Chapter 10 · NCERT Class 9 Mathematics

Heron’s Formula

Sometimes you know all three sides of a triangle but not its height — so ½ × base × height won’t work. Heron’s formula finds the area from the side lengths alone: first the semi-perimeter s = (a + b + c)/2, then Area = √(s(s−a)(s−b)(s−c)). Tap each idea to see it in action.

📐 3 topics⏱ ~25 min📝 12-question quiz

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The idea behind Heron’s formula

Heron’s formula has just a few moving parts. Tap each term to see what it means and when you reach for it.

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The three big ideas

The usual area of a triangle is ½ × base × height, where the height is the perpendicular distance from the base to the opposite vertex.
This formula needs a base and its matching height — for a right triangle the two perpendicular sides are the base and height, so it is easy.
The catch: if you are only told the three side lengths and the height is not known, you cannot use ½ × base × height directly.
That is exactly the gap Heron’s formula fills — it needs only the three sides.

First find the semi-perimeter: s = (a + b + c)/2 — half the perimeter of the triangle.
Then the area is Area = √(s(s−a)(s−b)(s−c)), using the three sides a, b and c.
It works for any triangle — scalene, isosceles or equilateral — straight from the side lengths.

Common mistake: the s in Heron’s formula is the semi-perimeter (HALF the perimeter), s = (a + b + c)/2 — not the full perimeter a + b + c. Using the whole perimeter is a very common slip and gives the wrong area.

Worked example. Find the area of a triangle with sides 3, 4 and 5.

Step 1 — semi-perimeter: s = (3 + 4 + 5)/2 = 12/2 = 6.

Step 2 — apply Heron’s formula: Area = √(s(s−a)(s−b)(s−c)) = √(6 × 3 × 2 × 1) = √36 = 6 square units.

Equilateral triangle — with all sides equal to a, Heron’s formula simplifies to Area = (√3 ÷ 4) × a².
Quadrilaterals — split the four-sided shape into two triangles with a diagonal, find each triangle’s area with Heron’s formula, and add them.
The same trick handles any many-sided field: cut it into triangles, Heron each one, and sum the areas.

Where you'll meet it

Heron’s formula, at work

Triangular fields & plots

A farmer or builder can measure the three sides of a triangular field with a tape and get its exact area from Heron’s formula — no need to climb to the apex to measure a height. That area decides seed, fencing or flooring cost.

Surveying irregular land

Real plots are rarely neat rectangles. Surveyors split an irregular boundary into triangles, measure each side, apply Heron’s formula to every triangle, and add the areas to estimate the whole parcel of land.