Heron’s Formula

Heron’s Formula gives the area of a triangle when we know all three sides.

The area is calculated from the semiperimeter of the triangle, s and the length of the sides (a, b, and c).

Exercise 1

Find the area of the given triangle.

Solution:

In this triangle we know the three sides: a=4 cm, b=5 cm and c=3 cm. We are going to find its area using Heron’s formula.

First, we calculate the semiperimeter s.

Now we apply Heron’s formula:

The area of the triangle is 6 cm².

Exercise 2

Find the altitude on side b of the given triangle.

Solution:

Applying Heron’s formula, we have seen that its area is 6 cm².

We also know that the area of a triangle is:

With this formula we can calculate the altitude:

That is h = 2.4 cm.

Circumscribed Circle in a Triangle

We have one more procedure to calculate the area of a triangle, if we know its three sides and the radius R of the circumscribed circle or circumcircle, without having to resort to Heron’s formula.

Inscribed Circle in a Triangle

Similarly, without Heron’s formula, we have one more procedure to calculate the area of a triangle, but now from the radius of the inscribed circle, or incircle.

Table of Triangle Area Formulas

You can see the table of triangle area formulas. Depending on the type of triangle you may need one element, like in the equilateral triangle, two (base and altitude) or three. In this latter case, provided it is not its three angles.