Lesson 4Parallelograms

Learning Goal

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.

Lesson Terms

  • parallelogram
  • quadrilateral

Warm Up: Features of a Parallelogram

Problem 1

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?

Activity 1: Area of a Parallelogram

Problem 1

  1. Find the area of the parallelogram and explain your reasoning.

    open applet in presentation mode

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

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  3. 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?

Print Version

Find the area of each parallelogram. Show your reasoning.

1
A parallelogram on a grid.
2
A parallelogram in a grid. The parallelogram has a base of 6 units and a height of 3 units.

Activity 2: Lots of Parallelograms

Problem 1

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

  1. a parallelogram on a grid with a height of 5 and a width of 3
  2. a parallelogram on a grid with a height of 6 and a width of 2.
  3. Parallelogram C has base 6, height 4, and diagonal length 4.5.

Lesson 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.

A parallelogram with side lengths of 4.24 and 5 and angles of 45 degrees and 135. A second parallelogram with side of 9.34 and 4 with angles of 27.2 and 152.8 degrees.

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.

    A blue parallelogram enclosed in a 3x6 rectangle on a grid. The triangles outside the parallelogram but in the rectangle are moved together to form a rectangle.

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.