Law of Sines Calculator

Three boxes, pre-filled with a = 10, A = 50°, B = 30°. Click Calculate. The law of sines calculator form takes one side and the two angles — the ASA or AAS setup. The law of sines calculator is built for exactly this.

• Side a is any known side; angle A must be the angle opposite it, in degrees.
• Angle B is any second angle; the law of sines calculator finds its opposite side for you.
• Seven answers come back: side b, the third angle C, side c, plus the perimeter, the area, and the two circle radii. The law of sines calculator solves the whole triangle in one run.

The law of sines calculator keeps full precision, so 10, 50, and 30 return b ≈ 6.53, C = 100°, c ≈ 12.86.

This page is more than a static law of sines calculator — it is also an AI tutor you can ask questions:

• Ask in plain words. "One side is 8, its opposite angle is 40°, another angle is 65° — solve the triangle." The AI fills the form and runs this law of sines calculator for you. Any sine law questions work this way.
• Take a photo of your homework. The AI reads the triangle and fills the law of sines calculator boxes for you.
• Ask about the ideas, not just the answer. "Why does the ratio stay constant?" The AI explains step by step — every number still comes from the law of sines calculator, so nothing is guessed.
• Keep asking follow-ups. Ask "why?" as often as you need — follow-up questions are welcome. The test questions below double as law of sines worksheet practice: try each before revealing steps.
• Two sides and a loose angle? That is the SSA ambiguous case — the AI checks for the second triangle with you.

Big angles face big sides — the fact behind every law of sines calculator question. Double the angle and the side across from it roughly doubles too. Pin that proportion down with sines and you get the law of sines. It is "based on proportions": each angle's sine over its opposite side gives one shared ratio for all three pairs.

The picture shows the setup the law of sines calculator solves: one side a = 10 with both its neighbors' angles, 30° and 50°. The third angle is 180 − 30 − 50 = 100°, and now every ratio is a two-knowns equation. Solving any oblique triangle — "finding the measurements of all three angles and all three sides" — is exactly three such steps.

Read the picture first: side a opposite angle A, and so on around. Now the symbols the law of sines calculator uses:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Every number the law of sines calculator returns is one cross-multiplied ratio — that is all the law of sines calculator does, three times.

Give the law of sines calculator its home game: a = 10, A = 50°, and B = 30°.

1. Third angle: $C = 180 - 50 - 30 = 100\degree$.
2. Side b: $b = 10\sin 30\degree / \sin 50\degree \approx 6.53$.
3. Side c: $c = 10\sin 100\degree / \sin 50\degree \approx 12.86$.

These exact numbers come pre-loaded in the law of sines calculator above — click Calculate and the law of sines calculator confirms all three instantly.

Why does the law of sines calculator ask for two angles — and where do the perimeter and area rows come from? Because of the SSA trap. Try a = 6, α = 35°, b = 8:

1. $\sin β = 8 \sin 35\degree / 6 \approx 0.7648$.
2. Arcsine gives $β \approx 49.9\degree$ — but the supplement $130.1\degree$ also fits.
3. Two triangles: $(γ, c) \approx (14.9\degree, 2.7)$ or $(95.1\degree, 10.4)$. The law of sines calculator's ASA form never splits like this.

"Remember that the sine function is positive in both the first and second quadrants." With two angles given, the third is forced and the ambiguity dies — that is the ASA form this law of sines calculator uses.

Two radar stations sit 20 miles apart, handing the law of sines calculator a real job and both see the same aircraft, at elevation angles 35° and 15°. How high does it fly?

1. Triangle angle at the plane: $180 - 35 - 15 = 130\degree$.
2. Distance from a station: $a = 20 \sin 35\degree / \sin 130\degree \approx 14.98$ miles.
3. Height: $h = 14.98 \sin 15\degree \approx 3.88$ — about 3.9 miles up.

No right triangle in sight, yet the law of sines calculator machinery — one side, all angles — nails it with the law of sines calculator's own three rows. That is the law of sines calculator's home turf.

Four slips the law of sines calculator catches every day:

1. Pairing a side with the wrong angle. Each ratio must hold a side over its own opposite angle. Cross-pair b with A and the law of sines calculator formula never had a chance — the law of sines calculator pairs them for you.
2. Feeding radians where degrees belong. The law of sines calculator angle boxes take degrees. Convert first: 50° is about 0.873 radians.
3. Forgetting the angle sum. Two angles known means the third is free: C = 180 − A − B. Grab it before touching a ratio — that is exactly that is exactly the law of sines calculator's second row.
4. Ignoring the SSA second solution. Given two sides and a non-included angle, arcsine may hide a supplement. Test the supplementary angle; if the three angles still total under 180°, a second triangle exists. That is the one spot where the law of sines calculator needs your judgment.