﻿ Herons Formula. Explained with pictures, examples and practice problems # Heron's Formula

Explained with examples and pictures

### The Formula

Heron's formula is named after Hero of Alexendria, a Greek Engineer and Mathematician in 10 - 70 AD. You can use this formula to find the area of a triangle using the 3 side lengths.

Therefore, you do not have to rely on the formula for area that uses base and height. Diagram 1 below illustrates the general formula where S represents the semi-perimeter of the triangle.

semi-perimeter is just the perimeter divided by 2 : $$\frac{perimeter}{2}$$ .

Diagram 1
Diagram 2

A specific example

### Heron's Formula Applet

 Choose Base Side AB Side BC Side AC Formula Type Conventional Heron's



Side Lengths of Triangle

Status: Calculator waiting for input ### Examples

##### Example 1
(Straight forward example)

Use Heron's formula to find the area of triangle ABC, if $$AB= 3, BC = 2 , CA = 4$$ .

Step 1
Calculate the semi perimeter, S

$s = \frac{ 3 + 2 +4}{2 } \\ s = 4.5$

Step 2
Substitute S into the formula.

$A= \sqrt{ 4.5( 4.5 - 3 ) (4.5-2)( 4.5-4) } \\ A= \sqrt{8.4375 } \\ A \approx 2.9$ Since Heron's formula relates the side lengths, perimeter and area of a triangle, you might need to answer more challenging question types like the following example.

##### Example 2

Given $$\triangle ABC$$, with an area of $$8.94$$ square units, a perimeter of $$16$$ units and side lengths $$AB = 3$$ and $$CA = 7$$, what is $$\red { BC }$$ ?

Step 1

Calculate the semi perimeter, S.

$S =\frac{ perimeter}{2} \\ S = \frac{16}{2} \\ S = 8$

Step 2

Substitute known values into the formula . Let $$\red x = \red{ BC}$$

$A= \sqrt{ S(S - AB)(S - \red { BC} )(S- CA) } \\ 8.94 = \sqrt{ 8(8 - 3)(8 - \red x )(8- 7) } \\ 8.94 = \sqrt{ 8(5)(8 - \red x )(1) }$

Step 3

Solve for $$\red x$$ (square both sides and go from there).

$8.94^{\blue 2} = \left( \sqrt{ 8(5)(8 - \red x )(1) } ) \right) ^{\blue 2} \\ 79.9236 = 8(5)(8 -\red x)(1) \\ 79.9236 = 40(8 -\red x) \\ \frac{ 79.9236}{40} = 8 -\red x \\ 1.999809 = 8 -\red x \\ x \approx 6.0$

### Practice Problems

##### Problem 1

This problem is similar to example 1.

Step 1

Calculate the semi perimeter, S.

$S = \frac{8+41+44}{2} \\ S= \frac{93}{2} \\ S = \blue { 46.5 }$

Step 2

Substitute S into the formula. Round answer to nearest tenth.

$A = \sqrt{ \blue{ 46.5} \cdot ( \blue{ 46.5} -8) \cdot ( \blue{ 46.5}- 41) \cdot ( \blue{ 46.5} - 44) } \\ A = \sqrt{ \blue{ 46.5} \cdot 38.5 \cdot 2.5 \cdot 5.5 } A \approx 156.9$

##### Problem 2

This problem is similar to example 1.

Step 1

Calculate the semi perimeter, S.

$S = \frac{ 7+6+ 8}{2} \\ S = \frac{21}{2} \\ S = \blue {10.5 }$

Step 2

Substitute S into the formula . Round answer to nearest tenth.

$A = \sqrt{\blue {10.5 } (\blue{10.5} - 7) (\blue{10.5} - 6 ))(\blue{10.5} - 8 )} \\ A \approx 20.3$

##### Problem 3

This problem is similar to example 1.

Step 1

Calculate the semi perimeter, S.

$S = \frac{ 11 + 12 + 5}{2 } \\ S = \frac{ 28}{2} \\ S = \blue { 14 }$

Step 2

Substitute S into the formula.

$A = \sqrt{ \blue{14} (\blue{14} - 11) (\blue{14} - 12) ( \blue{14} - 5) } \\ A \approx 27.5$ Problems 2 and onward are similar to example 2 .
##### Problem 4

This problem is like example 2.

Step 1

Calculate the semi perimeter, S.

$S = \frac{perimeter}{2} \\ S = \frac{32}{2} \\ S = \blue{16}$

Step 2

Substitute known values into the formula. Let $$\red {x} = \red {CA}$$ .

$A =\sqrt{ S (S- AB)( S- BC )( S - CA)} \\ 35.8 =\sqrt{ \blue{16} (\blue{16}- 14)( \blue{16}- 12 )( \blue{16} - \red x)}$

Step 3

Solve for x (square both sides and go from there).

$35.8^{\blue{2}} =\sqrt{ 16 ( 16- 14)( 16- 12 )( 16 - \red x)}^{\blue{2}} \\ 1281.64 =16 ( 16- 14)( 16- 12 )( 16 - \red x) \\ 1281.64 =16 (2 )( 4 )( 16 - \red x) \\ 1281.64 = 128( 16 - \red x) \\ \frac{ 1281.64}{128} = 16 - \red x \\ \frac{ 1281.64}{128} = 16 - \red x \\ 10.0128125 = 16 - \red x \\ \red x \approx 6$

##### Problem 5

This problem is like example 2.

Step 1

Calculate the semi perimeter, S.

S = perimeter / 2
S = 26 / 2 = 13

Step 2

Substitute known values into the formula. Let x equal side length CA.

Step 3

Solve for x (square both sides and go from there).