Triangle Calculator (Sides, Angles, Area)

Enter three sides (SSS), two sides and the included angle (SAS), or one side and its two adjacent angles (ASA), and calculate the remaining sides, angles, area, perimeter, circumradius, and inradius — plus the triangle's classification.

Tips

  • In SSS mode, values that violate the triangle inequality (any two sides must sum to more than the third) will show "not a valid triangle."
  • Whether a triangle is a right triangle is determined by checking if its largest angle is 90°, which also works as a quick check for Pythagorean triples like 3-4-5.
  • The circumradius is derived from the law of sines (a / (2 sin A)); the inradius comes from area divided by the semi-perimeter.
  • SAS and ASA modes are handy for checking surveying or geometry problems where only two pieces of information are known.

Frequently Asked Questions

Use the law of cosines: cos A = (b² + c² − a²) / 2bc. Enter the three side lengths in the "Three sides (SSS)" mode and this tool computes all three angles automatically.

For a triangle to exist, the sum of any two sides must be greater than the third (the triangle inequality). Values that violate this, like 1, 1, 10, cannot form a triangle on a flat plane, so no calculation is possible.

The circumradius is the radius of the circle passing through all three vertices; the inradius is the radius of the circle tangent to all three sides. Both come up in architectural and drafting work when fitting triangular parts into circular frames, and frequently in geometry proofs.

Yes. Given two sides and the included angle, the law of cosines determines the third side uniquely, fully fixing the triangle's shape — this is one of the standard triangle congruence conditions.
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Side Note — Why SSS, SAS, and ASA are enough to "fix" a triangle

The triangle congruence conditions taught in school — three sides, two sides and the included angle, one side and its two adjacent angles — really describe the minimum amount of information needed to pin down a triangle uniquely. Fixing three points in a plane would normally take six degrees of freedom (each point's x and y coordinates), but once you remove the three degrees of freedom for translation, rotation, and reflection, only three independent pieces of information — some combination of sides and angles — are left to determine the shape. SSS, SAS, and ASA are the standard ways of supplying exactly that.

By contrast, two angles plus a side that isn't adjacent to both of them (often called SSA) can be ambiguous — the same two pieces of information can sometimes correspond to two different triangles. This "ambiguous case" trips up a lot of students in the law-of-sines unit of a geometry course precisely because it looks so similar to the SAS condition that actually does determine a triangle uniquely.

The circumcircle and incircle are also consequences of a triangle being fully determined by its "three pieces of information." Any three non-collinear points determine exactly one circle passing through them, and its center can be constructed as the intersection of the perpendicular bisectors of the sides. This same geometric fact underpinned pre-GPS surveying: triangulation lets you locate an unknown point purely from two known points and a couple of measured angles.

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