Heron's formula gives the area of any triangle from its three sides alone — no height or angle required. Halve the perimeter to get the semi-perimeter s = (a + b + c)/2, then area = √(s(s−a)(s−b)(s−c)). This calculator runs Heron's formula, checks the triangle inequality, and also reports the perimeter, the three interior angles (law of cosines), and the triangle's type by side and by angle.
Reviewed: June 20, 2026 · Author: Naveen P N, Founder — AI Calculator · Verified against: Heron's formula and the law of cosines, recomputed in code.
Heron's formula, step by step
Heron's formula is ideal when you know all three sides but no height. After the semi-perimeter, multiply the three differences by s and take the square root. For the angles, the law of cosines isolates each one from the side lengths. The triangle is valid only if every side is shorter than the sum of the other two; otherwise the product under the root goes non-positive and there's no triangle.
Worked example — sides 3, 4, 5
Scenario: a triangle with sides a = 3, b = 4, c = 5.
The 3-4-5 triangle has a semi-perimeter of 6 and an area of exactly 6, with a perimeter of 12. The law of cosines gives angles of 36.87°, 53.13° and 90° — so it's a right scalene triangle (all sides different, one 90° angle). Scaling up to 6-8-10 keeps the same shape but quadruples the area to 24, while an isosceles 5-5-6 triangle has area 12.
Frequently Asked Questions
Area = √(s(s−a)(s−b)(s−c)) with s = (a+b+c)/2. Sides 3,4,5 → s=6, area √36 = 6. No height needed.
Halve the perimeter for s, multiply s by (s−a), (s−b), (s−c), take the square root. The sides must form a valid triangle.
Each side must be less than the sum of the other two. 3,4,5 works; 1,2,5 doesn't (5 ≥ 1+2), so it can't form a triangle.
Law of cosines: A = arccos((b²+c²−a²)/(2bc)), etc. For 3-4-5 the angles are 36.87°, 53.13° and 90°.
By sides: equilateral (all equal), isosceles (two equal), scalene (all different). Also classed acute, right or obtuse by angle.