Law of Sines Calculator
Solve oblique triangles with the law of sines. Enter two angles and a side, or two sides and an angle, and see every step, both SSA solutions and the area.
Law of Sines Calculator
The Law of Sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant for all three sides. Formula: a/sin(A) = b/sin(B) = c/sin(C)
Triangle Properties
Enter two angles and one side to find another side length.
The Law Of Sines Calculator Is Not For Right Triangles
A common assumption is that a law of sines calculator works like any other trig tool, but it is built for one specific job: solving oblique triangles. An oblique triangle has no 90° angle. If you are trying to find the missing side of a right triangle, the standard sine, cosine, or tangent ratios are faster and simpler. A law of sines calculator solves non‑right triangles when you know two angles and a side (AAS or ASA) or two sides and a non‑included angle (SSA). It computes the missing sides, missing angles, area, and perimeter, and shows the working so you can follow each step. The tool detects the SSA ambiguous case, flags it, and reports both possible triangles when they exist. Enter side lengths, choose degrees or radians, and pick decimal precision up to 4 places. The calculator then outputs triangle properties, a diagram, and step‑by‑step calculations.
- What It Solves: Oblique triangles with no 90° angle
- Input Cases: AAS, ASA, SSA (ambiguous case auto‑detected)
- Outputs: Missing sides, angles, area, perimeter, step‑by‑step working
- Angle Units: Degrees or radians, selectable before calculation
- Decimal Precision: 0 to 4 places; 3 or 4 places sufficient for homework
The Law Of Sines Formula
The law of sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is the same for all three sides. Written as a single proportion:
a / sin(A) = b / sin(B) = c / sin(C)
Where a, b, and c are the side lengths, and A, B, and C are the angles opposite those sides. This constant ratio is also equal to 2R, where R is the circumradius of the triangle, the radius of the circle that passes through all three vertices. In practice you do not need R to solve the triangle; the proportion alone is enough.
The derivation comes from either the circumcircle proof or by dropping altitudes inside the triangle. Both are standard in precalculus texts. What matters when you use this law of sines solver is that you know which side is opposite which angle. Mistaking the pair is the most frequent error in manual calculation.
How To Use The Calculator: AAS, ASA, SSA
The calculator provides three calculation modes. Choose the one that matches your given data.
Find A Side (AAS or ASA)
If you know two angles and one side, you can find another side. The tool asks for side a, angle A, and angle B. It then computes the third angle C = 180° − A − B and uses the law of sines to solve for side b and side c. This mode always produces one unique triangle because the sum of angles fixes the shape.
Find An Angle (SSA)
If you know two sides and an angle opposite one of them, the calculator finds the missing angle. Enter side a, angle A, and side b. The tool applies the law of sines to compute sin(B). If sin(B) > 1, no triangle exists. If sin(B) is valid, it calculates angle B and then checks for a second possible solution. This is the SSA ambiguous case, covered below.
Solve Entire Triangle (AAS or ASA)
This mode accepts two angles and one side, and returns all three sides, all three angles, area, and perimeter. You must specify whether the known side is opposite angle A (AAS) or between angles A and B (ASA). The calculator then finds angle C, applies the law of sines to find the missing sides, and computes area and perimeter.
| Step | Action | Calculation | Result |
|---|---|---|---|
| Given | Side a = 7.5 units, Angle A = 35°, Angle B = 65° | — | a = 7.5, A = 35°, B = 65° |
| 1 | Find angle C | C = 180° − 35° − 65° = 80° | Angle C = 80° |
| 2 | Apply law of sines to find side b | b = a × sin(B) / sin(A) = 7.5 × sin(65°) / sin(35°) | b ≈ 7.5 × 0.9063 / 0.5736 ≈ 11.85 units |
| 3 | Apply law of sines to find side c | c = a × sin(C) / sin(A) = 7.5 × sin(80°) / sin(35°) | c ≈ 7.5 × 0.9848 / 0.5736 ≈ 12.87 units |
| 4 | Calculate area | Area = ½ × a × b × sin(C) = 0.5 × 7.5 × 11.85 × sin(80°) | Area ≈ 43.8 square units |
| 5 | Calculate perimeter | Perimeter = a + b + c = 7.5 + 11.85 + 12.87 | Perimeter ≈ 32.22 units |
The Ambiguous Case (SSA) In One Paragraph
When you know two sides and a non‑included angle (SSA), the law of sines can produce zero, one, or two valid triangles. The second solution exists because the sine of an angle equals the sine of its supplement: sin(θ) = sin(180° − θ). If the given angle A is acute and side a is greater than the height h = b·sin(A) but less than side b, two distinct triangles satisfy the inputs. If side a exactly equals h, the result is a single right triangle. If side a is shorter than h, no triangle exists. If angle A is obtuse (≥ 90°) and side a is not longer than side b, no triangle exists either. The calculator performs these checks automatically and displays both triangles when they appear.
When To Use The Law Of Cosines Instead
The law of cosines is the correct tool when you know two sides and the included angle (SAS) or all three sides (SSS). The law of sines cannot solve SAS or SSS directly because the proportion a/sin(A) = b/sin(B) = c/sin(C) does not contain enough information when you do not have an angle‑side pair. For SAS, start with the law of cosines to find the third side, then use the law of sines to find the remaining angles. For SSS, use the law of cosines to find one angle first. If you attempt to use the law of sines on an SAS problem, you will find that you have two sides and the included angle but no matching angle‑side pair, and the proportion fails.
Common Questions
What is the law of sines used for?
It is used to find missing sides or angles in any oblique (non‑right) triangle when you know two angles and a side (AAS, ASA) or two sides and a non‑included angle (SSA). It does not work for right triangles, SAS, or SSS.
Can I use the law of sines on a right triangle?
Yes, it works mathematically, but it is unnecessary. For a right triangle, the standard trigonometric ratios (sine = opposite/hypotenuse, etc.) are simpler and faster. Use the law of sines only when the triangle has no 90° angle.
What is the ambiguous case (SSA) and why does it matter?
The ambiguous case occurs with two sides and a non‑included angle. Because sin(θ) = sin(180° − θ), there can be zero, one, or two possible triangles. The calculator flags this and shows both solutions when applicable. Ignoring it leads to missing the second valid triangle.
When should I use the law of cosines instead of the law of sines?
Use the law of cosines when you know two sides and the included angle (SAS) or all three sides (SSS). The law of sines cannot solve these cases because the proportion a/sin(A) = b/sin(B) = c/sin(C) requires a known angle‑side pair.
Does the calculator work with radians?
Yes. Select radians under Display Options before entering values. The calculator then outputs angles in radians and uses radian measure in all formulas. Switching units after entering numbers will recalculate the results in the new unit.
