Heron's formula

Heron's formula is a method for calculating the area of a triangle when the lengths of all three sides are known.

 

Formula

If the triangle has side lengths \( \large a \), \( \large b \), \( \large c \), one can first calculate the semiperimeter:

 

$$ \large s = \frac{a+b+c}{2} $$

 

The area is then given by:

 

$$ \large A = \sqrt{s \cdot (s-a) \cdot (s-b) \cdot (s-c)} $$

 

 

Why does the formula work?

The formula can be derived by combining the well-known area formula for triangles

 

$$ \large A = \frac{1}{2} \cdot a \cdot h_a $$

 

where \( \large h_a \) is the height to the side \( \large a \), with the Pythagorean theorem.

 

By substituting the height expressed in terms of the sides \( \large a, b, c \) and simplifying, the height can be eliminated, leading to a formula that depends only on the side lengths. The result is Heron's formula.

 

 

Example

A triangle has side lengths \( \large a = 5 \), \( \large b = 6 \), \( \large c = 7 \).

The semiperimeter is:

 

$$ \large s = \frac{5+6+7}{2} = 9 $$

 

The area is:

 

$$ \large A = \sqrt{9 \cdot (9-5) \cdot (9-6) \cdot (9-7)} = \sqrt{9 \cdot 4 \cdot 3 \cdot 2} = \sqrt{216} \approx 14.7 $$

 

So the area of the triangle is approximately 14.7.