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

Picture of Herons Formula

Heron's Formula Applet

See applet on its own page

Heron's Formula Calculator

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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

Round answer to nearest tenth

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

Problem 1

Use Heron's formula to find the area of the triangle pictured with the following side lengths

AB = 8
BC = 41
CA = 44

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

Problem 2

Determine the area of the triangle using Heron's formula to find the area of the triangle pictured with the following side lengths

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.
Round answer to nearest tenth

Problem 3

Determine the area of the triangle using Heron's formula to find the area of the triangle pictured with the following side lengths

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.

Round answer to nearest tenth

Problem 4

If the perimeter of a triangle is 32 units, its area is 35.8 units squared, and the lengths of AB = 14 and BC = 12, what is the length of the third side, side CA?

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)
Problem 5

If the perimeter of a triangle is 26 units, its area is 18.7 units squared, and the lengths of AB = 12 and BC = 4, what is the length of the third side, side CA?

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)
toHeron's Formula Interactive Applet

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