Triangle Calculator

Heron's formula calculates a triangle's area from its three side lengths: compute the semi-perimeter s = (a+b+c)/2, then area = √(s(s−a)(s−b)(s−c)). It requires no angles — only the three sides. Example: sides 3, 4, 5 → s=6, area = √(6·3·2·1) = 6.

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². The classic example is the 3-4-5 triple: 9+16=25. This theorem is used to find a missing side in any right triangle.

By angles: right (one 90° angle), acute (all angles <90°), obtuse (one angle >90°). By sides: equilateral (3 equal sides), isosceles (2 equal sides), scalene (all sides different). Any valid triangle has angles summing to exactly 180°.

Use the law of cosines: cos A = (b²+c²−a²)/(2bc), so A = arccos(…). Then B = arccos((a²+c²−b²)/(2ac)). Finally C = 180°−A−B. This works for any triangle as long as the three sides satisfy the triangle inequality.

Three lengths a, b, c form a triangle if and only if the triangle inequality holds: the sum of any two sides must exceed the third: a+b>c, a+c>b, and b+c>a. If any condition fails the triangle does not exist.

If you know sides a and b and the included angle C, the area = (1/2)·a·b·sin(C). Example: a=3, b=4, C=90° → area = 0.5·3·4·1 = 6. This is the SAS formula and works for any angle between the two known sides.

The law of sines states: a/sin(A) = b/sin(B) = c/sin(C) = 2R, where R is the circumradius. It is used when one side and two angles are known (AAS or ASA). Example: if a=5, A=60°, B=60° then b = 5·sin(60°)/sin(60°) = 5.

Height h to base a = 2·area / a. For the 3-4-5 triangle with area 6: height to side 5 is h = 2·6/5 = 2.4. Alternatively use trigonometry: h = b·sin(C) where C is the angle at the vertex opposite base a.

The perimeter is the total length around the triangle: P = a+b+c (in metres, cm etc.). The area is the size of the enclosed surface, measured in square units (m², cm²). Heron's formula computes area from three sides without needing the height.

Check whether the Pythagorean theorem holds for the longest side c: a²+b²=c². If yes, it is a right triangle and the angle opposite c is exactly 90°. In practice, check if |a²+b²−c²| < a very small epsilon to account for rounding.