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Estimation of Parallelogram Area in Scale Drawings

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Estimation of Parallelogram Area in Scale Drawings

Mrs. Vasquez, Miguel's Mother, is buying new carpet to redecorate her parallelogram-shaped store. To determine how much carpet she needs, she measured the length of the room and found it was 36 meters. Then she measured across the room with a line perpendicular to the first and found it was 16 meters across.

How many square meters of carpet does Mrs. Vasquez need to buy?

Use what you have learned to solve this problem. We will return to it at the end of the Concept.

Guidance

Sometimes, you will see a scale drawing. A scale drawing is a drawing that uses a small measurement to represent a real-world measurement. We can work with a scale drawing and estimate actual areas of a parallelogram given the scale.

Estimate the area of this garden.

To work on solving this problem, we need to look at what information is given to us in the drawing.

We know that the scale says that 1” is equal to 3 feet.

The base of the garden in the drawing is 8” and the height is 3”.

We want to estimate the area.

Let’s start by estimating the base. $8 \times 3$ . We know that $8 \times 3$ is 24. This gives us a total of 24 feet for the base of the garden.

Let’s estimate the height. $3 \times 3$ . We know that $3 \times 3 = 9$ . The height of the garden is about 9 feet.

Next, we estimate the area of the garden.

24 rounds down to 20 and 9 rounds up to 10

$20 \times 10 = 200 \ square \ feet$

How close it our estimate?

Let’s do the actual multiplying to figure this out.

$24 \times 9 = 216 \ square \ feet$

We have an estimate that is reasonable.

Now it's time for you to try a few on your own. Use a scale of 1" = 2 feet.

Example A

Base of 6 inches and a height of 4 inches

Solution: 48 square feet

Example B

Base of 9 inches and height of 5 inches

Solution: 90 square feet

Example C

Base of 12 inches and height of 9 inches

Solution: 216 square feet

Here is the original problem once again.

Mrs. Vasquez, Miguel's Mother is buying new carpet to redecorate her parallelogram-shaped store. To determine how much carpet she needs, she measured the length of the room and found it was 36 meters. Then she measured across the room with a line perpendicular to the first and found it was 16 meters across.

How many square meters of carpet does Mrs. Vasquez need to buy?

Let’s begin by figuring out what the problem is asking us to find. We need to find how much carpet Mrs. Vasquez needs to cover the floor of her store, so we have to find the area of the store. That means we will use the area formula to solve for A. In order to use the formula, we need to know the base and height of the store.

We know that one side of the store will be 36 meters. Let’s call this the base. We also know that Mrs. Vasquez made a perpendicular line in order to measure the height, or the distance across the room. The height given in the problem is 16 meters. Let’s put this information into the formula and solve for area.

$A & = bh\\A & = 36(16) \\A & = 576 \ m^2$

Mrs. Vasquez will need to buy 576 square meters of carpet.

Vocabulary

Parallelogram
a quadrilateral with opposite sides parallel.
Perimeter
the distance around a figure.
Area
the amount of space contained inside a two-dimensional figure.
Scale Drawing
a drawing of a life size image where the drawing is made smaller than the actual image using a scale.

Guided Practice

Here is one for you to try on your own.

If you use the scale 1" = 3 feet, what is the area of a parallelogram with a height of 3 inches and a base of 6 inches?

To figure this out, we can use the formula for finding the area of a parallelogram. We will work in inches first and then convert that into square feet.

$A = bh$

$A = (3)(6)$

$A = 18 \ sq. in$

Now if 1" = 3 feet, then we need to take the area of the room in inches and multiply it by three.

$A = 18 \times 3 = 54 \ sq. feet$

Practice

Directions: For each parallelogram, find each new area using the scale 1" = 2 feet.

1. Base of 6 inches, height of 4 inches

2. Base of 8 inches, height of 6 inches

3. Base of 4 inches, height of 4 inches

4. Base of 5 inches, height of 4 inches

5. Base of 6 inches, height of 6 inches

6. Base of 10 inches, height of 8 inches

7. Base of 11 inches, height of 12 inches

8. Base of 15 inches, height of 9 inches

9. Base of 15 inches, height of 12 inches

Directions: Solve each problem.

10. A parallelogram has an area of 390 square centimeters. If its height is 15 cm, what is its base?

11. What is the height of a parallelogram whose base is 28 inches and area is 1,176 square inches?

12. Donna wants to cover her parallelogram-shaped crafts box in fabric. The base of the lid is 32.7 cm and the height is 12.2 cm. What is the area of the lid?

13. John is planting grass in a patch of lawn that is shaped like a parallelogram. The height of the parallelogram is 34 feet. The other border is 65 feet. How many square feet of grass will John plant?

14. Kara and Sharice are in a quilting competition. Both are stitching parallelogram-shaped quilts. So far Kara’s has an area of 2,278 square inches and a height of 44 inches. Sharice’s quilt has an area of 2,276 square inches and a height of 47 inches. Whose quilt is longer? By how many inches is it longer?

15. Denise bought a picture frame in the shape of a parallelogram. The area of the picture frame is 36,795 square centimeters. If its height is 165 centimeters, what is its base?