Sine Rule Calculator
Find a missing side or angle in any triangle using the sine rule — full step-by-step working shown.
Corners = Angles A, B, C • Opposite edges = Side A, Side B, Side C
Try these examples
Try these examples
Step-by-step solution
Choose a tab, enter your values, and press Calculate to see the working.
Sine Rule Calculator Explained
Core concepts, classroom applications, and plain-English explanations for all audiences.
When would you actually use this?
A hill can't be measured directly. But if you stand 400 m away and measure the angle to the top (12°), and someone further back gives you a second angle, you can triangulate the exact height. Surveyors, foresters, and engineers do exactly this — the sine rule connects angles and sides across any triangle.
Worked example
"In triangle ABC: angle A = 48°, angle B = 72°, side a = 15 cm. Find side b."
a/sin(A) = b/sin(B)
15/sin(48°) = b/sin(72°)
b = 15 × sin(72°) / sin(48°)
b = 15 × 0.9511 / 0.7431 = 19.20 cm
Angle C = 180° − 48° − 72° = 60° — always find the third angle first to check your setup.
The ambiguous case
- → When finding an angle using sin⁻¹, there are always two possible values: θ and 180° − θ. Check both against the known angle sum before discarding one.
- → If both values produce a valid triangle (all angles sum to 180° with none negative), there are genuinely two solutions.
When to use the sine rule
You need a matching angle-side pair (e.g. angle A and side a) plus one more piece of information. If you only have sides and the angle between them, use the cosine rule instead.
Understanding the Sine Rule
When to use the sine rule
Use when you know two angles and any side (AAS or ASA), or two sides and a non-included angle (SSA). It does NOT work if you only know two sides and the included angle — use the cosine rule instead.
The ambiguous case (SSA)
When you know two sides and an angle NOT between them, there can be 0, 1, or 2 valid triangles. Always check if 180° − B also gives a valid triangle by testing whether A + (180° − B) < 180°.
Sine rule formula
a/sin(A) = b/sin(B) = c/sin(C). Rearranged as sin(A)/a = sin(B)/b = sin(C)/c (useful when finding angles). All three ratios equal 2R (circumdiameter).