Area of the Toilet Paper Roll

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Toilet Paper Geometry

What did people use before toilet paper?

People of many cultures and social classes have used a number of different things in place of the rolls of toilet paper that we use today. Things they use included newsprint, straw, hay, grass, corn cobs, leaves, sand, pages from books, coconut shells, lace, rocks, a sponge on a stick, snow and tundra moss.

Important Dates in Toilet Paper History

1391 – First toilet paper invented

1857 – First toilet tissue

1890 – First toilet paper on a roll

1942 – First soft toilet paper; two-ply

Toilet Paper Facts

Two-ply toilet paper is not twice as thick as one-ply toilet paper. One-ply is made of #13 thickness paper while two-ply is made of two layers of #10 thickness paper.

The size of a sheet of toilet paper can vary, but the standard size is 4.5 inches by 4.5 inches. Some manufactures have come out with “cheater sheets” that are as small as 4 inches by 3.8 inches and use about 15% less paper.

The average person uses 8.6 sheets of toilet paper per trip, 57 sheets per day, and 20,805 sheets per year. (So who did this study?? And how did they get their information???)

With over 6 billion people living on earth it is estimated that people need to produce 83,048,116 rolls of toilet paper each and every day, 30,600,000,000 rolls per year, and 2.7 rolls per second.

Toilet Paper Geometry

Name of Toilet Paper Brand:____________________________________

Use a file folder to set up a “scientific board” to show the area, both volumes, and surface area of the toilet paper roll. You must include drawings and pictures of each step of the project.

Your project will be DUE MARCH 5!!! No late projects accepted!!!!

Area of the Toilet Paper Roll

1.

Unroll the toilet paper and place the pieces together to create a large rectangle. Let any leftover pieces be separate rectangles. Sketch these rectangles in the box below. Label the length and width of each rectangle in inches. Label the area inside each rectangle in square inches. You must also include a picture of yourself with the full rectangles of toilet paper.

2.

Area of the roll of toilet paper in square inches: _________________

3.

Area of the roll of toilet paper in square feet: ___________________

4.

Number of rolls of toilet paper required to cover the following:

Make sure to show your work!

Basketball Court Football Field Baseball Diamond

94 ft. x 50 ft.

____________

100 yds. x 53 1

3

yds. 90 ft. x 90 ft.

____________ ____________

Toilet Paper Geometry

Method 1 – Volume of a Cylinder

Volume of the Toilet Paper Roll including the Tube

1.

Measure the radius (r) of the toilet paper roll in inches. r= ____________.

2.

Measure the height (h) of the toilet paper roll in inches. h= ____________.

3.

Use the formula for volume (V=Bh) to determine the volume of the roll of toilet paper including the tube that holds the paper. V= ____________.

Volume of the Toilet Paper Tube

4.

Measure the radius (r) of the tube (cylinder) that holds the toilet paper. r= __________.

5.

Use the height (h) of the toilet paper tube from #2 above. h= ____________.

6.

Use the formula for volume (V=Bh) to determine the volume of the toilet paper tube.

V=____________

Volume of the Toilet Paper Only (minus the tube)

7.

Volume of the toilet paper roll – Volume of the toilet paper tube = Volume of the toilet paper

(including the tube)

____________ -- ____________ = ____________

Method 2 – Volume of a Rectangular Prism

1.

Tear apart and stack all of the individual sheets of toilet paper on top of each other to form a rectangular prism. Measure the length, width, and height of the toilet paper prism to find the volume using the formula

V=Bh. Important note: Before measuring the height of the prism, press down on it with your hand to approximate the same density as the toilet paper roll.

Volume of the = Area of the Base of the x height of the toilet

toilet paper prism toilet paper prism paper prism

____________ = ____________ x ____________

Compare the Volume of the Cylinder and the Rectangular Prism

1.

Analyze your results above. What did you discover? Why did this happen? Include a detailed explanation with your pictures to explain your thinking on your scientific board.

Toilet Paper Geometry Extra Credit

Method 1 – Surface Area of a Cylinder

Surface Area of the Toilet Paper Roll including the Tube

1.

Measure the radius (r) of the toilet paper roll in inches. r= ____________.

2.

Measure the height (h) of the toilet paper roll in inches. h= ____________.

3.

Use the formula for lateral surface area (

𝑆𝐴

=2 𝜋𝑟 h) to determine the lateral surface area of the roll of toilet paper including the tube that holds the paper. S= ____________.

4.

Use the formula for area of a circle (

𝑆𝐴 = 𝜋𝑟 2 ) to determine the area of the two (2) bases of the toilet paper rolls including the tube that holds the paper. A= ____________.

5.

To find total surface area put the two formulas together. S=2 𝜋𝑟ℎ + 2𝜋𝑟 2 S= ____________.

Method 2 – Surface Area of a Rectangular Prism

6.

Tear apart and stack all of the individual sheets of toilet paper on top of each other to form a rectangular prism. Measure the length, width, and height of the toilet paper prism to find the surface area using the formula SA=Ph + 2B. Important note: Before measuring the height of the prism, press down on it with your hand to approximate the same density as the toilet paper roll.

7.

Find the lateral surface area using the formula SA = P(perimeter)h

8.

Find the area of the two bases using the formula A=lw then multiply by two.

9.

To find the total surface area add the two formulas together to get SA=Ph + 2B.

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