Law of Sines Calculator
Solve law sines problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations. Free to use with no signup required.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Law of Sines Calculator
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Formula: a/sin(A) = b/sin(B) = c/sin(C) = 2R
Worked example — b = 9.7980 | c = 10.9283 | Area = 37.8564 | Circumradius = 5.6569
Formula
a/sin(A) = b/sin(B) = c/sin(C) = 2R
The law of sines states the ratio of each side to the sine of its opposite angle is constant and equals the diameter of the circumscribed circle (2R).
Worked Examples
Example 1: Finding a Side (AAS)
Problem:In triangle ABC, angle A = 45 degrees, angle B = 60 degrees, and side a = 8. Find side b and side c.
Solution:Angle C = 180 - 45 - 60 = 75 degrees. Using law of sines: a/sin(A) = b/sin(B) = c/sin(C) 8/sin(45) = 8/0.707107 = 11.313708 b = 11.313708 x sin(60) = 11.313708 x 0.866025 = 9.798 (approx) c = 11.313708 x sin(75) = 11.313708 x 0.965926 = 10.928 (approx) Area = 0.5 x 8 x 9.798 x sin(75) = 37.856
Result:b = 9.7980 | c = 10.9283 | Area = 37.8564 | Circumradius = 5.6569
Example 2: Ambiguous Case (SSA)
Problem:Given side a = 10, angle A = 30 degrees, and side b = 15. How many triangles are possible?
Solution:sin(B) = b sin(A)/a = 15 x sin(30)/10 = 15 x 0.5/10 = 0.75 B1 = arcsin(0.75) = 48.590 degrees B2 = 180 - 48.590 = 131.410 degrees Solution 1: A=30, B=48.590, C=101.410, c = 10sin(101.410)/sin(30) = 19.615 Solution 2: A=30, B=131.410, C=18.590, c = 10sin(18.590)/sin(30) = 6.374 Both give valid triangles (positive angles summing to 180).
Result:Two solutions: c = 19.6154 or c = 6.3740 | Ambiguous case
Frequently Asked Questions
When should you use the law of sines?
The law of sines is most appropriate in two configurations: AAS (two angles and a non-included side known) and ASA (two angles and the included side known). In both cases, knowing two angles automatically gives you the third angle (since they sum to 180 degrees), and the law of sines efficiently finds the remaining sides. The law of sines can also be used in the SSA case (two sides and a non-included angle), but this is the ambiguous case that may yield zero, one, or two valid triangles. For SAS (two sides and included angle) or SSS (three sides), the law of cosines is the better choice. In navigation and surveying, the law of sines is frequently used in triangulation to determine distances from angular measurements.
What is the ambiguous case of the law of sines?
The ambiguous case occurs when you know two sides and a non-included angle (SSA configuration). Given side a, angle A, and side b, you calculate sin(B) = b sin(A)/a. If sin(B) > 1, no valid triangle exists. If sin(B) = 1, exactly one triangle exists with angle B = 90 degrees. If sin(B) < 1, angle B could be acute (B = arcsin(value)) or obtuse (B = 180 - arcsin(value)). You must check if each possibility gives a valid triangle (all angles positive and summing to 180). This can result in zero, one, or two valid triangles. The ambiguous case is the primary limitation of the law of sines and is why many textbooks recommend using the law of cosines when possible.
How is the law of sines derived?
The law of sines can be derived using the area of a triangle. The area can be expressed three ways: Area = (1/2)ab sin(C) = (1/2)ac sin(B) = (1/2)bc sin(A). Dividing the first expression by the second: ab sin(C)/(ac sin(B)) gives b sin(C)/(c sin(B)) = 1, so b/sin(B) = c/sin(C). Similarly, dividing the first by the third gives a/sin(A) = b/sin(B). Therefore a/sin(A) = b/sin(B) = c/sin(C). An alternative derivation uses the circumscribed circle: inscribe the triangle in a circle of radius R. Using the inscribed angle theorem and properties of the circumradius, each ratio a/sin(A) equals 2R (the diameter of the circumscribed circle). This geometric proof elegantly connects the law to circle geometry.
How does the law of sines relate to the circumradius?
The law of sines has a beautiful geometric connection to the circumscribed circle (circumcircle) of the triangle. The common ratio a/sin(A) = b/sin(B) = c/sin(C) equals exactly 2R, where R is the radius of the circle that passes through all three vertices of the triangle. This means R = a/(2 sin(A)). This relationship provides an elegant way to calculate the circumradius and also explains why the law of sines works: it is fundamentally a statement about the relationship between chords of a circle and the inscribed angles that subtend them. In practice, this means once you calculate the ratio from any known side-angle pair, you immediately know the circumradius as half that ratio.
What are real-world applications of the law of sines?
The law of sines is widely used in practical applications. In surveying and land measurement, triangulation uses the law of sines to determine distances by measuring angles from two known points. In navigation, sailors and pilots use it to calculate distances and bearings from angular observations. In astronomy, it helps determine distances to nearby stars using parallax measurements. In forensic science, accident reconstruction uses the law of sines to calculate vehicle positions and trajectories. In architecture, it aids in designing non-rectangular structures where angular relationships must be precisely calculated. In telecommunications, tower placement and signal coverage calculations rely on triangle solving using the law of sines.
How do you handle the ambiguous case systematically?
To systematically handle the SSA ambiguous case, follow these steps. Given side a, angle A, and side b: First, calculate h = b sin(A), which is the altitude from the vertex opposite side b. If a < h, no triangle exists (side a is too short to reach the base). If a = h, exactly one right triangle exists. If h < a < b, two triangles exist (the ambiguous case). If a >= b, exactly one triangle exists. When two solutions exist, calculate both B1 = arcsin(b sin(A)/a) and B2 = 180 - B1, then verify each gives a positive angle C = 180 - A - B. This systematic approach eliminates guesswork and ensures you find all valid solutions. Law of Sines Calculator automatically detects and displays both solutions when they exist.
Can the law of sines be used for right triangles?
Yes, the law of sines works perfectly for right triangles, and it simplifies nicely. If angle C = 90 degrees, then sin(C) = 1, so the ratio a/sin(A) = b/sin(B) = c/sin(90) = c/1 = c. This means c (the hypotenuse) equals the common ratio, and a = c sin(A) and b = c sin(B). Since sin(A) = opposite/hypotenuse = a/c, this is consistent with the basic SOH-CAH-TOA definitions. The circumradius R = c/2 (half the hypotenuse), which is a well-known property of right triangles. While the law of sines works for right triangles, standard SOHCAHTOA ratios are usually simpler to apply. The law of sines truly shines for non-right triangles where SOHCAHTOA cannot be directly applied.
What is the extended law of sines?
The extended law of sines adds the relationship to the circumradius: a/sin(A) = b/sin(B) = c/sin(C) = 2R, where R is the circumradius of the triangle. This extended form provides additional information beyond side and angle relationships. It is derived from the inscribed angle theorem: in a circle, an inscribed angle is half the central angle that subtends the same arc. This leads to the relationship between a chord (triangle side) and the inscribed angle (opposite angle). The extended law is useful in problems involving circumscribed circles, in proving geometric theorems, and in computational geometry algorithms. It also provides a direct formula for the circumradius without needing to compute the triangle area first.
How do you verify a solution found using the law of sines?
To verify a solution obtained using the law of sines, perform multiple checks. First, confirm all three angles sum to exactly 180 degrees. Second, verify that the ratio a/sin(A) is the same for all three side-angle pairs (they should match to within rounding error). Third, check that each side is positive and that the longest side is opposite the largest angle. Fourth, compute the area using the formula Area = (1/2)ab sin(C) and verify it matches for all three side-angle combinations. Fifth, optionally verify using the law of cosines as an independent check: c2 should equal a2 + b2 - 2ab cos(C). These cross-checks help catch errors in computation, especially in the ambiguous case where it is easy to select an incorrect angle value.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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