Sin Triangle
Solve any triangle via the Law of Sines online. Enter three values to find missing sides, angles, area, and circumradius with our free triangle calculator.
About This Calculator
The Sin Triangle (Law of Sines) calculator solves any triangle by applying the fundamental trigonometric relationship: the ratio of a side length to the sine of its opposite angle is constant for all three sides. Formally stated as a/sin(A) = b/sin(B) = c/sin(C), this law works for every triangle -- acute, obtuse, or right -- making it one of the most versatile tools in trigonometry and geometry.
To use this calculator, enter any three values from the four input fields: side a, angle A (the angle opposite side a), side b, and angle B (opposite side b). At least one complete side-angle pair is required. The calculator will compute the missing fourth value using the Law of Sines, then determine the third angle C (since A + B + C = 180 deg), side c, the triangle's area using the formula Area = 1/2ab·sin(C), and the circumradius R = a/(2·sin(A)).
This tool is essential for students learning trigonometry and geometry, for professionals in surveying, navigation, engineering, and architecture who need to solve triangle problems quickly. Whether you're working with AAS (angle-angle-side), ASA (angle-side-angle), or SSA (side-side-angle) configurations, this calculator handles all standard cases. Note that the SSA ambiguous case may have two possible solutions; this calculator returns the primary acute-angle solution.
Frequently Asked Questions
What is the Law of Sines?
The Law of Sines states that for 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 triangle's circumcircle (2R). The law applies to all triangles, not just right triangles, making it a fundamental tool in trigonometry.
How do I use the Law of Sines to solve a triangle?
To solve a triangle using the Law of Sines, you need to know at least two angles and one side (AAS or ASA), or two sides and an angle opposite one of them (SSA). Enter any three values including at least one complete side-angle pair into this calculator. It will compute all missing sides and angles automatically using the formula a/sin(A) = b/sin(B) = c/sin(C).
What is the ambiguous case (SSA) in the Law of Sines?
The ambiguous case (SSA) occurs when you know two sides and a non-included angle. Depending on the values, there may be zero, one, or two possible triangles. If side a is shorter than side b but longer than the altitude from angle B (a > b x sin(A)), two triangles are possible. This calculator returns the primary acute-angle solution. If your triangle may have an obtuse angle, verify with the supplementary angle (180 deg - computed angle).
What are the real-world applications of the Law of Sines?
The Law of Sines is used extensively in surveying and navigation to calculate distances and positions using triangulation. Engineers apply it in structural analysis for truss designs and bridge construction. Astronomers use it to calculate distances to celestial bodies. In geography, it helps measure land areas. The law is also fundamental in physics for vector decomposition and in computer graphics for 3D rendering and terrain mapping.
Can the Law of Sines be used on right triangles?
Yes, the Law of Sines works for all triangles, including right triangles. In a right triangle with angle C = 90 deg, sin(C) = 1, so the law simplifies to a/sin(A) = b/sin(B) = c. This is consistent with the basic trigonometric ratios (SOH CAH TOA), where sin(A) = opposite/hypotenuse = a/c.
What is the difference between Law of Sines and Law of Cosines?
Use the Law of Sines when you know two angles and one side (AAS, ASA), or two sides and a non-included angle (SSA). Use the Law of Cosines (c^2 = a^2 + b^2 - 2ab x cos(C)) when you know three sides (SSS) or two sides and the included angle (SAS). The Law of Cosines works for any triangle but is more complex -- the Law of Sines is simpler when you know opposite side-angle pairs.
How do you calculate the area of a triangle using the Law of Sines?
The area of a triangle can be calculated using two sides and the sine of their included angle: Area = 1/2 x a x b x sin(C), where C is the angle between sides a and b. This formula is derived from the basic Area = 1/2 x base x height, where height = b x sin(C). This calculator automatically computes the area for you.
What is the circumradius of a triangle?
The circumradius (R) of a triangle is the radius of the circle that passes through all three vertices. Using the Law of Sines, R = a / (2 x sin(A)) = b / (2 x sin(B)) = c / (2 x sin(C)). The circumdiameter (2R) equals the constant ratio from the Law of Sines. This calculator computes the circumradius from the known side-angle pair.