Heron's Formula

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 Calculator

Side Lengths of Triangle Side 1: Side 2: Side 3:

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

Show Picture

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

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

Show Area

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

Show Area

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

Answer

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

Show Area

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 5) 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?

Show Area

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)

Here is an accurate picture of a triangle with these side lengths, area and perimeter. (Image was made using our triangle maker)

Problem 6) 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?

Answer

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 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)