Heron’s Formula

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Drawing of any triangle

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).

Heron's formula. Formula for the area of a triangle with three known sides

Exercise 1

Exercise 1 drawing

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.

Calculation of the semiperimeter in exercise 1

Now we apply Heron’s formula:

Solution in exercise 1

The area of the triangle is 6 cm2.

Exercise 2

Exercise 1 drawing

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:

Formula for the area of a triangle with known base and altitude

With this formula we can calculate the altitude:

Solution in exercise 2

That is h = 2.4 cm.

Circumscribed Circle in a Triangle

Drawing of a triangle inscribed in a circle to calculate its area

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.

Formula for the area of a triangle inscribed in a circle

Inscribed Circle in a Triangle

Drawing a circumscribed triangle on a circle to find its area

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.

Formula for the area of a triangle circumscribed in a circle

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.

Table of triangle area formulas for the reasons we know

AUTHOR: Bernat Requena Serra

YEAR: 2020


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