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Area of Right Triangles

This lesson presents the idea that the area of any right triangle is exactly half of a certain rectangle, and contains varied exercises for students. To find the area of any right triangle, you simply multiply the lengths of the two sides that are perpendicular to each other, and then take half of that.

This rectangle is divided into two right triangles that are congruent. This means that
if you could flip one of them and move it on top of the other, they would match exactly.

The rectangle has an area of 2 × 4 = 8 square units.
Can you figure out what the area of just one of the triangles is?

       
Here the area of the whole rectangle is 3 × 5 = 15 square units.
How could you figure out the area of just one of the triangles?
 		   
Here the sides of the triangle are 6 and 3 units. The other two sides of the
rectangle are drawn with dotted lines. The area of the rectangle is 18 square
units. The area of just the triangle is half of that, or 9 square units.
      Let’s look closer at the last triangle above. To confirm that its area is 9 square units, we can count the little squares in the triangle. 

Notice that some of the parts do not cover a complete square, but by combining those we can make whole squares and then count them.

 

1. Find the area of these right triangles. To help you, trace the “helping rectangle” for the triangles.

    a. ________ square units

    b. ________ square units

    c. ________ square units

    d. ________ square units

    e. ________ square units

    f. ________ square units

    g. ________ square units

    h. ________ square units



To find the area of a right triangle, multiply the lengths of the two sides that are perpendicular
to each other (in other words, the two that form the right angle). Then take half of that.

This works because the area of a right triangle is exactly ___________ of the area of the rectangle.


2. Draw a right triangle whose two perpendicular sides are given below, and then find its area.

    a.  1.2 cm and 5 cm

    b.  2 1/2 inches and 1 1/4 inches

 

We can find the area of this house-shape in three parts.

1. The square has an area of 4 × 4 = 16 square units.

2. Triangle 2 has perpendicular sides of 3 and 2 units,
     so its area is (1/2) × 2 × 3 = 3 square units.

3. Triangle 3 is the same shape and size as triangle 2,
    so its area is also 3 square units.

Lastly, add the areas: 16 + 3 + 3 = 22 square units in total.

3. Find the areas of these compound shapes.

a.

 

 

 

 

 b.

 

 c.

4. Draw a right triangle whose area is 13 square centimeters.
    Can you only draw one right triangle with that area, or several different kinds?
    Explain.


 

5. In the grid, draw 3 different right triangles that each have an area of 6 square units.



This lesson is taken from Maria Miller's book Math Mammoth Geometry 2, and posted at www.HomeschoolMath.net with permission from the author. Copyright © Maria Miller.


Math Mammoth Geometry 2

A self-teaching worktext for 6th and 7th grades that covers the area of triangles, parallelograms, and polygons, pi, area of circle, nets, surface area, and volume of common solids.

Download ($5.70). Also available as a printed copy.

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