General | Math & Conversions Heron's Formula Calculator Heron's Formula Calculator (Area from 3 Sides) Heron's Formula is a method to find the area ($\mathcal{A}$) of any triangle when only the three side lengths ($a$, $b$, and $c$) are known. Input the side lengths below to calculate the area and the semi-perimeter ($s$). Enter the Three Side Lengths Side a Side b Side c Calculate Area ($\mathcal{A}$) Key Results Perimeter ($P$) Semi-Perimeter ($s$) Area ($\mathcal{A}$) Step-by-Step Solution Heron's Formula ($\mathcal{A}$) Heron's Formula, attributed to Heron of Alexandria, calculates the area ($\mathcal{A}$) of a triangle using only the lengths of its sides ($a, b, c$). It is expressed in two sequential parts: Part 1: Calculate the Semi-Perimeter ($s$) The semi-perimeter is half the perimeter of the triangle: $$s = \frac{a + b + c}{2}$$ Part 2: Calculate the Area ($\mathcal{A}$) The area is then found by multiplying the semi-perimeter by the difference between the semi-perimeter and each side, and taking the square root of the final product: $$\mathcal{A} = \sqrt{s(s-a)(s-b)(s-c)}$$ The Triangle Inequality Rule Before applying Heron's formula, it is essential to ensure that the sides can actually form a triangle. If the sides do not satisfy the **Triangle Inequality Theorem**, the number inside the square root will be negative, and the area is non-real. The rule requires that the sum of the lengths of any two sides must be greater than the length of the third side: $a + b > c$ $a + c > b$ $b + c > a$ If these conditions are not met, the calculator will indicate an invalid triangle. Frequently Asked Questions (FAQ) What is Heron's Formula used for? Heron's Formula (also called Hero's Formula) is used to find the area of any triangle when the lengths of all three sides are known. It is particularly useful when the height of the triangle is unknown or difficult to determine. What is the semi-perimeter? The semi-perimeter ($s$) is simply half the perimeter of the triangle. It is the intermediate variable required to calculate the area using Heron's Formula: $s = \frac{a + b + c}{2}$. What is the Triangle Inequality Rule? The Triangle Inequality Theorem states that for any triangle to be valid, the sum of the lengths of any two sides must be greater than the length of the third side. For sides a, b, and c, this means: $a+b > c$, $a+c > b$, and $b+c > a$. Does Heron's Formula work for right triangles? Yes, Heron's Formula works for *all* types of triangles, including right triangles, acute triangles, obtuse triangles, and equilateral triangles. For a right triangle, the area can also be found more simply using $A = \frac{1}{2}ab$. Heron's Key Formulas Semi-Perimeter $$s = \frac{a+b+c}{2}$$ Area $$\mathcal{A} = \sqrt{s(s-a)(s-b)(s-c)}$$ Related Triangle Tools Triangle Solver (SSS, SAS, AAS) Area Calculator (General) Perimeter Calculator Law of Cosines Calculator Law of Sines Calculator