Heron's Formula

Heron's Formula is a method to find the area of any triangle when all three sides are known. It does not require the height of the triangle.

This formula is extremely useful when the height is difficult to measure or not given.

Triangle with sides a, b, c → Use Heron's Formula

Heron's Formula uses two steps:

Step 1: Find the semi-perimeter

\( s = \dfrac{a + b + c}{2} \)

Step 2: Use the area formula

\( \text{Area} = \sqrt{s(s - a)(s - b)(s - c)} \)

The semi-perimeter is simply half the perimeter of the triangle.

If the sides are \(a\), \(b\), and \(c\):

\( s = \dfrac{a + b + c}{2} \)

This value helps in calculating the area without needing height.

For a triangle with sides:

• \( a = 7 \)
• \( b = 8 \)
• \( c = 9 \)

First compute the semi-perimeter:

\( s = \dfrac{7 + 8 + 9}{2} = 12 \)

Now apply Heron's Formula:

\( \text{Area} = \sqrt{12(12 - 7)(12 - 8)(12 - 9)} \)

\( = \sqrt{12 \cdot 5 \cdot 4 \cdot 3} = 12\sqrt{5} \)

Use Heron’s Formula when:

• The height is not known,
• Only the three sides are given,
• The triangle is irregular (scalene),
• You want to avoid perpendicular constructions.

Heron's Formula is derived from geometric and algebraic relationships among triangle sides. It works for all types of triangles — scalene, isosceles, acute, obtuse, or right.

It provides an elegant way to compute area using just side lengths.