90
°
17.320508
20
86.60254
47.320508
20
90
°
17.320508
20
86.60254
47.320508
20
The Law of Sines Calculator solves triangles using the fundamental trigonometric relationship that connects side lengths to their opposite angles. The Law of Sines states that in any triangle, the ratio of a side to the sine of its opposite angle is constant: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$. This powerful tool finds unknown sides and angles when you know one side, its opposite angle, and one additional angle.
Given side $$a$$, angle $$A$$ (opposite to side $$a$$), and angle $$B$$, the calculator first determines the third angle using $$C = 180° - A - B$$, then computes the missing sides: $$b = \frac{a \sin B}{\sin A}$$ and $$c = \frac{a \sin C}{\sin A}$$. This AAS (Angle-Angle-Side) configuration always yields a unique triangle, making it the most straightforward application of the Law of Sines.
The Law of Sines is indispensable in surveying and navigation. Surveyors use triangulation to measure distances to inaccessible points by measuring angles from two known positions and applying the Law of Sines. Similarly, celestial navigation uses the law to determine positions from star observations. In engineering, it helps analyze force systems where forces act at known angles.
One important consideration is the ambiguous case (SSA configuration), where you know two sides and an angle opposite one of them. In this scenario, zero, one, or two valid triangles may exist. This calculator avoids the ambiguous case by taking two angles and a side, which always produces a unique solution. For SSA problems, additional analysis is needed to determine if multiple solutions exist.
The calculator also computes the triangle area using the formula $$A_{\text{area}} = \frac{1}{2} a b \sin C$$, which is derived from the standard base-height area formula combined with trigonometry. This provides a complete solution: all three sides, all three angles, and the area from just three inputs.
The Law of Sines also connects to the circumradius of the triangle. The common ratio equals the diameter of the circumscribed circle: $$\frac{a}{\sin A} = 2R$$. This relationship is fundamental in circle geometry and provides an elegant link between a triangle and its circumscribing circle.
The Law of Sines states:
$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$
Step 1: Find the third angle:
$$C = 180° - A - B$$
Step 2: Solve for side $$b$$:
$$b = \frac{a \cdot \sin B}{\sin A}$$
Step 3: Solve for side $$c$$:
$$c = \frac{a \cdot \sin C}{\sin A}$$
Step 4: Compute the area:
$$\text{Area} = \frac{1}{2} \cdot a \cdot b \cdot \sin C$$
Side b is the length opposite angle B, computed proportionally from the known side-angle pair. Angle C is the remaining angle (180° minus the two given angles). Side c is the length opposite angle C. The area uses the two-side-included-angle formula. Ensure that A + B < 180°, otherwise no valid triangle exists. If the output shows negative or infinite values, the input angles are invalid.
Inputs
Results
A 30-60-90 triangle with shortest side 10. Side b = 10·sin(60°)/sin(30°) = 10·0.866/0.5 = 17.32. Hypotenuse c = 20.
Inputs
Results
All angles < 90°. b = 8·sin(70°)/sin(45°) ≈ 10.63. Area ≈ 38.55 square units.
The Law of Sines states that in any triangle, the ratio of a side length to the sine of its opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C). This ratio also equals the diameter of the circumscribed circle (2R).
Use the Law of Sines when you know an angle-side pair (AAS, ASA, or SSA configurations). Use the Law of Cosines when you know three sides (SSS) or two sides and the included angle (SAS). The Law of Cosines is also preferred to avoid the ambiguous case.
The ambiguous case occurs in SSA (Side-Side-Angle) configurations where you know two sides and an angle opposite one of them. Depending on the values, zero, one, or two valid triangles may exist. This calculator uses AAS input, which always gives a unique solution.
No. Since all three angles of a triangle must sum to exactly 180°, the two input angles must sum to less than 180°. If A + B ≥ 180°, no valid triangle exists and the calculator will produce invalid results.
The Law of Sines ratio equals the diameter of the circumscribed circle: a/sin(A) = 2R, where R is the circumradius. This means R = a/(2·sin(A)), providing a direct way to find the circumscribed circle's radius.
The calculator uses JavaScript's built-in Math.sin and Math.PI with double-precision floating-point arithmetic (about 15-16 significant digits). Results are displayed to 6 decimal places for sides and 4 for angles, providing more than sufficient accuracy for practical applications.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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