Formula for Finding the Area of a Parallelogram

Instructor: Joseph Vigil
In this lesson, you'll learn what a parallelogram is and how to determine a parallelogram's area. You'll also discover the special relationship between squares, rectangles, and parallelograms. Then, you can test your new knowledge with a brief quiz.

Attracting Customers

Lewis runs his own mechanic shop, and he's designing a sign to attract customers. He wants to make his sign stand out to attract attention, so he decides to make a slanting sign, and he wants it to have a blue background.

The sign Lewis makes is really a parallelogram.
blue parallelogram

Before he can make the sign, Lewis needs to know how much paint he'll need if he makes the sign six feet wide and two feet high.

Without realizing it, Lewis has made a parallelogram. A parallelogram is a quadrilateral (or four-sided polygon) that meets three criteria:

  1. Its opposite sides are parallel.
  2. Its opposite sides are congruent, or equal in size.
  3. Its opposite angles are congruent.

Let's check out Lewis's sign. Its opposite sides are parallel because they'll never intersect if extended, and they're the same size:

congruent opposite sides
Parallelogram with congruent sides indicated

And its pairs of opposite angles are equal to each other:

congruent opposite angles
Parallelogram with congruent angles indicated

So his sign is clearly a parallelogram.

Area of a Parallelogram

To determine how much paint he'll need, Lewis will need to find the sign's area. But how would he find the area of such an odd shape?

It's not really such an odd shape, though. In fact, it's a variation on a rectangle.

converting a rectangle to a parallelogram
Converting a rectangle to a parallelogram

If we had the rectangle indicated by the brown lines, all Lewis has done is cut out a triangular portion (in light gray) and pasted it onto the rectangle's other side (in light blue) to create the new shape:

The resulting parallelogram
parallelogram

Since the area of a rectangle is simply length times width, and a parallelogram is just a rearranged rectangle, it has a similar area formula:

A = b * h

Where b is the length of the parallelogram's base and h is the parallelogram's height.

In the case of Lewis's sign, the base is 6 feet long, and the height is 2 feet. We can plug those values in for b and h in our area formula:

A = 6 * 2

A = 12

The sign has an area of 12 square feet. Lewis will have to buy enough paint to cover that much surface.

Land Length

Victor dreams of a country lifestyle and wants to buy a large plot of land. There's a plot he has his eye on. The owner has sent him this diagram of the land:

parallelogram with area 1,200,000 feet and length 6,000 feet

This diagram gives the plot's area, but Victor wants to know how long it is.

We've discovered that the formula for a parallelogram's area is:

A = b * h

In this case, we have the values for A and h. Since we have two out of the three values, we can solve for b:

1,200,000 = b * 6,000

1,200,000 / 6,000 = b

200 = b

This plot of land is 200 feet long.

Parallelograms, Rectangles, and Squares. Oh My!

Let's look at the criteria for a parallelogram again:

  1. Its opposite sides are parallel.
  2. Its opposite sides are congruent, or equal in size.
  3. Its opposite angles are congruent.

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