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Heron's Formula Area

Heron's Formula · Class IX

Enter the three sides; Heron computes the area without any height.

5
type a value
120
6
type a value
120
7
type a value
120
Live values
  • Semi-perimeter s9
  • Area14.6969
  • Perimeter18
xy
  • Area as side c varies

Formulas in this lab

  • Semi-perimeter s
    s=(a+b+c)/2s = (a+b+c)/2
  • Area
    s(s−a)(s−b)(s−c)\sqrt{s(s-a)(s-b)(s-c)}
  • Perimeter
    a+b+ca + b + c
Tip: Area is zero outside the triangle inequality and peaks somewhere between |a-b| and a+b.

Frequently asked questions

▶What is Heron's formula?

If a triangle has sides a, b, c and semi-perimeter $s = \frac{a+b+c}{2}$, its area is $A = \sqrt{s(s-a)(s-b)(s-c)}$. The formula needs no height; just the three side lengths are enough.

▶How do I use this lab?

Slide the three sides and the lab shows s and the area instantly. Try a = 13, b = 14, c = 15 to get the classic area 84 square units. Make sides equal to see how an equilateral triangle's area emerges.

▶Common mistake on this topic

Students often forget to halve the perimeter to get s, plugging in the full perimeter instead. Also, square-rooting at the end is a must - many leave the answer as $s(s-a)(s-b)(s-c)$ and lose marks.

▶Where is Heron's formula useful in India?

Farmers with triangular plots use Heron's formula because measuring the height is hard on uneven ground. A surveyor can just chain the three boundary sides and compute the area for land records.