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, the formula is :









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

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:


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:


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