How To Find The Area of a Right Triangle

Area of a Right Triangle (Formula & Examples)

Video Definitions Formula

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, $△CAS$, 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.

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:

$A = \frac{1}{2} bh$

$A = \frac{1}{2} (24 \times 18)$

$A = \frac{1}{2} \left(432\right)$

$A = 216 yd{s}^2$

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

$A = \frac{1}{2} (18 \times 24)$

$A = \frac{1}{2} \left(432\right)$

$A = 216 yd{s}^2$

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]

$A = \frac{1}{2} bh$

$A = \frac{1}{2} (30 \times 14.4)$

$A = \frac{1}{2} \left(432\right)$

$A = 216 yd{s}^2$

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.

Next Lesson:

How To Find The Perimeter of a Triangle