
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 semiperimeter. 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 cm^{2}
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 cm^{2}
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 cm^{2}

