6th Grade – Angle Relationships, Area, Perimeter and Circumference

Shape within a Shape

1)
Jason and his brother purchased a round rug to lay in their living room. The living room is 13ft x 11ft, and the rug is 4 feet in diameter. What is the area of the room NOT covered by the rug? / 2)
What is the area of the shaded region if the length of this square is 2.2 cm?

3)
The room for the wedding is 30 ft by 32 ft. There are 5 round tables for guests. Each table has a diameter of 6 ft. What is the area of room that is available for dancing and walking? / 4)
What is the area of the shaded region if the radius of this circle is 3.4 in?

5)
Rachel put her coffee table on top of a circular rug. The circular rug has a diameter of 4 feet. The round coffee table has a diameter of 3 feet. How much of the rug is NOT covered by the coffee table? / 6)
The triangle’s base is 2.5 cm and is equal to the radius of the circle. What is the area of the shaded region?
7)
A circular table with a diameter of 4 feet has four circular place mats on it, each with a diameter of 6 inches. What is the area of the table in inches, NOT covered with place mats? / 8)
What is the area of the shaded region if the radius of the circle is 10.4 ft?

9)
Calculate the area of the shaded section in the picture below:
15 cm


9 cm / 10)
Mary’s father put a garden in their backyard that had an area of 5 ft. by 9 ft. He put a sidewalk around the garden that had an area of 7 ft. by 12 ft. What is the area of the sidewalk around the garden?
11)
Calculate the area of the shaded section in the picture below: 12 yd


18 yd. / 12)
The area of a local school is 3,844 sq. meters. When they built the school they put a sidewalk around the school. The area of that sidewalk is 72 meters by 70 meters. What is the area of the space between the school and sidewalk?
13)
Calculate the area of the shaded section in the picture below:
12 cm

11 cm / 14)
Bob built his very own lemonade stand in front of his house. His lemonade stand was 8 feet by 12 feet. He decided that he needed to make it look nicer by planting flowers all the way around the stand. The area of the planted flowers was 130 sq. feet. How much space was there between his lemonade stand and the flowers?
15)
Calculate the area of the shaded section in the picture below:
15 ft.


12 cm / 16)
N.C. State’s basketball court has an area of 150 sq. feet. There are chairs around the entire court that make up an area of 16 feet by 20 feet. How much space is there between the court and the chairs?

Name ______Date ______

Constructing a Deck

Mr. and Mrs. Smith are building a deck adjacent to their house. The diagram below shows the dimensions of the deck. Mr. Smith will be laying the decking to keep the cost to a minimum.

On sale, pre-stained 2 in. x 6 in. brown treated lumber is advertised at the following prices

Length (ft) / 2 / 4 / 5 / 7 / 8 / 10 / 12 / 14 / 16
Price (dollars) / 2.99 / 3.64 / 4.29 / 4.99 / 5.79 / 6.99 / 8.49 / 9.99 / 12.29

Since these prices are lower than the current prices for special brown treated or green treated deck boards, the Smiths decided to use the 2 in. x 6 in. brown treated lumber as deck boards.

Problem: Using the prices for brown treated lumber and any variety of lengths of lumber, determine the minimum cost and minimum waste of materials of the deck design. (The actual measure of 2 in. x 6 in. stock is 1 ½ in. x 5 ½ in.)

Credit: PBS Teacher Source. Mathline Activity II- Pricing a Deck

Name ______Date ______

Fencing a Yard

A couple is planning to erect a fence on three sides of their property. Figure 1 shows the shape of their property and provides the dimensions of the three sides they want to fence.

Figure 1: The couple's property, with the dimensions of the sides to be fenced.

The couple can purchase sections of fencing they have chosen in standard eight-foot lengths that are six feet high. They can also use a catalog to custom-order shorter lengths of this fencing. The manufacturer of the fencing materials recommends using 4 in. x 4 in. treated lumber 10 ft. long for the fence posts and 6 in. x 6 in. treated lumber 10 ft. long for the corner posts.

In their design, the couple plans to use as many eight-foot standard pieces as possible to keep costs to a minimum, but they want to maintain symmetry so that the fencing looks good.

Figure 2: Example of the symmetry the couple wants.

To determine the number of standard 8-ft. sections they need, the couple divided the length of each side of the lot by 8. When they divided the 100-foot side by 8, they found they would need 12 eight-foot sections and 1 four-foot section. To be symmetrical, the fence has to be constructed with equal amounts on each end of the fencing along the 100-foot length. Because 2-foot sections are not available in the catalog and to maintain symmetry on the 100-foot side, the couple decides to use 11, 8-foot sections and 2, 6-foot sections.

If they have a budget of $5500 for the fencing, which of the designs can the couple buy? Use the information in the tables to help you make your determination.

Description / Standard / Price / Custom / Price
Solid 1 x 4 Red Cedar / 6 ft. x 8 ft.. / $76.50 / 6 ft. x 6 ft. / $60.00
Peek-A-Boo 1 x 4 Red Cedar / 6 ft. x 8 ft.. / $62.25 / 6 ft. x 6 ft. / $48.75
Board by Board 1 x 4 Red Cedar / 6 ft. x 8 ft.. / $85.00 / 6 ft. x 6 ft. / $65.50
Basket Weave 1 x 4 Red Cedar / 6 ft. x 8 ft.. / $70.00 / 6 ft. x 6 ft. / $55.50
Basket Weave with Topper / 6 ft. x 8 ft.. / $88.00 / 6 ft. x 6 ft. / $69.50
4 in. x 4 in. treated lumber / 10 ft. / $10.89
6 in. x 6 in. treated lumber / 10 ft. / $23.60
Description / Custom / Price / Custom / Price
Solid 1 x 4 Red Cedar / 6 ft. x 5 ft. / $49.50 / 6 ft. x 4 ft. / $40.00
Peek-A-Boo 1 x 4 Red Cedar / 6 ft. x 5 ft. / $40.00 / 6 ft. x 4 ft. / $33.50
Board by Board 1 x 4 Red Cedar / 6 ft. x 5 ft. / $55.50 / 6 ft. x 4 ft. / $45.50
Basket Weave 1 x 4 Red Cedar / 6 ft. x 5 ft. / $46.50 / 6 ft. x 4 ft. / $38.00
Basket Weave with Topper / 6 ft. x 5 ft. / $60.00 / 6 ft. x 4 ft. / $48.00

Use the chart below to help you solve the problem. Don’t forget to include the corner posts and the fence posts.

Quantity / Description / Price / Cost

With a budget of $5500 for the fencing, which of the designs can the couple buy? Explain.

______


ANSWER KEY FOR Shape within a Shape

1)
Jason and his brother purchased a round rug to lay in their living room. The living room is 13ft x 11ft, and the rug is 4 feet in diameter. What is the area of the room NOT covered by the rug?
130.44 sq. ft / 2)
What is the area of the shaded region if the length of this square is 2.2 cm?

3.7994 sq. cm
3)
The room for the wedding is 30 ft by 32 ft. There are 5 round tables for guests. Each table has a diameter of 6 ft. What is the area of room that is available for dancing and walking?
818.7 sq. ft / 4)
What is the area of the shaded region if the radius of this circle is 3.4 in?

9.9416 sq. in
5)
Rachel put her coffee table on top of a circular rug. The circular rug has a diameter of 4 feet. The round coffee table has a diameter of 3 feet. How much of the rug is NOT covered by the coffee table?
5.495 sq. ft / 6)
The triangle’s base is 2.5 cm and is equal to the radius of the circle. What is the area of the shaded region?
16.5 sq. cm
7)
A circular table with a diameter of 4 feet has four circular place mats on it, each with a diameter of 6 inches. What is the area of the table in inches, NOT covered with place mats?
37.68 sq. in / 8)
What is the area of the shaded region if the radius of the circle is 10.4 ft?

108.16 sq. ft
9)
Calculate the area of the shaded section in the picture below: 99 sq. cm
15 cm


9 cm / 10)
Mary’s father put a garden in their backyard that had an area of 5 ft. by 9 ft. He put a sidewalk around the garden that had an area of 7 ft. by 12 ft. What is the area of the sidewalk around the garden?
39 sq. ft
11)
Calculate the area of the shaded section in the picture below: 12 yd


18 yd.
189 sq. yd / 12)
The area of a local school is 3,844 sq. meters. When they built the school they put a sidewalk around the school. The area of that sidewalk is 72 meters by 70 meters. What is the area of the space between the school and sidewalk?
1196 sq. m
13)
Calculate the area of the shaded section in the picture below:
12 cm
11 cm
92 sq. cm / 14)
Bob built his very own lemonade stand in front of his house. His lemonade stand was 8 feet by 12 feet. He decided that he needed to make it look nicer by planting flowers all the way around the stand. The area of the planted flowers was 130 sq. feet. How much space was there between his lemonade stand and the flowers?
34 sq. ft
16)
Calculate the area of the shaded section in the picture below:
15 ft.

12 cm
162 sq. ft / 16)
N.C. State’s basketball court has an area of 150 sq. feet. There are chairs around the entire court that make up an area of 16 feet by 20 feet. How much space is there between the court and the chairs?
170 sq. ft

Answer Key for Constructing a Deck

Assume that the benches are all straight cuts.

Solution:

To determine the number of boards necessary to cover the deck, divide the length of the deck in inches by 5 ½ in., the actual width of a 2 in. x 6 in. board.

Deck:

14’ x 12 ÷ 5.5 = 31 boards 14’ long

62 boards at 7’ long is the cheapest = $309.38

Benches:

2’ x 12” = 24” 24” ÷ 5.5 = 5 boards

15 8’ boards is the cheapest = $86.85

$309.38 + $86.85 = $396.23 with 4” of waste

Credit: PBS Teacher Source. Mathline Activity II- Pricing a Deck

Answer Key for Fencing a Yard

This activity can be modified by providing the number of sections and posts required for each pattern of fencing.

Solution:

The total cost of the fence posts regardless of the style fencing is 64 x $10.89 = $696.96 plus 2 x $23.60 = $47.20 for a grand total of $744.16.

To determine the cost of the fence sections, students can use the information from the table in the activity to create another table.

Qty / Description / Standard / Price / Cost
61 / Solid 1 x 4 Red Cedar / 6 ft. x 8 ft.. / $76.50 / $4666.50
61 / Peek-A-Boo 1 x 4 Red Cedar / 6 ft. x 8 ft.. / $62.25 / $3797.25
61 / Board by Board 1 x 4 Red Cedar / 6 ft. x 8 ft.. / $85.00 / $5185.00
61 / Basket Weave 1 x 4 Red Cedar / 6 ft. x 8 ft.. / $70.00 / $4270.00
61 / Basket Weave with Topper / 6 ft. x 8 ft.. / $88.00 / $5185.00

The Basket Weave and the Peek-a-Boo fencing are within the couple’s budget.

Basket Weave: 1 x 4 Red Cedar, the cost is $744.16 for posts + $4270.00 for the 8-ft section + $46.50 for a 5-ft section + $38.00 for a 4-ft section + $111.00 for the two 6-ft sections for a total cost of $5209.66

Peek-a-Boo: 1 x 4 Red Cedar, the cost is $744.16 for posts + $3797.25 for the 8-ft section + $40.00 for a 5-ft section + $33.50 for a 4-ft section + $97.50 for the two 6-ft sections for a total cost of $4712.41

Credit: PBS Teacher Source. Mathline. Activity III: Fencing the Yard

Wake County Public Schools, 2013Page 1