Area of a triangle

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Finding the area of a triangle is depending on the type of triangle or which sides and angles we know.

The most common formula for finding the area of a triangle is:

Area of an Equilateral Triangle

An equilateral triangle has three equal sides and angles. As in any type of triangle, its area is equal to half of the product of its base and height. So if the altitude of an equilateral triangle is:

The area of an Equilateral Triangle it will be defined by the following formula:

Area of an Isosceles Triangle

Like in any triangle, the area of an isosceles triangle is determined by multiplying the base b and the height h and then divides it by 2. In an isosceles triangle the area is:

Area of a Scalene Triangle

The area of a scalene triangle can be calculated using Heron’s formula if all its sides (a, b and c) are known.

It is also possible to calculate the area of a triangle if we know the length of one side (b) and the altitude h related to that side.

Area of a Right Triangle

A right triangle has one right angle (90°), so its height coincides with one of its sides (a). Its area is half the product of the two sides that form the right angle (legs a and b).

Area of a Triangle Knowing One Side and the Altitude Related to that Side

The area of any triangle can be calculated by knowing one side and the height (or altitude), associated with that side. This side serves as the base.

In fact the most common way to calculate the area of a triangle is to take half of the base (b) times the height (h). That’s:

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).

How to Use Trigonometry to Find the Area of a Triangle

A triangle has six parts, three sides and three angles. Given almost any three of them (three sides, two sides and an angle, or one side and two angles) you can find the other three values.

So, any triangle can be solved (solution of triangles) if three of its elements are known, but at least one of them must be a side.

In particular, by knowing two of its sides and the angle they form you can calculate the area of a triangle.
Therefore, three formulas can be applied to calculate the area depending on the two sides that are known:

• a and b
• a and c
• or b and c

Triangle Inscribed in a Circle

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.

Triangle Circumscribed about a Circle

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 ( equilateral triangle), two (base and height) or three (as long as they are not the three angles).

Practice Exercises

Exercise 1: Area of an Equilateral Triangle

Find the area of an equilateral triangle in which its three equal sides have the length a=5 cm.

What is its area?

Applying the above formula:

The area is 10.83 cm².

Exercise 2: Area of an Isosceles Triangle

Determine the area of a isosceles triangle knowing its two equal sides (a=3 cm) and the unequal one, whose length is 2 cm (b=2 cm).

What is its area?

Calculate the area using the above formula by multiplying the base by the height:

The isosceles triangle’s area is 2.83 cm².

Exercise 3: Area of a Scalene Triangle

Find the area of the scalene triangle given its three sides: a=2 cm, b=4 cm and c=3 cm.

What is its area?

We can calculate the area using Heron’s formula. First, we have to determine the semiperimeter s:

Now, we can apply the Heron’s formula:

Exercise 4: Area of a Right Triangle

Find the area of a right triangle given its two legs, which form the right angle: a = 3 cm and b = 4 cm.

Solution:

Apply the above formula:

And we have that the area is 6 cm².

Exercise 5: Area of a Triangle Given Three Sides

Find the area of a triangle ABC with three known sides: a = 6 cm, b = 10 cm, and c = 8 cm. Try to do it without using Heron’s Formula.

First we need to find the angles.

To do it, apply the Law of Cosines whose formula is:

Substitute values:

Repeat the step to solve for the other missing angles:

And:

And check whether the sum of the three angles is equal to 180°:

Since the B angle is a right angle, sides a and c are the legs. So, we can apply the most common formula for the area of a triangle: