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. 

Heron's formula

Let us call the lengths of the three sides a, b, and c. 

Let s be called the semi-perimeter.

The name makes sense because it is the perimeter divided by two.

Then, the formula to use to find the perimeter is the following.

Notice that you must first compute s or the semi-perimeter

Heron's formula

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

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