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 ($s$).

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:

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:

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)

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s = \frac{a + b + c}{2}

\mathcal{A} = \sqrt{s(s-a)(s-b)(s-c)}

s = \frac{a+b+c}{2}

$s = \frac{a + b + c}{2}$

$\mathcal{A} = \sqrt{s(s-a)(s-b)(s-c)}$