Math Doubts

Heron’s formula

Formula

$A = \sqrt{s(s-a)(s-b)(s-c)}$

Introduction

A mathematician Heron (or Hero) of Alexandria derived a geometrical proof to express area of a triangle in algebraic form in terms of lengths of three sides and half-perimeter of the triangle. Hence, this formula is called as Heron’s formula or Hero’s formula.

heron's formula triangle

$a$, $b$ and $c$ are lengths of three sides of a triangle and its perimeter is denoted by $2s$.

$2s \,=\, a+b+c$

$\implies$ $s \,=\, \dfrac{a+b+c}{2}$

The area of the triangle is denoted by either $A$ or $\Delta$.

$\Delta = \sqrt{s(s-a)(s-b)(s-c)}$

This formula in algebraic form is called as Hero’s (or Heron’s) formula in geometry.

Example

Find area of a triangle, if $a = 5cm$, $b = 6cm$ and $c = 7cm$.

Firstly, find the semi perimeter of the triangle.

$s = \dfrac{5+6+7}{2}$

$\implies$ $s = \dfrac{18}{2}$

$\implies$ $\require{cancel} s = \dfrac{\cancel{18}}{{2}}$

$\implies$ $s = 9cm$

Now, find the area of the triangle.

$A = \sqrt{9(9-5)(9-6)(9-7)}$

$\implies$ $A = \sqrt{9(4)(3)(2)}$

$\implies$ $A = \sqrt{9 \times 4 \times 3 \times 2}$

$\implies$ $A = \sqrt{36 \times 6}$

$\,\,\, \therefore \,\,\,\,\,\,$ $A = 6\sqrt{6} \, cm^2$

Proof

Learn how to derive the hero’s formula in geometrical approach to find the area of a triangle.

Math Questions

The math problems with solutions to learn how to solve a problem.

Learn solutions

Math Worksheets

The math worksheets with answers for your practice with examples.

Practice now

Math Videos

The math videos tutorials with visual graphics to learn every concept.

Watch now

Subscribe us

Get the latest math updates from the Math Doubts by subscribing us.

Learn more

Math Doubts

A free math education service for students to learn every math concept easily, for teachers to teach mathematics understandably and for mathematicians to share their maths researching projects.

Copyright © 2012 - 2023 Math Doubts, All Rights Reserved