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Illustrative Mathematics Unit 6.1, Lesson 6: Area of Parallelograms

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Learn more about finding the area of a parallelogram using its base and height. After trying the questions, click on the buttons to view answers and explanations in text or video.

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Area of Parallelograms
Let's practice finding the area of parallelograms.

6.1 - Missing Dots

An arrangement of dots.

How many dots are in the image? How do you see them? Can you do this without counting the dots?

  • Hints

    The dots are regularly-spaced and are all the same size. Think of the dots as shaded squares in a grid showing the area of a figure.

  • Answers

    The dots make up a square that is 6 by 6 dots, minus the space in the middle that is 3 by 2 dots. Hence, there are (6 × 6) - (3 × 2) = 30 dots.

    Notice that this is similar to what you would do to find the area of a figure with these dimensions.

  • See Video 1 for Whole Lesson
  • See Video 2 for Whole Lesson




6.2 - More Areas of Parallelograms

1. Open the applet.

A. Calculate the area of the given figure in the applet. Then, check if your area calculation is correct by clicking the Show Area checkbox.
B. 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.
C. 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.

  • Hints

    First find the area of the parallelogram using the base and height that are provided. The area of the parallelogram remains the same when a different side is used as a base.

  • Answers

    The area of the parallelogram is 15 cm × 8 cm = 120 square cm.

    The area of the parallelogram remains the same when a different side is used as a base, so 120 square cm = 10 cm × h

    h = 120 square cm ÷ 10 cm
    h = 12 cm


3. Open the applet to see two different parallelograms with the same area.

A: Explain why their areas are equal.
B: 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. (Note that the Check button in the applet is affected by minor mismatches of your parallelograms with the grid.)

  • Hints

    The formula to find the area of a parallelogram is A = b · h.To draw parallelograms with the same area, find pairs of different bases and heights that have the same product.

  • See Possible Answers

    A: Calculating the areas of the two parallelograms using their bases and heights gives the same result:
    Left: 6 × 4 = 24 square units
    Right: 3 × 8 = 24 square units

    Another way to show this is that parallelogram on the right can be decomposed and rearranged into a parallelogram with the same base and height as the one on the left.

    Two parallelograms. The one on the right has been decomposed and rearranged to be a mirror image of the one on the left.

    B: These are examples of two parallelograms that are not identical but have the same area as each other.

    Two parallelograms. On the left, a parallelogram with base 3 units and height 4 units. On the right, a parallelogram with base 2 units and height 6 units.

    Left: 3 units × 4 units = 12 square units
    Right: 2 units × 6 units = 12 square units


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

A parallelogram composed of 4 smaller identical parallelograms and an unshaded region in the middle. Each of the 4 smaller parallelograms has base 5, height 2.4, and side length 3.

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

  • Hints

    One way to find the area of the unshaded region is to find the total area of the small parallelograms, and then subtract this from the area of the large parallelogram. What information do you need to find the area of the large parallelogram?

    To help you find this information, find the area of one of the small parallelograms. When you know the area, you can choose another side of a small parallelogram to be the base and find the new height. Mark all the heights of the small parallelograms.

  • Answers

    One way to find the area of the unshaded region is to find the total area of the small parallelograms, and then subtract this from the area of the large parallelogram. The area of a small parallelogram is 5 × 2.4 = 12 square inches, so the total area of the shaded region is 4 × 12 = 48 square inches.

    To find the area of the large parallelogram, we need the base and height.

    A parallelogram composed of 4 smaller identical parallelograms and an unshaded region in the middle. Each of the 4 smaller parallelograms has base 5, height 2.4, and side length 3. Dashed lines have been added to show how this can be used to find the height of the large parallelogram.

    If a small parallelogram has an area of 12 square inches, and we choose the 3-inch side as the base, the red dashed line is the new height. This dashed line is 12 ÷ 3 = 4 inches.

    To this red line, we can add the first height we used for the small parallelograms (2.4 inches), labeled in blue. This gives us a full height for the large parallelogram: 4 + 2.4 = 6.4 inches.

    The base of the large parallelogram is 5 + 3 = 8 inches. Hence, the area of the large parallelogram is 8 × 6.4 = 51.2 square inches.

    The area of the unshaded region is 51.2 - 48 = 3.2 square inches.




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.

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.

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.

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.



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.

  • Answers

    A, B, and D have the same area, as they have the same values for their bases and heights.

    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. The bases and heights have been labeled to show which parallelograms have the same bases and heights.

    A: 5 × 3 = 15 square units
    B: 3 × 5 = 15 square units
    C: 4 × 4 = 16 square units
    D: 3 × 5 = 15 square units


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

A: b = 4, h = 3.5
B: b = 0.8, h = 20
C: b = 6, h = 2.25
D: b = 10, h = 1.4

  • Answers

    A: 4 cm × 3.5 cm = 14 square cm
    B: 0.8 cm × 20 cm = 16 square cm
    C: 6 cm × 2.25 cm = 13.5 square cm
    D: 10 cm × 1.4 cm = 14 square cm

    B produces the greatest area.


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

Three parallelograms labeled A, B, and C. A has an area of 10 square units and a base of 5 units. B has an area of 21 square units, a height of 7, and a base extension of 1. C has an area of 25 square units, a height of 5, and a base extension of 2.5.

  • Answers

    A: 10 square units ÷ 5 units = 2 units
    B: 21 square units ÷ 7 units = 3 units
    C: 25 square units ÷ 5 units = 5 units

    Note that while dashed lines have been added to extend the bases of B and C to meet the height, the length of the base extension is not necessary to find the base.


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

A photograph of a parallelogram-shaped building via Max Pixel.

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?

  • Answers

    86 meters × 55 meters = 4730 square meters


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.

  • Answers

    e, f, j, and k are all perpendicular to the base m and could represent its corresponding height.


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.

  • Hints

    First, find the area of the large rectangle. Then, is it possible to find the area of the unshaded regions?

  • Answers

    The area of the large rectangle is (12 + 2) × (6 + 4) = 140 square centimeters.

    The unshaded regions are triangles which can be rearranged into rectangles.

    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. Unshaded regions have been rearranged into rectangles.

    The total area of the unshaded regions is (2 × 6) + (12 × 4) = 60 square centimeters.

    Hence, the area of the shaded region is 140 - 60 = 80 square centimeters.



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