Unit Circle Lab
Law of cosines Unit circle

Law of Sines Calculator

This law of sines calculator solves AAS, ASA and SSA triangles. For SSA, the ambiguous case, it tells you whether there are zero, one or two triangles, explains why, and draws them to scale.

try

side a is opposite angle A, b opposite B, c opposite C · angles in degrees

h = 5A 30°Cb = 10B₁B₂Arc of radius a = 6 meets the base twiceA 30°B 56.44°C 93.56°a = 6b = 10c = 11.977Triangle 1A 30°B 123.56°C 26.44°a = 6b = 10c = 5.344Triangle 2

Ambiguous case (SSA)

2 triangles c = 11.977 or 5.344 h = b·sin A = 5

Triangle 1

a
6
b
10
c
11.977
A
30°
B
56.44°
C
93.56°
Area
29.942
Perimeter
27.977

Triangle 2

a
6
b
10
c
5.344
A
30°
B
123.56°
C
26.44°
Area
13.359
Perimeter
21.344

Steps

rounded to 3–4 decimals
  1. This is the ambiguous case: you know angle A, its opposite side a, and side b.
  2. Height from C to the base: h = b · sin A = 10 · sin 30° = 5.
  3. h = 5 < a = 6 < b = 10, so side a can swing to two positions: two triangles.
  4. sin B = b · sin A / a = 10 · sin 30° / 6 = 0.833, so B = arcsin(0.833) = 56.44°.
  5. The supplement also works: B₂ = 180° − 56.44° = 123.56°, because A + B₂ = 153.56° < 180°.
  6. Triangle 1: C = 180° − A − B = 93.56°, and c = a · sin C / sin A = 11.977.
  7. Triangle 2: C = 180° − A − B = 26.44°, and c = a · sin C / sin A = 5.344.

How it works

Pick what you know, then read every side and angle.

  1. 01 · case

    Choose the case

    AAS: two angles and a side opposite one of them. ASA: two angles and the side between them. SSA: two sides and an angle opposite one of them.

  2. 02 · solve

    Use the ratio

    The law of sines links each side to the sine of its opposite angle. One complete pair unlocks the rest.

  3. 03 · check

    Watch the ambiguous case

    With SSA the calculator compares side a with the height h = b·sin A and draws the arc that finds 0, 1 or 2 triangles.

§1The law of sines

In any triangle, the ratio of a side to the sine of the angle opposite it is the same for all three sides:

a / sin A = b / sin B = c / sin C

Here a is the side opposite angle A, b is opposite B and c is opposite C. If you know one complete pair, a side and its opposite angle, plus one more side or angle, you can find everything else. The angles also add up to 180°, which gives the third angle for free when two are known.

§2When to use it

  • AAS: two angles and a side that is not between them. Find the third angle, then use the ratio.
  • ASA: two angles and the side between them. The third angle is opposite the known side, so that gives the complete pair.
  • SSA: two sides and an angle opposite one of them. This is the ambiguous case below.

If you know three sides (SSS) or two sides and the angle between them (SAS), there is no complete pair yet. Use the law of cosines calculator instead.

§3The ambiguous case (SSA)

Suppose you know angle A, its opposite side a and another side b. Draw angle A with side b along one of its rays; that fixes the vertex C. Side a now has to reach from C to the other ray, and it can swing like a compass. Whether it reaches depends on the height from C to that ray, h = b · sin A.

Number of triangles when A is acute
ConditionTrianglesWhy
a < h0a is too short to reach the base
a = h1a touches the base at a right angle
h < a < b2a meets the base at two points
a ≥ b1the second point falls behind vertex A

If A is right or obtuse, side a must be the longest side: there is one triangle when a > b and none when a ≤ b. The dashed arc in the diagram is side a swinging around C; each point where it meets the base is a possible vertex B.

§4Worked example with two triangles

A = 30°, a = 6, b = 10. h = 10 · sin 30° = 5, and 5 < 6 < 10 → two triangles

sin B = b · sin A / a = 10 · 0.5 / 6 ≈ 0.8333 → B₁ ≈ 56.44°

B₂ = 180° − 56.44° = 123.56° (allowed, because 30° + 123.56° < 180°)

C₁ ≈ 93.56°, c₁ ≈ 11.977 · C₂ ≈ 26.44°, c₂ ≈ 5.344

Your calculator’s arcsin only returns the acute angle B₁. The obtuse angle 180° − B₁ has the same sine, so you always have to check it yourself.

§5Area and the circumradius

Once the triangle is solved, its area is ½ · a · b · sin C (any two sides and the angle between them work). The common ratio has a meaning too: a / sin A equals 2R, the diameter of the circle that passes through all three vertices.

§6Common mistakes

  • Missing the second triangle in SSA. Always test 180° − B₁.
  • Pairing a side with the wrong angle. Each side goes with the angle opposite it, never an adjacent one.
  • Calculator in radian mode: sin 30 in radians is about −0.988, not 0.5.
  • Rounding too early. Keep full precision until the last step; this calculator rounds only what it shows.

§7FAQ

Why can SSA give two triangles?

Because sin B = sin(180° − B). An acute angle and its supplement have the same sine, so the law of sines alone cannot choose between them; both work when the angles still add up to less than 180°.

Can the law of sines solve a triangle from three sides?

No. With three sides you have no angle, so there is no complete pair. Start with the law of cosines.

Does it work for right triangles?

Yes. With C = 90°, sin C = 1, and the ratios become the usual sine definitions, such as sin A = a / c.

Which units does the calculator use?

Angles in degrees and sides in any unit, as long as all sides use the same one. Results are rounded to two decimals for angles and three for sides.