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Triangle Calculator

Calculators · Added

Three independent measurements fix a triangle — with one famous exception, where they fix two. Choose which three you know, enter them, and this solver returns every remaining side and angle along with the area, the three altitudes and both circle radii. In the exceptional case it returns both triangles, because both are correct and a calculator that shows one has quietly thrown away half the answer.

All three side lengths are known.

Opposite angle A

Opposite angle B

Opposite angle C

Try:

Sides are unitless — enter centimetres, inches or anything else, and every length in the answer comes back in the same unit. Angles are in degrees.

How to use the triangle calculator

  1. 1Choose which three parts you know from the list: three sides, two sides and the angle between them, two angles and a side, or two sides and an angle opposite one of them.
  2. 2Enter the three values. The labels change with the case, and each says exactly which side or angle it means — a is always opposite A.
  3. 3Angles are in degrees. Sides carry no unit, so enter centimetres, inches or anything else and every length in the answer comes back the same way.
  4. 4Press Solve triangle.
  5. 5Read the working underneath the result: it names the law used at each step, which is usually what the answer is being checked against.

Examples

Three sides, a familiar triangle

Input
SSS with a = 3, b = 4, c = 5
Result
A = 36.87°, B = 53.13°, C = 90°, area 6

The right angle appears on its own rather than being assumed. The arccosine of (9 + 16 − 25) ÷ 24 is the arccosine of zero, which is exactly 90°.

Two sides and the angle between them

Input
SAS with b = 7, A = 40°, c = 9
Result
a = 5.80, B = 50.87°, C = 89.13°, area 20.25

This case never has more than one answer: fixing the angle between two known sides leaves nothing free.

The ambiguous case, with both answers

Input
SSA with a = 8, b = 10, A = 40°
Result
Two triangles — one with B = 53.46° and c = 11.87, one with B = 126.54° and c = 4.10

Side a is longer than the height b·sin A = 6.43 but shorter than b, so it can meet the base at two different points. Both triangles have exactly the measurements given.

About the triangle calculator

Which three parts are enough

A triangle has six parts — three sides and three angles — and any three of them determine the rest, with two exceptions. Three angles are not enough: they fix the shape but not the size, so AAA describes an infinite family of similar triangles and there is nothing to solve. And two sides with an angle opposite one of them can describe two different triangles at once, which is the case set as an exam question precisely because it catches people out.

The other four combinations behave. Three sides give every angle through the law of cosines. Two sides and the angle between them give the third side the same way. Two angles give the third immediately, since they sum to 180°, and any known side then scales the whole triangle through the law of sines. In each of these the answer is unique, and the tool says so in its working rather than leaving you to wonder whether something was missed.

That distinction between shape and size is worth holding on to. Knowing angles alone tells you what a triangle looks like; you need at least one length before it has a size. It is why the ASA and AAS cases feel different from SSS — in those you are scaling a known shape, and in SSS you are discovering one.

The ambiguous case, and why calculators get it wrong

Take a = 8, b = 10 and A = 40°. Drop a perpendicular from the far vertex to the base: its length is b·sin A, about 6.43. Side a has to reach that base. At 8 it is longer than 6.43, so it reaches — but it is shorter than b, which means it reaches at two different points, one either side of the foot of the perpendicular. Two triangles, both with a side of 8, a side of 10 and a 40° angle opposite the 8.

The arithmetic reflects this exactly. The law of sines gives sin B = b·sin A ÷ a, and any value of sin B between 0 and 1 has two angles behind it: one acute, one obtuse, adding to 180°. A calculator that reports arcsin and stops has reported the acute one and silently discarded the other, and it will do this without any indication that a second answer existed. That is why both are shown here, side by side, with a note explaining that the measurements alone cannot decide between them.

Deciding usually needs information from outside the measurements — a sketch, a physical constraint, a statement that the triangle is acute. Where such information exists the ambiguity dissolves. Where it does not, two answers is the correct answer, and any tool presenting one is being confident rather than being right.

Frequently asked questions

Why does SSA sometimes give two answers?
Because an arcsine cannot tell an angle from its supplement — sin 53.46° and sin 126.54° are the same number. When you know two sides and an angle that is not between them, the known side can swing to meet the base at two different points, and both closures are genuine triangles with exactly the measurements you supplied. Whether it happens depends on how the known side compares with the height b·sin A: shorter and nothing closes, equal and there is one right triangle, longer but still shorter than b and there are two.
Why does it say no triangle exists?
Two reasons, depending on the case. With three sides it is the triangle inequality: the two shorter sides have to add up to more than the longest, or they cannot reach across it — 2, 3 and 9 describes nothing. With two sides and an opposite angle it is that the known side is shorter than the perpendicular distance to the base, so it never reaches. The error message names which of the two happened and gives the figure it failed against.
Which formula is used for the area?
Half of a·b·sin C, not Heron's formula. The two agree mathematically, but Heron subtracts each side from the semi-perimeter, and on a long thin triangle those are two nearly equal numbers whose difference loses most of its significant figures to floating-point error. The sine version has no such cancellation, so it stays accurate on exactly the shapes where accuracy is hardest.
What are the inradius and circumradius for?
The inradius is the radius of the largest circle that fits inside the triangle, touching all three sides; the circumradius is the radius of the circle through all three vertices. They come up constantly in geometry coursework and in layout work — the inradius is the biggest round thing that fits in a triangular space, and the circumradius is the circle you would need to draw to place three points at those distances.
Can I use this for a right triangle?
Yes, and there is no separate mode for it. Enter what you know as three sides, or as two sides with the 90° between them, or as one side with two angles including the 90°. The right angle comes out of the general solution rather than being special-cased, and the classification line will name it as a right triangle when it is one.

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