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)


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.

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 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, A is the right 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 A to Point H on side SC. Triangle and altitude form figure CASH.

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 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 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:

A = 12 (base × height)

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 = 12 bh

A = 12 (24 × 18)

A = 12 (432)

A = 216 yds2

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

A = 12 (18 × 24)

A = 12 (432)

A = 216 yds2

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 = 12 bh

A = 12 (30 × 14.4)

A = 12 (432)

A = 216 yds2

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

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