What is the law of sines?

The law of sines states that in any triangle, each side divided by the sine of its opposite angle equals the same constant: a divided by sin(A) equals b divided by sin(B) equals c divided by sin(C). This common ratio is twice the circumradius of the triangle. The law applies to all triangles, not just right-angled ones.

When can I use the law of sines?

The law of sines is useful when you know two angles and any one side (AAS or ASA), or when you know two sides and an angle opposite one of them (SSA). For the AAS and ASA cases the triangle is uniquely determined. The SSA case can sometimes produce two valid triangles (the ambiguous case), which this calculator does not resolve automatically.

What is the circumradius?

The circumradius is the radius of the circle that passes through all three vertices of the triangle (the circumscribed circle or circumcircle). From the law of sines, the circumradius R equals any side divided by twice the sine of its opposite angle: R = a / (2 sin A).

Law of Sines Calculator

This calculator applies the law of sines to solve a triangle when you know two angles and the side opposite to the first angle (AAS configuration). The law of sines is the relationship a / sin(A) = b / sin(B) = c / sin(C), where a, b and c are the side lengths of a triangle and A, B and C are the angles opposite those sides respectively. Because the three angles of any triangle must sum to 180 degrees, knowing two angles immediately gives you the third. Once you have all three angles, the law of sines lets you scale the side lengths using the known side as an anchor. This covers AAS (angle-angle-side where the known side is not between the two known angles) and ASA (angle-side-angle) triangles. You enter angle A in degrees, angle B in degrees and the length of side a (opposite angle A). The calculator returns side b, side c, angle C, the triangle area using the formula (1/2) times a times b times sin(C), and the circumradius R which equals a divided by (2 times sin(A)). The calculator flags invalid inputs such as angles that do not leave room for a positive third angle, or an angle of zero. It is useful for students solving NCEA geometry problems, surveyors working with triangulated measurements, and anyone who needs to find unknown dimensions of a triangular shape when two angles and one side are known. For the ambiguous SSA case where two solutions may exist, use the law of cosines instead.

Angle A (degrees)

Angle B (degrees)

Side a (opposite A)

8.66

side b (opposite B)

Side c10.00

Angle C90.00°

Area21.65

Circumradius R5.00

Angles A + B must be less than 180 degrees. This calculator uses the AAS case; for SSA input use the Law of Cosines Calculator.

How it works

Angle C = 180 − A − B. The common ratio k = a / sin(A). Side b = k × sin(B). Side c = k × sin(C). Area = (1/2) × a × b × sin(C). Circumradius R = a / (2 × sin(A)). All angles are converted to radians before applying Math.sin. If A + B is 180 degrees or more, the triangle is invalid and the calculator shows an error.

Worked example

Enter A = 30 degrees, B = 60 degrees, a = 5. Angle C = 180 − 30 − 60 = 90 degrees. The common ratio k = 5 / sin(30) = 5 / 0.5 = 10. Side b = 10 × sin(60) = 10 × 0.8660 = 8.66. Side c = 10 × sin(90) = 10.00. Area = 0.5 × 5 × 8.66 × sin(90) = 21.65. Circumradius = 5 / (2 × 0.5) = 5.00. These match the default values pre-filled above.

Related calculators

Law of Cosines Calculator: solve SAS or SSS triangles using the cosine rule.
Triangle Angle Calculator: find the third angle from two known angles.
Sin Cos Tan Calculator: look up all six trig values for any angle.
Arc Length and Sector Area Calculator: find arc length and sector area from radius and angle.

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