 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.

## Area of a Right Triangle (Formula & Examples)

The right triangle comes along frequently in geometry. It is used over and over for examples, since it offers a readymade right angle, a hypotenuse, and other great parts. Finding the area of a right triangle is easy and fast.

## What you'll learn:

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

• Identify a right triangle
• Manipulate a right triangle to find all its altitudes or heights
• Recall and apply the formula ½ base x height to find the area of a right triangle

## Right Triangle

Let's create a right triangle, $△CAS$, with $\angle A$ as the right angle. It measures $90°$ and has the hypotenuse, or longest side, opposite it. That is side $SC$, long. It is a right triangle because it has a right angle, not because it is facing to the right.

Insert first drawing: right triangle, $△CAS$, $\angle A$ is the right angle, $\angle C$ ≈53.1, ∠S ≈36.8. Drawing oriented with sides $CA$ and $AS$ forming vertical and horizontal. Rotate for second drawing so $AS$ is vertical and $CA$ is horizontal. Rotate for third drawing so side $SC$ is horizontal and introduce altitude from $\angle A$ to Point $H$ on side $SC$. Triangle and altitude form figure C$AS$H.

The word "right" refers to the Latin word rectus, which means upright. A right angle shows one line or line segment upright from another; a right triangle has an upright angle.

## Base

The base of any triangle is the side used to create an altitude, or height, to an opposite vertex. The way our $△CAS$ is sitting, you can easily see that side $AS$ is the base for the altitude $CA$ to $\angle C$. Side $CA$ is the base for the altitude $AS$.

[insert second drawing]

For a right triangle, the sides adjacent to the right angle serve double duty as bases and altitudes, making the calculations of area really, really easy.

## Altitude or Height

The altitude of any triangle is the perpendicular from a base (side) to the opposite vertex. For our $△CAS$ we can rotate the figure one more time and place side $SC$ on the horizontal, then construct an altitude from $\angle A$ down to the base. The point where it intersects $SC$ can be labelled $H$. Now we have $△CAS$ and Point $H$; $CASH$!

## Area of a Right Triangle Formula

The area of a right triangle is the measure of its interior space, in square units. For any triangle, the formula is:

For a right triangle, this is really, really easy to calculate using the two sides that are not the hypotenuse. In our $△CAS$ we can use side $AS$, 24 yards, as the base, making side $CA$, 18 yards, the altitude:

If we rotate the right triangle and use side $CA$ as the base, $AS$ becomes the altitude, and you get the same answer!

Rotate it one more time and use side $SC$, 30 yards, as the base, with constructed altitude AH, which is 14.4 yards in height:

[insert third drawing]

All three sides, all three altitudes or heights, and all three answers the same! Remember the area is always in square units of the linear measurement (yards, in our triangle).

## Lesson Summary

After reviewing this lesson, you are now able to identify a right triangle, manipulate a right triangle to find all its altitudes or heights, and recall and apply the formula ½ base x height to find the area of a right triangle.

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