Triangle Calculator: Finding Missing Sides, Angles and Area

Three numbers is the magic count. Fewer and the triangle is not pinned down. More and you are probably about to feed it a contradiction. The trick is knowing which three, because not every set of three has a single answer.

Which three values actually work

Four combinations solve cleanly. Three sides (SSS). Two sides and the angle between them (SAS). Two angles and the side between them (ASA). Two angles and a side that is not between them (AAS). Three angles on their own give the shape but not the size: a triangle with angles of 40, 60 and 80 degrees can be a centimetre across or a kilometre.

The case that has two answers

Two sides and an angle that is not between them is the awkward one. Say side a is 7, side b is 10, and angle A is 40 degrees. The sine rule gives sin B = 10 x sin 40 / 7 = 0.918, and two angles share that sine: 66.7 degrees and 113.3 degrees. Both produce a real triangle. A good calculator shows you both instead of silently picking one. When an answer looks wrong, this is usually why.

A worked example with no right angle

Sides of 7, 9 and 12. Start with the largest angle, the one opposite the 12, using the cosine rule: cos C = (49 + 81 - 144) / (2 x 7 x 9) = -0.111, so C = 96.4 degrees. A negative cosine means an obtuse angle, which is worth noticing on its own.

Then the sine rule for the rest: sin A = 7 x sin 96.4 / 12 = 0.580, so A = 35.4 degrees, and B is whatever is left, 48.2 degrees. Area comes from Heron's formula. The semi-perimeter is (7 + 9 + 12) / 2 = 14, and the area is the square root of 14 x 7 x 5 x 2 = 980, which is 31.3 square units.

Doing that by hand takes five minutes and one dropped minus sign to get wrong. That is the argument for the calculator.

Using the triangle calculator

Fill in exactly three boxes and leave the rest blank. Sides are lower case a, b and c; angles are capital A, B and C, each sitting opposite the side of the same letter. Getting that pairing backwards is the single most common mistake, and it produces answers that look plausible but are not.

Check the angle unit first. Degrees is what most people want. Radians hands you a triangle whose angles sum to 180 radians, which is nonsense, and the tool will say so.

Reading the output

You get three sides, three angles, the area and usually the perimeter. Sanity check two things: the angles should sum to 180, and the longest side should sit opposite the largest angle. If either fails, an input is in the wrong box. For a real room or garden, convert units before you start, because a triangle in feet and a triangle in metres are different triangles even when the digits match. Our note on measuring square footage covers the odd shapes that get split into triangles.

Common questions

Why does it say no triangle exists? The triangle inequality. Any two sides must add to more than the third. Sides of 3, 4 and 9 fail, because 3 + 4 is only 7 and those pieces will never close into a shape. Sides of 3, 4 and 7 fail more subtly: they lie flat in a straight line with zero area.

How do I find the area from just the three sides? Heron's formula. Take half the perimeter, subtract each side from it in turn, multiply the four results together and take the square root. It works for every triangle, right angled or not, and needs no angles at all. Useful when you have a tape measure and nothing else. It is the same reasoning behind the general area calculations for irregular plots.

Can a triangle have two right angles? No. Two right angles use the full 180 degrees between them, leaving nothing for the third, and the two sides would run parallel forever without ever meeting.

Is a 6-8-10 triangle the same as a 3-4-5? Same shape, double the size. Both have angles of 36.87 and 53.13 degrees, but the area goes from 6 to 24. Doubling every side quadruples the area, which catches people out when pricing materials.