Triangle Calculator

Solve any triangle from 3 known values. Compute all sides, angles, area, perimeter, and draw diagrams.

Triangle Calculator tool

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Example

Input: Side a = 3, side b = 4, side c = 5 (three sides known)

Result: Angles A 36.8699, B 53.1301, C 90 degrees; area 6; perimeter 12; scalene, right

Law of cosines: cos(C) = (3^2 + 4^2 - 5^2) / (2 x 3 x 4) = 0, so C = 90 degrees; area = 0.5 x 3 x 4 = 6.

Which Rule Solves Which Triangle

You knowCaseHow it is solvedSolutions
3 sidesSSSLaw of Cosines for the angles1, or none if a side is too long
2 sides and the angle between themSASLaw of Cosines for the third side1
2 angles and the side between themASAAngle sum, then Law of Sines1
2 angles and a side not between themAASAngle sum, then Law of Sines1
2 sides and an angle not between themSSALaw of Sines0, 1 or 2
3 angles onlyAAACannot be solved (size unknown)Infinite
  1. Enter any three known values from the six triangle properties (three sides and three angles).
  2. The calculator detects the case type (SSS, SAS, ASA, AAS, or SSA) automatically.
  3. View all missing sides, angles, area, and perimeter computed instantly.
  4. For the ambiguous SSA case, both possible triangle solutions are shown.
  5. Review the visual triangle diagram that updates with your values.
  6. Click the copy button to copy any calculated value.

Solving a Triangle From Three Values

A triangle is fully fixed by three of its six measurements, as long as at least one of them is a side. Enter any three values, and this calculator finds the missing sides and angles, the area and the perimeter. For sides 3, 4 and 5, it returns angles of 36.87, 53.13 and 90 degrees and an area of 6.

The labels follow the usual convention. Side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. Angles are in degrees. If you have radians, convert them first with the Angle Converter. The calculator picks its method from what you enter. With three sides, it uses the Law of Cosines to find each angle. With two sides and the angle between them, the Law of Cosines gives the third side. With two angles and a side, it finds the last angle from the 180-degree sum and uses the Law of Sines for the rest. The area comes from half of a x b x sin(C). The table on this page sums this up.

Two sides and an angle that is not between them can be ambiguous. For example, angle A = 30 degrees with a = 6 and b = 10 fits two different triangles, one with angle B of about 56.4 degrees and one with about 123.6 degrees. The calculator shows both, each with its own diagram. When no triangle fits, it says so.

Results show up to four decimal places. You also get the side type (equilateral, isosceles or scalene), the angle type (acute, right or obtuse), and a scaled drawing with every side and angle labeled. For right triangles, the Pythagorean Theorem Calculator is quicker.

Formula / How It Works

Law of Cosines: c^2 = a^2 + b^2 - 2ab*cos(C) | Law of Sines: a/sin(A) = b/sin(B) = c/sin(C) | Area = 0.5*a*b*sin(C)

Frequently Asked Questions

Any three values that include at least one side: three sides (SSS), two sides and the angle between them (SAS), two angles and a side (ASA or AAS), or two sides and an angle not between them (SSA). Enter exactly three values. If you fill in a fourth, the calculator asks you to clear the extra one.
It happens when you know two sides and an angle that is not between them (SSA). The known angle can open two ways, so there may be two different triangles, one, or none. For A = 30 degrees, a = 6 and b = 10 there are two. The calculator checks every option and shows each valid triangle.
Three angles fix the shape but not the size. A triangle with angles of 30, 60 and 90 degrees can have a short side of 1 cm or 1 km. You need at least one side length to pin down the scale, which is why the calculator asks for one.
The calculator uses Area = 0.5 x a x b x sin(C), where a and b are two sides and C is the angle between them. It works for every triangle, once the missing values are known. For a right triangle it reduces to half the base times the height.
Each side must be shorter than the other two added together. Sides of 2, 3 and 6 fail, because 2 + 3 is less than 6, so the short sides cannot meet. Check your measurements, or swap in the correct longest side, and the calculator will solve it.

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