﻿ 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. The picture below illustrates the general fro mu la where S represents the semi-perimeter of the triangle

### Heron's Formula Applet

See applet on its own page

### Heron's Formula Calculator

Side Lengths of Triangle


### Examples

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

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

Step 1
Calculate the semi perimeter, S

S = (3+2+4) /2
S = 9/2 = 4.5

Step 2
Substitute S into the formula

Since Heron's formula relates the side lengths, perimeter and area of a triangle, you might need to answer questions the like the following example.

##### Example 2
(a more challenging problem type)

Given a triangle, with an area of 8.94 square units, a perimeter of 16 and side lengths AB = 3 and CA = 7, what is the length of side BC?

Step 1

Calculate the semi perimeter, S

S = perimeter /2
S = 16/2 = 8

Step 2

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

Step 3

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

(Image was made using our partner site's triangle maker)

### Practice Problems

This problem is similar to example 1

Step 1

Calculate the semi perimeter, S

S = (8 + 41 + 44) /2
S = 93/2 = 46.5

Step 2

Substitute S into the formula. Round answer to nearest tenth

This problem is similar to example 1.

Step 1

Calculate the semi perimeter, S

S = (7 + 6 + 8) /2
S = 21/2 = 10.5

Step 2

Substitute S into the formula.

This problem is similar to example 1

Step 1

Calculate the semi perimeter, S

S = (11 + 12 + 5) /2
S = 28/2 = 14

Step 2

Substitute S into the formula.

This problem is like example 2

Step 1

Calculate the semi perimeter, S

S = perimeter /2
S = 32/2 = 16

Step 2

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

Step 2

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

Picture of Triangle

Here is an accurate picture of a triangle with these side lengths, area and perimeter.

(Image was made using our triangle maker)

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)

Picture of Triangle

Here is an accurate picture of a triangle with these side lengths, area and perimeter.

(Image was made using our triangle maker)

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