Heron’s formula helps us find the area of a triangle when all three sides are known but the height is not given. This makes it one of the most useful tools in geometry and real-life measurements where height is difficult to calculate. Whether solving exam problems or dealing with land measurements, this formula gives a reliable and accurate method for calculating the triangle’s area.
Formula for Calculating the Area of a Triangle using Heron’s Formula – Overview
| Formula | Variables & Meaning | When It Is Used |
|---|---|---|
| A = √[s(s – a)(s – b)(s – c)] | a, b, c = sides; s = semi-perimeter | When base and height are not directly known |
What is Heron’s Formula in Maths?
Heron’s formula is a mathematical method for calculating the area of a triangle when all three sides are given. Instead of requiring height, Heron’s formula uses just the three side lengths to find the area.
To use it, first calculate the semi-perimeter (s) using:
s = (a + b + c) / 2
Then substitute values of a, b, c, and s into the formula:
A = √[s(s – a)(s – b)(s – c)]
This formula works for scalene, isosceles, and equilateral triangles. It is widely used in geometry, mensuration, engineering, architecture, land surveys, and competitive exams like CUET, JEE, SSC, and Banking.
Examples to Calculate Area Using Heron’s Formula
Example 1: Sides = 7 cm, 8 cm, 9 cm
Given sides: 7 cm, 8 cm, 9 cm
Formula: A = √[s(s – a)(s – b)(s – c)]
Step 1: s = (7 + 8 + 9) / 2 = 12
Step 2: A = √[12(12 – 7)(12 – 8)(12 – 9)]
Step 3: A = √[12 × 5 × 4 × 3]
Step 4: A = √720 = 26.83 cm²
So, the area of the triangle is 26.83 cm².
Example 2: Sides = 6 m, 5 m, 5 m
Given sides: 6 m, 5 m, 5 m
Formula: A = √[s(s – a)(s – b)(s – c)]
Step 1: s = (6 + 5 + 5) / 2 = 8
Step 2: A = √[8(8 – 6)(8 – 5)(8 – 5)]
Step 3: A = √[8 × 2 × 3 × 3]
Step 4: A = √144 = 12 m²
So, the area of the triangle is 12 m².
FAQs about Heron’s Formula
Q1. What is Heron’s formula used for?
It is used to calculate the area of a triangle when all three side lengths are known.
Q2. Can Heron’s formula be applied to any triangle?
Yes, it works for scalene, isosceles, and equilateral triangles.
Q3. What is the semi-perimeter in Heron’s formula?
It is half the sum of the three sides of a triangle.
Q4. Why is Heron’s formula important for exams?
It helps solve triangle problems where height is not given, making it common in CUET, SSC, JEE, and other exams.
Q5. Is Heron’s formula used in the real world?
Yes, it is used in land surveying, architecture, engineering, and design calculations.
Q6. Who introduced Heron’s formula?
The formula was given by Hero (Heron) of Alexandria, a Greek mathematician.