can satisfy those side and angle values, so the result is capped at 90°.
Do the units matter? Sides can be any consistent length unit, since only their ratio is used. The angle must be entered in degrees.
