Let's investigate the features and areas of parallelograms.
Exercise \(\PageIndex{1}\): Features of a Parallelogram
Figures A, B, and C are
parallelograms
. Figures D, E, and F are
not
parallelograms.
Study the examples and non-examples. What do you notice about:
the number of sides that a parallelogram has?
opposite sides of a parallelogram?
opposite angles of a parallelogram?
Exercise \(\PageIndex{2}\): Area of a Parallelogram
Find the area of the parallelogram and explain your reasoning.
Change the parallelogram by dragging the green points at its vertices. Find its area and explain your reasoning.
If you used the polygons on the side, how were they helpful? If you did not, could you use one or more of the polygons to show another way to find the area of the parallelogram?
Exercise \(\PageIndex{3}\): Lots of Parallelograms
Find the area of each parallelogram. Show your reasoning.
Summary
A
parallelogram
is a quadrilateral (it has four sides). The opposite sides of a parallelogram are parallel. It is also true that the opposite sides of a parallelogram have equal length, and the opposite angles of a parallelogram have equal measure.
There are several strategies for finding the area of a
parallelogram
.
We can decompose and rearrange a parallelogram to form a rectangle. Here are three ways:
We can enclose the parallelogram and then subtract the area of the two triangles in the corner.
Both of these ways will work for any parallelogram. However, for some parallelograms the process of decomposing and rearranging requires a lot more steps than if we enclose the parallelogram with a rectangle and subtract the combined area of the two triangles in the corners.
Glossary Entries
Definition: Parallelogram
A parallelogram is a type of quadrilateral that has two pairs of parallel sides.
Here are two examples of parallelograms.
Definition: Quadrilateral
A quadrilateral is a type of polygon that has 4 sides. A rectangle is an example of a quadrilateral. A pentagon is not a quadrilateral, because it has 5 sides.
Practice
Exercise \(\PageIndex{4}\)
Select
all
of the parallelograms. For each figure that is
not
selected, explain how you know it is not a parallelogram.
Exercise \(\PageIndex{5}\)
Decompose and rearrange this parallelogram to make a rectangle.
What is the area of the parallelogram? Explain or your reasoning.
Exercise \(\PageIndex{6}\)
Find the area of the parallelogram.
Exercise \(\PageIndex{7}\)
Explain why this quadrilateral is
not
a parallelogram.
Exercise \(\PageIndex{8}\)
Find the area of each shape. Show your reasoning.
(From Unit 1.1.3)
Exercise \(\PageIndex{9}\)
Find the area of the rectangle with each set of side lengths.