Area of Parallelograms

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.

We draw a line from one vertex of the parallelogram in order to form a right triangle. Then we move the triangle to the other side, as shown. Look! We get a rectangle! The rectangle’s area is 6 × 4 = 24 square units, and that is also the area of the original parallelogram.

It works here, as well. The area of the rectangle and of the parallelogram are the same: both have the area of 4 × 4 = 16 square units.

The area of a parallelogram is the same as the area of the corresponding rectangle. You construct the rectangle by moving a right triangle from one side of the parallelogram to the other.

1. Imagine moving the marked triangle to the other side as shown. What is the area of the original parallelogram?

2. Draw a line in each parallelogram to form a right triangle. Imagine moving that triangle to the other side so that
    you get a rectangle, like in the examples above. Find the area of the rectangle, thereby finding the area of the
    original parallelogram.

a. _________ sq. units     b. _________ sq. units     c. _________ sq. units     d. _________ sq. units

One side of the parallelogram is called the base. You can choose any of the four sides to be the base, but people often use the “bottom” side. A line segment that is perpendicular to the base and goes from the base to the opposite side of the parallelogram is called the altitude. When we do the trick of “moving the triangle,” we get a rectangle. One of its sides is congruent (has the same length) to the parallelogram’s altitude. The other side is congruent to the parallelogram’s base. That is why you can simply multiply  BASE × ALTITUDE to get the area of a parallelogram.   →   →

3. Draw an altitude to each parallelogram. Highlight or “thicken” the base. Then find the areas.

a. _________ sq. units     b. _________ sq. units     c. _________ sq. units     d. _________ sq. units     e. _________ sq. units

4. a. Draw the altitudes to the parallelograms and mark their bases. One parallelogram's altitude is already
        marked. Notice how that altitude does not “reach” the base, but instead ends at the continuation of
        the base. That is no problem—what matters is that the altitude is perpendicular to the base.

    b. Find the areas of these parallelograms. What do you notice?

5. Draw as many differently shaped parallelograms as you can that all have an area of 12 square units.

8. Find the area of the parallelogram in square meters.

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.

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