Herons Formula Calculator
Free Herons formula Calculator for triangle. Enter values to get step-by-step solutions with formulas and graphs. Free to use with no signup required.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
Area = sqrt(s(s-a)(s-b)(s-c)), where s = (a+b+c)/2
Herons formula calculates triangle area from three side lengths a, b, c. First compute the semi-perimeter s = (a+b+c)/2, then Area = sqrt(s(s-a)(s-b)(s-c)). No height or angle measurement is needed.
Worked Examples
Example 1: Area of a Scalene Triangle
Problem:Find the area of a triangle with sides 13, 14, and 15 using Herons formula.
Solution:Semi-perimeter s = (13 + 14 + 15) / 2 = 21\ns - a = 21 - 13 = 8\ns - b = 21 - 14 = 7\ns - c = 21 - 15 = 6\nArea = sqrt(21 x 8 x 7 x 6) = sqrt(7056) = 84\nInradius = 84 / 21 = 4\nCircumradius = (13 x 14 x 15) / (4 x 84) = 2730 / 336 = 8.125
Result:Area = 84 sq units | Inradius = 4 | Circumradius = 8.125
Example 2: Verifying with a Right Triangle
Problem:Verify Herons formula for a 5-12-13 right triangle.
Solution:Standard formula: Area = (1/2) x 5 x 12 = 30\nHerons formula: s = (5 + 12 + 13) / 2 = 15\ns - a = 10, s - b = 3, s - c = 2\nArea = sqrt(15 x 10 x 3 x 2) = sqrt(900) = 30\nBoth methods give Area = 30 sq units\nInradius = 30 / 15 = 2\nCircumradius = (5 x 12 x 13) / (4 x 30) = 780 / 120 = 6.5
Result:Area = 30 sq units (verified) | Inradius = 2 | Circumradius = 6.5
Frequently Asked Questions
How do you calculate the semi-perimeter for Herons formula?
The semi-perimeter s is simply half the perimeter of the triangle. If the three sides are a, b, and c, then s = (a + b + c) / 2. For example, a triangle with sides 7, 8, and 9 has perimeter 24 and semi-perimeter 12. The semi-perimeter is a convenient intermediate value that simplifies several triangle formulas beyond just Herons formula. It appears in the inradius formula (r = Area/s), in angle bisector calculations, and in Eulers formula relating the circumradius and inradius. Computing the semi-perimeter first makes the main area calculation much cleaner.
Can Herons formula be used for any type of triangle?
Yes, Herons formula works for any valid triangle, including acute, right, obtuse, equilateral, isosceles, and scalene triangles. The only requirement is that the three side lengths satisfy the triangle inequality: the sum of any two sides must be greater than the third side. If the triangle inequality is violated, the expression under the square root becomes negative, indicating that no triangle with those side lengths exists. For degenerate triangles (where three points are collinear and the area is zero), Herons formula correctly returns zero since one of the factors (s-a), (s-b), or (s-c) equals zero.
How does Herons formula compare to the standard base-height area formula?
The standard area formula (Area = 1/2 times base times height) requires knowing a base and its corresponding perpendicular height. If the height is not given, you must calculate it, often using trigonometry or the Pythagorean theorem. Herons formula only requires the three side lengths, making it more direct in many situations. However, the base-height formula is computationally simpler and may be preferred when the height is known. For right triangles, the two legs serve as base and height, making the standard formula trivial. Herons formula is most valuable for oblique triangles where the height is unknown.
How do you derive Herons formula?
Herons formula can be derived from the cosine rule and the standard area formula. Start with Area = (1/2)ab sin(C). From the law of cosines, cos(C) = (a^2 + b^2 - c^2)/(2ab). Using the identity sin^2(C) = 1 - cos^2(C), substitute and simplify: Area^2 = (1/4)a^2 b^2 sin^2(C) = (1/4)a^2 b^2(1 - cos^2(C)). After algebraic manipulation using the difference of squares, this becomes Area^2 = s(s-a)(s-b)(s-c). The derivation involves factoring a quartic polynomial into the product of four linear terms, which is why the semi-perimeter appears naturally.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy