Law of Sines Calculator

What the law of sines says

The law of sines connects the three sides of a triangle to the sines of the three angles facing them. Using the usual labels, side a sits opposite angle A, b opposite B and c opposite C, and the rule reads a / sin(A) = b / sin(B) = c / sin(C). All three ratios come out to the same number. The Wolfram MathWorld entry on the law of sines states it with these same labels and adds a detail worth knowing: the shared value equals 2R, twice the radius of the circle drawn through the three corners. The NIST Digital Library of Mathematical Functions, section 4.14 defines the sine function the formula leans on.

The shared ratio is worth watching while you work, because it doubles as a check. In the triangle with a 7 in side at 30 degrees opposite it and 45 degrees at the far end of that side, every ratio comes to 14: 7 / sin(30) = 9.8995 / sin(45) = 13.5230 / sin(105) = 14. If one ratio drifts away from the others, a keying slip happened somewhere.

Two of the three ratios are enough to solve anything. Pick the ratio that contains only known values, set it equal to the one holding the unknown, and cross multiply. That single move, repeated, fills in every missing side and angle of the triangle.

AAS worked example: a 7 in side, 30 and 45 degrees

Take a triangle where side a measures 7 in, the angle opposite it, A, is 30 degrees, and a second angle, B, is 45 degrees. The third angle needs no trigonometry at all: C = 180 - 30 - 45 = 105 degrees, because the three angles of any triangle add to 180.

Side b comes from pairing the ratio you know with the one you want. Both a / sin(A) and b / sin(B) equal the same constant, so b = a x sin(B) / sin(A) = 7 x 0.70711 / 0.5 = 9.8995 in. Side c uses the third angle: c = a x sin(C) / sin(A) = 7 x 0.96593 / 0.5 = 13.5230 in. The 105 degree angle has the longest side facing it, which is what the numbers show, and that ordering is a fast sanity check: the biggest angle always faces the biggest side.

ASA input is the same problem wearing different clothes. If the givens are two angles and the side between them, subtract to find the third angle, treat the given side as opposite that third angle, and the steps above run unchanged.

ASA: the surveyor's case

Surveyors made this calculation famous. Measuring across a river or a canyon directly is out of the question, but measuring a baseline along the bank and sighting two angles at the far marker is easy, and that is an ASA triangle. The NOAA geodesy tutorial on triangulation describes how crews mapped entire coastlines this way, one measured baseline and a chain of angle sightings at a time.

Worked case: the baseline along one bank runs 100 ft between two observers. Observer A sights the far tree at 40 degrees off the baseline; observer B, at the other end, sights it at 65 degrees. The triangle's third angle is 180 - 40 - 65 = 75 degrees. The side opposite the 75 degree angle is the 100 ft baseline, so the shared ratio is 100 / sin(75) = 103.5276. The distance from A to the tree is 103.5276 x sin(65) = 93.8279 ft, and from B it is 103.5276 x sin(40) = 66.5463 ft. Two tape measurements on dry land produced two distances nobody had to swim for.

The same trick sizes flagpoles, building lots and the width of a lake behind a house. One side and two angles is enough, as long as the angles sit at the ends of the known side, or the problem can be relabelled that way.

SSA: the case that can hide two triangles

SSA looks harmless: two sides and an angle not trapped between them. It is the treacherous case, because the given numbers can fit one triangle, two triangles, or none. The reason sits in the sine curve: 53.1301 degrees and 126.8699 degrees share the same sine, 0.8, so when solving for an angle returns a sine of 0.8, both answers stay alive until a geometric check kills one.

Worked case: a = 5, angle A = 30 degrees, b = 8. Start the usual way: sin(B) = b x sin(A) / a = 8 x 0.5 / 5 = 0.8. The arcsine hands back 53.1301 degrees. The supplement, 126.8699 degrees, has the same sine, and both survive the check that A + B must stay under 180 degrees: 30 + 53.13 = 83.13, and 30 + 126.87 = 156.87. Two triangles, then, and each needs its own third angle, third side and area.

Change the numbers and the answer count changes with them. With a = 10, A = 30 degrees and b = 7, the sine works out to 0.35, the acute answer is 20.4873 degrees, and the supplement dies because 30 + 159.5127 = 189.5127 exceeds 180 degrees: one triangle. Push a down to 3.9 and sin(B) = 8 x 0.5 / 3.9 = 1.0256, past the ceiling of 1, so no real angle exists and the three measurements describe nothing. Land on the ceiling, a = 4 with the same angle and side, and sin(B) = 1 gives B = 90 degrees, a single right triangle with c = 6.9282.

A ladder summarizes the textbook setup where A is acute. Compare a against b x sin(A): below it, no triangle; equal to it, one right triangle; between it and b, two triangles; at b or beyond, one triangle. Keeping that ladder in mind turns the ambiguous case from a trap into a routine.

Area from two sides and the sine between them

Once two sides and their included angle are known, the area is half their product times the sine of the angle between them, and the law of sines supplies whichever piece is missing. The worked AAS triangle from earlier has sides 7 in and 9.8995 in around the 105 degree angle, so area = 0.5 x 7 x 9.8995 x sin(105) = 33.4676 square inches. Any of the three corners gives the same value, which makes the formula a third kind of check: 0.5 x 7 x 13.5230 x sin(45) and 0.5 x 9.8995 x 13.5230 x sin(30) both land on 33.4676 too.

For the two-headed SSA case the two triangles are genuinely different shapes, so their areas differ as well: 19.8564 square units for the acute answer against 7.8564 for the obtuse one. When all three sides are known instead, Heron's formula takes over from the sine route: sides 7, 9.8995 and 13.5230 have semiperimeter 15.2112, and Heron returns 33.47, matching the sine route to rounding. The triangle calculator runs that side-only case directly.

When the law of cosines is the right tool

The law of sines needs an angle and its opposite side somewhere in the givens. When the numbers arrive as two sides and the angle squeezed between them, or as three sides with no angle at all, the law of cosines takes over. Two sides a and b with included angle C give the third side through c^2 = a^2 + b^2 - 2ab cos(C). Three known sides give an angle through cos(A) = (b^2 + c^2 - a^2) / (2bc), after which the remaining angles fall to the law of sines one at a time.

The division of labor is worth memorizing. SAS and SSS belong to the cosine rule; AAS, ASA and the first pass of SSA belong to the sine rule. Many full solutions use both, the cosine rule to break into the triangle and sines to finish it. The trigonometry calculator covers the ratio side of the subject, and the right triangle calculator skips the oblique cases entirely, since a 90 degree corner lets the ordinary opposite over hypotenuse definitions do the work.

Three mistakes that bend a triangle

The first is degree mode. Every angle on this page is in degrees, and a calculator left in radian mode turns sin(30) into -0.98803 instead of 0.5, producing sides that look plausible and are quietly wrong. One glance at that single value settles the mode question before any damage spreads: the sine of 30 degrees is 0.5, the sine of 30 radians is -0.98803.

The second is a mislabelled side. The formula pairs each side with the angle facing it, and after a paper rotation the labels wander. In the worked AAS triangle, feeding 7 and the 45 degree angle into the wrong slots gives 7 x sin(30) / sin(45) = 4.9497 instead of 9.8995, and the shared ratio exposes it: 4.9497 / sin(45) = 7, not 14.

The third is rounding the sine before inverting it. The SSA case hit a sine of exactly 0.8, but a measured value of 0.78 is close to real work, and arcsin(0.78) = 51.2606 degrees against arcsin(0.8) = 53.1301 degrees, a drift of almost 2 degrees from two digit rounding. Carry at least four decimals through the middle of a calculation and round only the displayed result. The arcsin calculator shows the same sensitivity near the top of its range, and the arctan calculator is the safer inverse whenever the two known legs are the adjacent pair instead.