Heron's Formula

📅 Published on Thursday, November 20, 2025 👤 By Smart CBSE AI Tutor
Infographic: The Power of Heron's Formula

The Triangle's Secret

Finding Area Without Knowing the Height

The Height Dilemma

For centuries, the area of a triangle was found using one simple formula. This formula is fast and effective, but it has a critical weakness: you must know the vertical height.

h = ? base

Area = ½ × base × height

But what if you can't measure the height?

Enter Heron's Formula

Heron of Alexandria (c. 10-75 C.E.) devised a brilliant way to find the area of any triangle using only the lengths of its three sides ($a$, $b$, and $c$).

Step 1: Find the Semi-Perimeter (s)

$s = (a + b + c) / 2$

Step 2: Use the Master Formula

Area = √$s(s-a)(s-b)(s-c)$

How to Use It: A 4-Step Process

Step 1

Get side lengths:
$a, b, c$

→
Step 2

Calculate Semi-Perimeter:
$s = (a+b+c) / 2$

→
Step 3

Find the differences:
$(s-a), (s-b), (s-c)$

→
Step 4

Calculate Area:
$\sqrt{s(s-a)(s-b)(s-c)}$

Case Study 1: The Triangular Park

A park has sides of 40m, 32m, and 24m. Let's find its area.

  • Sides: $a=40$, $b=32$, $c=24$
  • Semi-Perimeter (s): (40 + 32 + 24) / 2 = 48
  • Differences:
    • $(s-a):$ 48 - 40 = 8
    • $(s-b):$ 48 - 32 = 16
    • $(s-c):$ 48 - 24 = 24
  • Area: $\sqrt{48 \times 8 \times 16 \times 24}$ = $\sqrt{147,456}$ = 384 m²

Interestingly, this is a right triangle ($24^2 + 32^2 = 40^2$). The traditional formula ($\frac{1}{2} \times 24 \times 32$) also gives 384 m². They match!

Case Study 2: Any Triangle, Any Size

Heron's formula works for all triangles. Let's compare the perimeter and area of different types from the textbook.

Note: Two Y-axes are used to compare Perimeter (left) and Area (right) on different scales.

Case Study 3: The Flyover Ad

A triangular flyover wall with sides 122m, 120m, and 22m is used for advertising.

  • Semi-Perimeter (s): (122 + 120 + 22) / 2 = 132
  • Area: $\sqrt{132(132-122)(132-120)(132-22)}$
  • Area: $\sqrt{132 \times 10 \times 12 \times 110}$ = 1320 m²

A company rents this wall for 3 months at a rate of ₹5000 per m² per year.

Total Rent for 3 Months:

₹1,650,000

(1320 m² × ₹5000/yr) / 4 Quarters

The Power of Three Sides

Heron's Formula is a powerful tool for mathematics, engineering, and design. It bypasses the need for height, allowing anyone to find the area of any triangle, no matter how irregular, using only the lengths of its three sides.

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