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9.10: Unknown Dimensions of Parallelograms

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Have you ever measured banners in a stadium? Have you ever seen one up close?

One afternoon, Miguel’s assignment is to hang a banner in the stadium. The banner say’s “Go Wildcats” and is the shape of a parallelogram with a yellow background and dark blue border. He has two different places that he can put it. One spot is only 11 feet wide and one spot is 13 feet wide.

Miguel isn’t sure which spot is the best spot for the “Go Wildcats” banner. Wherever he chooses to hang the banner it needs to fit with a little space on each side so that people don’t bump it as they go past.

The banner is rolled up in a tube. Miguel looks at the label to see that the square footage of the banner is 42 square feet. The height of the banner is 3.5 feet. The length of the banner is missing. It seems to have been covered up with a shipping label.

Miguel is a bit frustrated. He isn’t sure that he has enough information to figure out the length of the banner. He knows that he needs to figure out the length of the banner in order to choose the correct spot to hang it. Miguel takes out a piece of paper to try to figure things out.

Does Miguel have enough information? Can he figure out the length of the banner using the given information? How long is the banner? Which spot is the best spot?

All of these questions can be answered by using parallelograms and this Concept will teach you all that you need to know to figure this one out.

Guidance

Sometimes, you will be given the area of the parallelogram and one other dimension such as the base or the height. Then you will have to use the formula and your problem solving skills to figure out the missing dimensions.

A parallelogram has an area of 105 \ m^2 . The height of the parallelogram is 7 m. What is its base?

In this problem, we know the area and the height of the parallelogram. We put these numbers into the formula and solve for the base, b .

A &= bh\\105 &= b(7)\\105 \div 7 & = b\\15 \ m & = b

By solving for b , we have found that the base of the parallelogram is 15 meters. Let’s check our calculation to be sure. We can check by putting the base and height into the formula and solving for area

A  &=  bh\\A  &=  15 (7)\\A  &=  105 \ m^2

We know the area is 105 \ m^2 , so our calculation is correct.

Let’s look at another one.

The area of a parallelogram is 184 square yards and its base is 23 yards. Find its height.

This time we know the area of the parallelogram and its base. We can put these into the formula and solve for the height, h .

A  &=  bh\\184  &=  23h\\184  \div  23  &=  h\\8  \ yd  &=  h

We have found that the parallelogram has a height of 8 yards. Again, let’s use the formula to check our work.

A  &=  bh\\A  &=  23 (8)\\A  &=  184 \ yd^2

Our calculation is correct! Whenever we are given two pieces of information about a parallelogram, we can use the formula for area to find the third measurement.

Find the missing dimension in each parallelogram.

Example A

Base = 8 inches, Area = 32 sq. inches. What is the height?

Solution: 4 inches

Example B

Base = 9.5 inches, Area = 57 sq. inches. What is the height?

Solution: 6 inches

Example C

Height is 2.5 ft, Area = 20 sq. feet. What is the base?

Solution: 8 feet

Here is the original problem once again.

One afternoon, Miguel’s assignment is to hang a banner in the stadium. The banner say’s “Go Wildcats” and is the shape of a parallelogram with a yellow background and dark blue border. He has two different places that he can put it. One spot is only 11 feet wide and one spot is 13 feet wide.

Miguel isn’t sure which spot is the best spot for the “Go Wildcats” banner. Wherever he chooses to hang the banner it needs to fit with a little space on each side so that people don’t bump it as they go past.

The banner is rolled up in a tube. Miguel looks at the label to see that the square footage of the banner is 42 square feet. The height of the banner is 3.5 feet. The length of the banner is missing. It seems to have been covered up with a shipping label.

Miguel is a bit frustrated. He isn’t sure that he has enough information to figure out the length of the banner. He knows that he needs to figure out the length of the banner in order to choose the correct spot to hang it. Miguel takes out a piece of paper to try to figure things out.

To start working on this problem, we can use the formula for finding the area of a parallelogram.

A = bh

Next, we can fill in the area and the height. The base is the length of the parallelogram and that is unknown.

42=b(3.5)

Now to solve for b , we divide both sides by 3.5.

42 \div 3.5 &= 12\\b&=12 \ feet

The length of the banner is 12 feet. Given this length, Miguel should put it in the spot that is 13 feet wide.

Vocabulary

Here are the vocabulary words in this Concept.

Parallelogram
a quadrilateral with opposite sides parallel.
Perimeter
the distance around a figure.
Area
the amount of space contained inside a two-dimensional figure.

Guided Practice

Here is one for you to try on your own.

Christie is making a banner for the school talent show in the shape of a parallelogram. She used 59.5 square yards of paper. If the base of the banner is 17 yards, what is its height?

Answer

First, let’s make sure we understand the question in the problem. We need to find the height of the parallelogram. This time we are given the area and the base. As we’ve seen, we simply put these numbers into the formula and solve for the height, h .

A  &=  bh\\59.5  &=  17h\\59.5  \div  17  &=  h\\3.5 \ yd  &=  h

The height of the banner is 3.5 feet.

As long as we have any two pieces of the area formula, we can solve for the third.

Video Review

Here is a video for review.

This is a James Sousa video on the area of parallelograms.

Practice

Directions: Given the area and one dimension, find the missing dimension.

1. Base = 9 feet, area = 45 sq. ft.

2. Base = 8 in, area = 20 sq. in.

3. Base = 11 feet, area = 99 sq. feet

4. Base = 12 inches, area = 120 sq. inches

5. Base = 4.5 feet, area = 11.25 sq. ft.

6. Height = 3 inches, area = 36 sq. inches

7. Height = 4 feet, area = 72 sq. feet

8. Height = 5 meters, area = 80 sq. meters

9. Area = 22 sq. inches

Base = 11 inches

10. Area = 50 sq. miles

Base = 10 miles

11. Area = 48 sq. inches

Base = 8 inches

12. Area = 30 sq. meters

Base = 15 meters

13. Area = 45 sq. feet

Height = 3 feet

14. Area = 88 sq. feet

Height = 8 feet

15. Area = 121 sq. feet

Height = 11 feet

16. Area = 160 sq. miles

Height = 20 miles

17. Area = 90 sq. meters

Height = 30 meters

18. Area = 100 sq. feet

Base = 25 feet

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Difficulty Level:

At Grade

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Date Created:

Dec 21, 2012

Last Modified:

Oct 28, 2014
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