Rectangles

The distance around a rectangle is called the perimeter, obtained by adding the measure of the four sides. Because opposite sides are equal, a rectangle’s perimeter is (2 × length) + (2 × width).

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4.1.2: Using perimeter.

1)

The part where a floor and wall meet usually has a baseboard, for protection and to hide gaps. Baseboard (shown below on the right) is a sturdy attractive strip nailed to the wall. The length of baseboard needed for a room is the room’s perimeter. A room is 15 feet long by 10 feet wide. How much baseboard is needed?

Answer

50

Since the room is a rectangle, the perimeter can be found using 2×length + 2×width.

2×15 + 2×10 = 30 + 20 = 50 feet

2)

A police officer must run caution tape around a store where a crime occurred. The store is 30 feet long and 20 feet wide. How much tape is needed?_{Photo source: wikimedia.org}

Answer

1002×30 + 2×20 = 60 + 40 = 100 feet

3)

Mindy wants to put garden trim around her backyard gardens to prevent the grass from growing into the garden. She has 3 gardens, each 10 feet long and 5 feet wide. How much trim is needed?

Answer

90

The perimeter of one garden is 2×10 + 2×5 = 30 feet.

So, the perimeter of all 3 gardens is 3 × 30 = 90 feet.

4)

A square room is 10 feet long and 10 feet wide. What is the room’s perimeter?

Answer

40

2×10 + 2×10 = 40 feet

A rectangle with 4 equal sides is a square. Since the length and width are equal, the perimeter of a square can be found using 4×length = 4×10 = 40 feet. This shortcut can reduce the amount of measuring needed from two walls to just one wall.

5)

Three bedrooms each measure 15 feet long and 10 feet wide. A good baseboard costs about 4 dollars per foot. How much will the baseboard cost for all 3 bedrooms?

Answer

600

The perimeter of each room is 2×15 + 2×10 = 50 feet.

So, the perimeter for all 3 rooms is 3×50 = 150 feet.

At 4 dollars per foot, the cost for baseboard in all 3 rooms is 4×150 = 600 dollars.

Cost for home improvement can really add up. Cheap baseboard only costs about 0.60 dollars, making the total cost only 0.60×150 = 90 dollars, a difference of 600 – 90 = 510 dollars.

6)

Some houses install “crown molding”, which is like a baseboard but installed at the ceiling instead of the floor, for a look of elegance. Crown molding may cost about 9 dollars per foot. If 3 bedrooms each measure 15 feet long and 10 feet wide, how much is the cost for crown molding for all three rooms?

Answer

1350

The perimeter of each room is 2×15 + 2×10 = 50 feet. So, the total perimeter for all 3 rooms is 3 × 50 = 150 feet. Since crown molding costs 9 dollars per foot, the total cost is 9 × 150 = 1,350 dollars.

Many people underestimate the expense of such aspects of home remodeling projects.

Circles

The perimeter of a circle has a special name: circumference.

Measuring the distance around a circle can be hard. Fortunately, for any size circle, the circumference is always about 3.14 times longer than the diameter: The distance across a circle through the center. So circumference = 3.14 × diameter, meaning a person can measure a circle’s diameter and then multiply by 3.14 to know the circumference.

PARTICIPATION ACTIVITY

4.1.3: Using diameter to find circumference.Start2x speed

A quarter:

circumference =

diameter =

3.01 inches

0.96 inches

≈ 3.14

Earth:

circumference =

diameter =

24,901 miles

7,917 miles

≈ 3.14

circumference

diameter

≈ 3.14

circumference

diameter

≈ 3.14

×

How much stone for a round garden?

diameter = 12 feet

circumference ≈ 3.14 × 12

≈ 37.7 feet

keyboard_arrow_upCaptions

1. The ratio circumference/diameter for any circle is about 3.14. For this coin, measuring the circumference yields 3.01 inches, and diameter 0.96 inches. Ratio is about 3.14.
2. The value isn’t exactly 3.14 but close, so ≈ is used instead of =. The ≈ symbol means “about equal”.
3. For this different-sized circle (the Earth), measuring the circumference would yield 24,901 miles, and diameter 7,917 miles. The ratio is again about 3.14.
4. The ratio can be rewritten as circumference = 3.14 × diameter.
5. For some circles, the diameter is easy to measure. For this garden, the diameter is 12 feet, so the circumference is easily computed to be 3.14 × 12 = 37.7 feet.

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_{Earth photo source:…}

The value 3.14 is approximate. The exact ratio of a circle’s circumference to diameter is a number called pi, written using the symbol π, which has infinite decimal places. Pi to six decimal places is π = 3.141593.

PARTICIPATION ACTIVITY

4.1.4: Finding the circumference of a circle.

Use 3.14 as the approximation of pi.

1)

A circular garden has a diameter of 9 feet. How many feet of trim are needed to surround the garden?

Answer

28.26Circumference = 3.14 × diameter = 3.14 × 9 = 28.26 feet of trim.

2)

A kids basketball game will be played in a cafeteria. Mia is asked to put white tape on the floor to make the basketball court, so Mia calculates the needed amount of tape. At court’s center is a circle with a diameter of 12 feet. How many feet of white tape is needed for the circle?

Answer

37.68

3.14 × 12 = 37.68 feet of tape.

3)

Sometimes a circumference is known and the diameter is desired. The equation diameter = circumference / 3.14 can be used.

Mo wants to swim across a circular lake, but first wants to know the distance (diameter). Unable to measure distance across a lake, Mo walks the circumference and counts 500 steps, where each step is 1 yard. What is the lake’s diameter?

Round to the nearest whole yard.

Answer

159Circumference is 500 yards, so diameter = circumference / 3.14 = 500 / 3.14 = 159.24 yards, which rounds to 159. That distance is a long way to swim; Mo better be an excellent swimmer.

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PARTICIPATION ACTIVITY

4.1.5: Painting lines on a soccer field.

Many soccer organizations ask parent volunteers to paint the field before a game. A volunteer wants to estimate how many spray paint cans to buy the night before. The picture below shows a soccer field’s line measures for 5-6 year old kids. Note: “Radius” is half the diameter of a circle.

1)

How much paint is required for the outer border?120 yards of paint60 yards of paint

2)

About how much paint is required for the center circle? Use 3.14 for pi.18.8 yards of paint9.4 yards of paint

3)

How much paint is required for the entire field?177.6 yards of paint157.6 yards of paint

4)

If one can of spray paint can spray 50 yards, how many cans are required to paint the entire field? Round up to the nearest whole can.5 cans of paint4 cans of paint

Jetway auto-leveler

Ever wonder what that wheel is touching a jet plane from the jetway? That wheel is called an “auto-level”. As people board, the plane gets heavier: If 100 passengers board weighing 170 pounds each with 30 pounds of luggage, the plane gains 20,000 pounds. Because the planes tires/wheels are on a shock-absorbing suspension (like a car), that weight lowers the plane. The jetway needs to lower too, else soon people would have to step down onto the plane

The wheel is pressed against the plane’s body and has non-slip rubber around the wheel’s circumference. As the plane lowers, the wheel turns a bit. A computer detects the amount the wheel turned, and knowing the wheel’s circumference, knows by how much to lower the jetway. If the wheel’s circumference is 10 inches, and the wheel turned 1/4 the way around, that’s 1/4 × 10 = 2.5 inches.

Measuring wheels.

A “measuring wheel” is used by architects, contractors, and other people to measure distances. The wheel below has a circumference of exactly 12 inches. The display counts up an inch (the far right number with white background) as the wheel rotates 1/12 around, and upon reaching 12 inches increments the feet (the numbers with black background) meaning the wheel went all the way around. Below, the measured distance is 3 feet 2 inches. Measuring wheels are useful for measuring the size of house’s property, for example.

Sometimes a person wants to mark a specific distance, like 100 yards on the street so the kids can be timed running the 100-yard dash, or various distances to mark off a soccer field’s lines. A simple and clever approach uses a bike. First, a person measures the bike’s front wheel diameter, such as 20 inches, and computes the circumference: 3.141593 × 20 = 62.83 inches. 100 yards converted to wheel rotations is: 100 yards × (3 feet)/(1 yard) × (12 inches)/(1 foot) × (1 rotation)/(62.83 inches) = 57.3 rotations. To measure out 100 yards, a person can put tape on the side of the wheel’s tire, and then ride the bike and count 57 rotations, plus another 1/3 (which is about 0.3). The same technique can be used in reverse: To measure the distance to the end of a cul de sac, a person can ride the bike and count the rotations. If 70.5 rotations occurred, the distance is 70.5 rotations × (62.83 inches)/(1 rotation) × (1 foot)/(12 inches) × (1 yard)/(3 feet) = 123 yards.

Other linear distances

Shipping companies may limit a package’s size based on girth. Girth is the distance around the center of an item, like around a box or a person’s waist. For a box, the girth is 2×width + 2×height. The company may limit length + girth to under a certain number of inches.

Airlines often restrict the size of checked bags. Because girth confuses some people, airlines commonly just limit the length + width + height. Ex: In 2019, Southwest limited each bag’s length + width + height to 65 inches, charging extra for over 65 inches.

PARTICIPATION ACTIVITY

4.1.6: Package and bag requirements.Start2x speed

girth + length must be under 160 inches

50 in

10 in

60 in

girth = 50 + 10 + 50 + 10 = 120

length = 60

girth + length = 120 + 60 = 180

x

length + width + height must be under 65 inches

20 in

13 in

31 in

length + width + height

13 + 20 + 31 = 64

[Math Processing Error]✓keyboard_arrow_upCaptions

1. A shipping company may require packages to have a girth and length total under 165 inches.
2. Girth is the distance around the center of an item.
3. Airlines often just limit the length + width + height of a checked bag to under 65 inches, for a simpler calculation.

PARTICIPATION ACTIVITY

4.1.7: Shipping packages.

The following chart from easyshipping gives package measurement requirements for shipping services.

1)

A box has a width of 30 inches, height of 13 inches, and a length of 15 inches. Could the box be shipped using Priority Mail?YesNo

2)

Luke would like to use the box shown below for shipping. Will he be able to use Priority Mail or will he have to use Parcel Select Ground?_{Photo source: uline.com}

Priority MailParcel Select Ground

3)

Find length plus girth for a cardboard tube that has a diameter of 12 inches and a length of 25 inches. Round to the nearest inch._{Photo source: newpig.com.com}

37 inches63 inches

Expressing feet and inches together

The symbol ‘ is commonly used for feet, 10 feet may be labeled as 10’. The symbol ” is commonly used for inches, 15 inches may be labeled as 15”. A size that includes both feet and inches may use both these symbols.

Figure 4.1.1: Using symbols to express feet and inches.

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PARTICIPATION ACTIVITY

4.1.8: Labeling feet and inches.

1)

Which notation can be used for 6 feet and 11 inches?6′ 11”6” 11′

2)

What is the length of the phone?_{Photo source: imore.com}

1.334 inches4.7 inches

3)

The measurements of this tabletop involve feet, inches, and fractions of an inch. What are the measurements of the tabletop?

3 feet and 7 3/4 inches by 4 feet and 8 1/2 inches37 feet and 3/4 of an inch by 48 feet and 1/2 of an inchFeedback?

Spinal Tap movie: Confused inches and feet.

In the movie “Spinal Tap”, a comedic fake documentary of a rock band, a band member sketches an idea for a giant Stonehenge prop to be dramatically lowered onto stage. But, he wrote 18″ in his sketch instead of 18′. The prop designer built exactly what was requested, which turned out to be a less dramatic stage effect than desired.

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