Heron's formula

You can use Heron's formula to calculate the area of any triangle when you know the lengths of the three sides. If you call the lengths of the three sides a, b, and c, then here is the formula. 

S is called the semi-perimeter. The name makes sense because it is the perimeter divided by two.

A few examples showing how to use Heron's formula to calculate the area of triangles

Example #1:

Use Heron's formula to find the area of a triangle when a = 3 cm, b = 5 cm, and c = 4 cm

s = (3 + 5 + 4)/2 = 12/2 = 6

s − a = 6 − 3 = 3

s − b = 6 − 5 = 1

s − c = 6 − 4 = 2

s × (s − a) × (s − b) × (s − c) = 6 × 3 × 1 × 2 = 36

√(36) = 6

The area of this triangle is 6 cm2


Example #2:

Use Heron's formula to find the area of a triangle when a = 4 cm, b = 6 cm, and c = 8 cm

s = (4 + 6 + 8)/2 = 18/2 = 9

s − a = 9 − 4 = 5

s − b = 9 − 6 = 3

s − c = 9 − 8 = 1

s × (s − a) × (s − b) × (s − c) = 9 × 5 × 3 × 1 = 135

√(135) = 11.61

The area of this triangle is 11.61 cm2

Example #3:

Use Heron's formula to find the area of a triangle when a = 3/2 cm, b = 5/2 cm, and c = 2 cm

s = (3/2 + 5/2 + 2)/2 = (3/2 + 5/2 + 4/2)/2 = (12/2)/2 = 6/2 = 3

s − a = 3 − 3/2 = (6/2 − 3/2) = (6 − 3)/2 = 3/2

s − b = 3 − 5/2 = (6/2 − 5/2) = (6 − 5)/2 = 1/2

s − c = 3 − 2 = 1

s × (s − a) × (s − b) × (s − c) = 3 × 3/2 × 1/2 × 1 = 9/4

√(9/4) = 3/2

The area of this triangle is 1.5 cm2


Recent Articles

  1. Discrete and Continuous Data

    Sep 17, 20 03:57 PM

    Learn clearly the difference between discrete and continuous data with good examples.

    Read More

New math lessons

Your email is safe with us. We will only use it to inform you about new math lessons.

                                 Follow me on Pinterest

Real Life Math Skills

Learn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball.