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.
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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).
Last reviewed: December 2025
Worked Examples
Example 1: Finding a Side (AAS)
Example 2: Ambiguous Case (SSA)
Background & Theory
The Law of Sines Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Law of Sines Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
Sources & References
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.\nUsing law of sines: a/sin(A) = b/sin(B) = c/sin(C)\n8/sin(45) = 8/0.707107 = 11.313708\nb = 11.313708 x sin(60) = 11.313708 x 0.866025 = 9.798 (approx)\nc = 11.313708 x sin(75) = 11.313708 x 0.965926 = 10.928 (approx)\nArea = 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\nB1 = arcsin(0.75) = 48.590 degrees\nB2 = 180 - 48.590 = 131.410 degrees\n\nSolution 1: A=30, B=48.590, C=101.410, c = 10sin(101.410)/sin(30) = 19.615\nSolution 2: A=30, B=131.410, C=18.590, c = 10sin(18.590)/sin(30) = 6.374\nBoth 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.
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.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy