Law of Sines Calculator - Two Angles, One Side

A law of sines calculator is a tool that takes two interior angles of a triangle and the length of the side opposite one of them, then returns the third angle, the other two sides, the area, and the perimeter from those three numbers. It uses the identity a / sin(A) = b / sin(B) = c / sin(C) so that, once the third angle falls out of 180 - A - B, the two missing sides come straight from the sine ratio. The same identity covers acute, right, and obtuse triangles with one formula.

• • Find a distance across a river or canyon: Stand at one bank, sight the far bank at two known angles from a baseline, and the law of sines gives the distance you cannot walk to.
• • Resolve a force or velocity into two unknowns: Given one known side opposite a known angle and a second known angle, the sine rule returns the other two sides of the force or velocity triangle.
• • Check AAS homework problems: Type in the two given angles and the side opposite one of them, then read the third angle, the two missing sides, the area, and the perimeter together.

The native use case is the AAS case (two angles and a non-included side), because the third angle is 180 - A - B and the other two sides follow from one ratio each. A true ASA problem, where the known side sits between the two given angles, must be reduced to an AAS shape first: compute the third angle, derive a new side from the law of sines, then enter the pair into this panel.

For two sides with the included angle (SAS) or all three sides (SSS), the Law of Cosines Calculator takes the same triangle and finishes it with c^2 = a^2 + b^2 - 2ab*cos(C).

Behind the panel sit one ratio identity, the angle-sum rule, and the standard area and perimeter formulas. The form takes two angles in degrees and the side opposite one of them, then returns five derived values.

• a: Length of the side opposite angle A, in any consistent length unit. The form asks for this single side, and it must be opposite one of the two angles.
• b: Length of the side opposite angle B, computed in the same unit as side a.
• c: Length of the side opposite angle C, computed in the same unit as side a.
• A, B, C: Three interior angles of the triangle, in degrees, with A + B + C = 180.
• R: Circumradius. The shared ratio a / sin(A) equals 2R.

The first step is to convert angles A and B from degrees to radians because the JavaScript sin function expects radians.

Once sin(A), sin(B), and sin(C) are in hand, side b = a * sin(B) / sin(A) and side c = a * sin(C) / sin(A) are two simple ratios. The area is 0.5 * b * c * sin(A) and the perimeter is a + b + c.

According to Wikipedia: Law of sines, the law of sines states that for any triangle with sides a, b, c opposite angles A, B, C, the ratios a/sin A, b/sin B, and c/sin C are all equal to the same constant.

As published by Wolfram MathWorld: Law of Sines, the identity holds for any triangle, with the shared constant equal to the diameter 2R of the circumscribed circle.

Four ideas carry the whole identity. Once you can name them, the AAS pattern this form expects feels mechanical.

The same identity also explains the SSA ambiguous case: when you know two sides and an angle that is not between them, the missing angle can have one or two geometrically valid answers. That is why this form sticks to the AAS case, where the answer is unique.

When the triangle happens to have a 90-degree angle, the law of sines reduces to the same ratio as Right Triangle Calculator uses for the legs and the hypotenuse.

Five steps from typing the first angle to reading the perimeter at the bottom of the results panel. The form expects an AAS pattern: two angles and the side opposite one of them.

1. 1 Pick a side and call it side a: Choose the side whose length you already know, label it side a, and find the angle opposite it. That opposite angle is angle A.
2. 2 Enter angle A and angle B: Type both known interior angles in degrees. A + B must come to less than 180. For a true ASA problem, compute 180 - A - C first and enter that as the second angle.
3. 3 Enter side a: Type the length of the side you chose, in any consistent unit (cm, m, in, ft, ...).
4. 4 Read the third angle C: The first row of the results panel shows the remaining angle as 180 - A - B. If it reaches 180, the panel flags a validation error.
5. 5 Read the missing sides, area, and perimeter: The next rows show side b, side c, the area, and the perimeter, each computed from the ratio of sines.

Try A = 50, B = 70, a = 12. The panel shows C = 60, b = 14.7179, c = 13.8691, area = 73.0604, perimeter = 40.5870. For a true ASA problem, compute the third angle first.

If you would rather let a general solver pick the AAS, SAS, or SSS case for you, Triangle Calculator chains the law of sines and the law of cosines in one workflow.

Five practical reasons to let the panel do the work instead of doing the sine rule by hand.

• • One identity, five answers: A single a / sin(A) = b / sin(B) = c / sin(C) call returns the third angle, both missing sides, the area, and the perimeter together.
• • Works for acute, right, and obtuse triangles: The same formula covers every shape because it does not require a right angle, and the panel handles angles greater than 90 without any branch logic.
• • No unit conversion needed for the sides: Side a, side b, and side c inherit the same length unit. The area reports in the squared version, the perimeter in the linear version.
• • Doubles as a circumradius readout: Any one side-angle pair from the results panel is the circumdiameter 2R. Divide by 2 for the radius, or compare 2R across triangles of the same family.
• • Fast sanity check for AAS homework: Typing the two given angles and the side opposite one of them takes a few seconds, and the panel prints all five derived values together.

Watching the two input angles and the input side change and seeing all five output rows update together makes the relationship between angles and opposite sides easier to internalize than reading a textbook diagram.

For a separate pass at the area with three different methods (base-height, three-sides, SAS), Triangle Area Calculator runs the same input through Heron's formula and the standard 0.5 * b * c * sin(A) form.

Five things that move the numbers in the panel, plus two honest caveats about the underlying math.

• • The form targets the AAS case. A true ASA problem must be reshaped into an AAS pair first, and the SSA case has zero, one, or two valid triangles and is not auto-branched here.
• • The law of sines cannot detect a typo. Enter 60 in the angle A box when you meant 6 and the panel happily shows a result for 60 degrees.

Sanity-check the third angle (it must be positive and less than 180) and the unit on side a before reading the rest of the panel.

According to Britannica: Law of sines, the law of sines relates each side of a triangle to the sine of the opposite angle, and the identity holds for any triangle including obtuse ones.

When you have a side ratio and want a quick second opinion on the angle that produced it, Arcsin Calculator returns arcsin in degrees or radians in one step.