# How to find the area of a right triangle

## Area of a right triangle

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.

## Right triangle

Let's create a **right triangle**, $\triangle CAS$, with $\angle A$ as the right angle. It measures **90° **and has the hypotenuse, or longest side, opposite it. That is side * SC*,

**30 yards**long. It is a right triangle because it has a right angle, not because it is facing to the right.

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 $\triangle CAS$ is sitting, you can easily see that side * AS* is the base for the altitude

*to $\angle C$. Side*

**CA***is the base for the altitude*

**CA***.*

**AS**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 $\triangle CAS$ 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

*can be labelled*

**SC***. Now we have $\triangle CAS$ and Point*

**H***;*

**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 $\triangle CAS$ 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,

*becomes the altitude, and you get the same answer!*

**AS**Rotate it one more time and use side * SC*,

**30 yards**, as the base, with constructed altitude

*, which is*

**AH****14.4 yards**in height:

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 $\frac{1}{2}$** (base x height) **to find the area of a right triangle.