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Lesson 6Area of Parallelograms

Let's practice finding the area of parallelograms.

Learning Targets:

  • I can use the area formula to find the area of any parallelogram.

6.1 Missing Dots

An arrangement of dots.

How many dots are in the image?

How do you see them?

6.2 More Areas of Parallelograms

  1. open applet in presentation mode
    1. Calculate the area of the given figure in the applet. Then, check if your area calculation is correct by clicking the Show Area checkbox.
    2. Uncheck the Area checkbox. Move one of the vertices of the parallelogram to create a new parallelogram. When you get a parallelogram that you like, sketch it and calculate the area. Then, check if your calculation is correct by using the Show Area button again.
    3. Repeat this process two more times. Draw and label each parallelogram with its measurements and the area you calculated.
  2. Here is Parallelogram B. What is the corresponding height for the base that is 10 cm long? Explain or show your reasoning.
    A parallelogram with side lengths 15 centimeters and 10 centimeters. An 8-centimeter perpendicular segment connects one vertex of the 15-centimeter side to a point on the other 15-centimeter side.
  3. Here are two different parallelograms with the same area.
    open applet in presentation mode
    1. Explain why their areas are equal.
    2. Drag points to create two new parallelograms that are not identical copies of each other but that have the same area as each other. Sketch your parallelograms and explain or show how you know their areas are equal. Then, click on the Check button to see if the two areas are indeed equal.

Are you ready for more?

Here is a parallelogram composed of smaller parallelograms. The shaded region is composed of four identical parallelograms. All lengths are in inches.

Here is a parallelogram composed of smaller parallelograms. The shaded region is composed of four identical parallelograms. All lengths are in inches.

What is the area of the unshaded parallelogram in the middle? Explain or show your reasoning.

Lesson 6 Summary

Any corresponding pair of base and height can help us find the area of a parallelogram, but some base-height pairs are more easily identified than others.

  • 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. The first parallelogram has horizontal sides that are each 8 units long with angled sides that rise 2 vertical units over 4 horizontal units. The bottom horizontal side of the shape is labeled “b”. A 2-unit perpendicular segment labeled “h” connects the horizontal sides. The second parallelogram has two vertical sides that are each 6 units long, with angles 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 a base and a corresponding height are known.

    A parallelogram with side lengths 10 units and 8 units. An 8-unit perpendicular segment connects one vertex of the 8 unit side to a point on the other 8 unit side.

    In this parallelogram, the corresponding height for the side that is 10 units long is not given, but the height for the side that is 8 units long is given. This base-height pair can help us find the area. 

Regardless of their shape, 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 some parallelograms with the same pair of base-height measurements.

Four different parallelograms. Each parallelogram has a base labeled 3 and a height labeled 4.

Lesson 6 Practice Problems

  1. Which three of these parallelograms have the same area as each other?
    Four parallelograms, labeled A, B, C, and D. In figure A, the top and bottom are each 5 units long, and the sides descend 3 units and move right 1 unit. In figure B, the top and bottom are each 3 units long and the left and right sides ascend 5 units while moving right 6 units. Figure C is a square of 4 units on each side. In figure D, the left and right sides ascend 3 units, and the top and bottom descend 1 unit as they move right 5 units.

  2. Which of the following pairs of base and height produces the greatest area? All measurements are in centimeters.

    1. b = 4 , h=3.5
    2. b = 0.8 , h=20
    3. b = 6 , h=2.25
    4. b = 10 , h=1.4

  3. Here are the areas of three parallelograms. Use them to find the missing length (labeled with a "?") on each parallelogram.

    A: 10 square units

    B: 21 square units

    C: 25 square units

    three parallelograms are shown

  4. The Dockland Building in Hamburg, Germany is shaped like a parallelogram.

    an image of the Architecture Port Elbe Hamburg Building
    “Architecture Port Elbe Hamburg Building” via Max Pixel. Public Domain.

    If the length of the building is 86 meters and its height is 55 meters, what is the area of this face of the building?

  5. List all segments that could represent a corresponding height if the side m is the base.

    A parallelogram with a bottom side labeled m and a right side labeled n. Dashed lines e, f, j, and k are drawn perpendicular to side m, and dashed lines g and h are drawn perpendicular to side n.

  6. Find the area of the shaded region. All measurements are in centimeters. Show your reasoning.

    A shaded rectangle located at an angle within a larger rectangle. The sides of the larger rectangle are divided where the smaller rectangle contacts them. The longer sides are labeled 2 and 12 on each side of the divide, and the shorter sides are labeled 6 and 4 on each side of the divide.