Lesson 4Parallelograms

Let’s investigate the features and area of parallelograms.

Learning Targets:

  • I can use reasoning strategies and what I know about the area of a rectangle to find the area of a parallelogram.
  • I know how to describe the features of a parallelogram using mathematical vocabulary.

4.1 Features of a Parallelogram

Figures A, B, and C are parallelograms. Figures D, E, and F are not parallelograms. 

Six figures on a grid labeled A--F.

Study the examples and non-examples. What do you notice about:

  1. the number of sides that a parallelogram has?
  2. opposite sides of a parallelogram?
  3. opposite angles of a parallelogram?

4.2 Area of a Parallelogram

  1. Find the area of the parallelogram and explain your reasoning.
  1. Change the parallelogram by dragging the green points at its vertices. Find its area and explain your reasoning.
  1. 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?

4.3 Lots of Parallelograms

Find the area of the following parallelograms. Show your reasoning.

Two figures on a grid: parallelogram A and parallelogram B.
Parallelogram C has base 6, height 4, and diagonal length 4.5.

Lesson 4 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.
  • The opposite angles of a parallelogram have equal measure.
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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:
    Three identical parallelograms with horizontal sides that are four units long, drawn in grids. The first parallelogram has a perpendicular segment extending from 2 units in from the top left down to the vertex of the bottom horizontal side. An arrow extends from the resulting triangle to the opposite side of the parallelogram to create a rectangle measuring 4 units wide and 3 units high. The second parallelogram has a perpendicular segment extending from 2 units in from the bottom right up to the vertex of the top horizontal side. An arrow extends from the resulting triangle to the opposite side of the parallelogram to create a rectangle measuring 4 units wide and 3 units high. The third parallelogram has a perpendicular segment extending from 3 units in from the bottom right up to the vertex of the top horizontal side. An arrow extends from the resulting shape to the opposite side of the parallelogram to create a rectangle measuring 4 units wide and 3 units high.
  • We can enclose the parallelogram and then subtract the area of the two triangles in the corner.
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Both of these ways will work for any parallelogram.

For some parallelograms, however, 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. Here is an example.

A shaded parallelogram drawn on a grid, with a base of three units angled sides that decline 6 vertical units over 9 horizontal units. The parallelogram is divided by dashed segments into six equal right triangles, triangle has one side that is 2 units and another that is 3 units. Arrows extend to the left from each of the lower 5 triangles. The resulting shape is a rectangle that is 6 units tall by 3 units wide.

Glossary Terms

parallelogram

A parallelogram is a type of quadrilateral that has two pairs of parallel sides.

Here are two examples of parallelograms.

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Lesson 4 Practice Problems

  1. Select all of the parallelograms. For each figure that is not selected, explain how you know it is not a parallelogram. 

  2. a. Decompose and rearrange this parallelogram to make a rectangle.

    b. What is the area of the parallelogram? Explain your reasoning.

  3. Find the area of the parallelogram.

    A parallelogram with one side labeled 3.2 centimeters, and another side labeled 10 centimeters. A dashed line perpendicular to the 10 centimeter sides is labeled 3 centimeters
  4. Explain why this quadrilateral is not a parallelogram.

    A quadrilateral with a bottom side length of 8 units, a top side length of 4 units. The left side ascends 5 units while moving right 13 units, and the right side ascends 5 units while moving right 9 units.
  5. Find the area of each shape. Show your reasoning.

    A shape with eight sides. Four sides are straight sides and extend left, right, up and, down for 2 units each. The remaining sides are angled sides connecting each of the straight sides to the next. The shape is a total of 6 units tall and 6 units wide.
    A shape with six sides. It is 9 units long and six units wide at it’s widest point. Two vertical sides connect four sloped sides, which meet at either end of the shape.
  6. Find the areas of the rectangles with the following side lengths.

    1. 5 in and \frac13 in

    2. 5 in and \frac 43 in

    1. \frac 52 in and \frac 43 in

    2. \frac 7 6 in and \frac 67 in