Unit 1, Lesson 5
Area of Parallelograms
Accelerated 6
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Elena and Tyler were finding the area of this parallelogram:
Here is how Elena did it:
Here is how Tyler did it:
How are the two strategies for finding the area of a parallelogram the same? How they are different?
For each parallelogram:
| parallelogram | base (units) | height (units) | area (sq units) |
|---|---|---|---|
| A | |||
| B | |||
| C | |||
| D | |||
| any parallelogram |
In the last row of the table, write an expression for the area of any parallelogram, using and .
Here are two different parallelograms with the same area. Explain why their areas are equal.
Two different parallelograms P and Q both have an area of 20 square units. Neither of the parallelograms are rectangles.
On the grid, draw two parallelograms that could be P and Q. Explain how you know.
Any pair of a base and a corresponding height can help us find the area of a parallelogram.
If we draw any perpendicular segment from a point on the base to the opposite side of the parallelogram, that segment will always have the same length. We call that value the height. There are infinitely many segments that can represent the height!
When a parallelogram is drawn on a grid and has horizontal sides, we can use a horizontal side as the base.
When it has vertical sides, we can use a vertical side as the base.
The grid can help us find (or estimate) the lengths of the base and of the corresponding height.
Two parallelograms drawn on two grids. First parallelogram, 2 horizontal sides each 8 units long, 2 slanted sides that rise 2 vertical units over 4 horizontal units. Bottom horizontal side labeled, b. A 2-unit perpendicular segment labeled, h, connects the horizontal sides. Second parallelogram, 2 vertical sides each 6 units long, 2 slanted sides that rise 4 vertical units over 4 horizontal units. The left vertical side is labeled, b. A 4-unit perpendicular segment labeled, h, connects one vertex of the vertical side to a point on the other vertical side.
When a parallelogram is not drawn on a grid, we can still find its area if we know a base and a corresponding height.
No matter which side is chosen as the base, the area of the parallelogram is the product of that base and its corresponding height.
We often use letters to stand for numbers. If is a base of a parallelogram (in units), and is the corresponding height (in units), then the area of the parallelogram (in square units) is the product of these two numbers:
Notice that we write the multiplication symbol with a small dot instead of a symbol. This is so that we don’t get confused about whether means multiply, or whether the letter is standing in for a number.
Parallelograms that have the same base and the same height will have the same area; the product of the base and height will be equal. Here are 4 different parallelograms with the same pair of base-height measurements.
Any side of a parallelogram or triangle can be chosen its base. The length of this side is also called the base.
The height is the shortest distance from the base of the shape to the opposite side (for a parallelogram) or to the opposite vertex (for a triangle).
The height can be shown in more than one place. It is always perpendicular to the chosen base.