 Instructor: Malcolm M.
Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

## How To Find The Area of a Triangle (Formula & Examples)

An old saying has it that good things come in threes: Life, Liberty, and the Pursuit of Happiness; three wishes from a genie; three little pigs. And, of course, three sides to a triangle. Today you can learn how to find the area of a triangle, using its three sides.

## What you'll learn:

After working your way through this lesson and video, you will be able to:

• Define an oblique triangle
• Recall the meaning of, and identify, the base and altitude (or height) of a triangle
• Recall, identify and apply the formula for finding the area of a triangle

First, let's cover a few definitions that will help us on our quest to finding the area of a triangle.

## Area

The area of a triangle is the interior space enclosed by its three straight sides. Though always triangle-shaped, it is measured in square units.

## Base

The base of a triangle is the side used to create the altitude, or height. Below we have $△RCK$, which is clearly an oblique triangle:

[insert drawing △$RC$K with $\angle R$=31°, $\angle C$=47°,$\angle K$=102°; orient so side $RC$, identified as 8 cms long, is vertical and on left side; side $RK$ must be 5.98 cms and side $KC$ must be 4.21 cms]

Normally, geometry textbook writers give you a triangle with a base drawn horizontally for you, but that is deceptive! No deceptions here -- any side can be a base!

## Height or Altitude

Add a perpendicular from side $RC$ over to $\angle K$. That perpendicular is the altitude, or height, of the triangle from that base. We know that side $RC$ is $8$ cms long, and we can calculate the height to be about $3.08$ cms. Label the point where the altitude intersects with $RC$ as . This triangle $ROCKs$!

Notice that our altitude is perpendicular to side $RC$, even when side $RC$ is vertical, as in our drawing. The base does not have to be drawn horizontally for you.

For our same triangle, we could pick a different side to be the base. Let's use side $RK$, $5.98$ cms, and construct an altitude to $\angle C$. That altitude is $4.12$ cms.

[Same drawing but with new altitude constructed from side $RK$ to $\angle C$]

Try the last side and construct an altitude from side $KC$, $4.21$ cms long, and have it meet $\angle R$. That altitude is $5.85$ cms in height.

[Same drawing but with new altitude constructed from side $KC$ to $\angle R$]

## Oblique Triangle

You worked hard to recall equilateral, isosceles and scalene triangles. They are named by the lengths of their sides. Then someone told you about right, acute and obtuse triangles. They are named by their interior angles. A right triangle, for example, has one right angle in it.

Now we have to let you in on a secret about acute and obtuse triangles.

They are both types of oblique triangles, or triangles with no right angles. Acute triangles have three acute interior angles (each is less than $90°$). An equilateral triangle is an example of an acute triangle. Obtuse triangles have one angle greater than $90°$ and two acute angles.

Can you see why you can never have a triangle with two interior angles greater than $90°$? Two of the three sides would diverge, or move away from each other, instead of converging.

Now we're ready to unlock our formula!

## Area of a Triangle Formula

A handy formula, , gives you the area in square units of any triangle.

We already have $△RCK$ ready to use, so let's try the formula on it:

But we also found the altitude using the other two sides as bases. So let's see if the formula works on all three sides:

That is within $0.0019$ of our first calculation!

That is within $0.0065$ of the first! These are insignificant simple rounding errors, with more exact measurements than you could achieve with a ruler.

## Lesson Summary

Now that you have worked your way through this lesson, you now know how to define a right triangle (having one right angle), an oblique triangle (having no right angles), the base of a triangle, and the altitude or height of a triangle.

You also should be able to recall, identify and apply the formula for finding the area of triangles: .

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