Andre The Giant

Similar Figures: Finding Area and Volume

Have you read any comic books with unbelievably large creatures? Have you seen any movies with enormously large people? Have you ever seen something true to life gigantic?

Although comic books and horror films often prey on fears of the gigantic, many giants do exists. Be sure to note that "giant" humans are real people with real feelings. Many "giants" are credited with having larger than life compassion, gentleness, and love for people.


Robert Wadlow

 

What people said about these two giants:

Andre the Giant weighed in at 500 pounds and stood 7 feet 4 inches tall.

"His drive, talent and ambition, however, proved to be as big as Andre himself, and the wrestler became legendary for his achievements in and out of the ring."—Andrethegiant.com

Robert Wadlow grew to a height of 8 feet 11.1 inches.

"Robert was the world’s tallest man, but also a kind, thoughtful, spiritual person. He was a successful student, a Boy Scout and even attended college for a year."—AltonMuseum.com

 

Obviously humans are not proportionate to each other. Andre the Giant weighed more than Robert Wadlow even though Robert Wadlow was 1½ feet taller!
What would happen if they were similar? How much do you think Robert Wadlow would have weighed if he had the same "physique" as Andre the Giant?

Let's investigate some principles of similar figures, then we will answer this question about how much Robert Wadlow would have weighed.

Open up a new file and save it as assignment5.1b, and complete the following.
Be sure to copy/paste in the original questions and show your responses in purple or blue font for easy reading.
  1. Consider the two similar rectangles below.

    The scale factor is 2 since the larger rectangle is 2 times as long and 2 times as tall. That is, the lengths are in a 1:2 ratio. Yet the area of the larger rectangle is not 2 times as much.
    What is the ratio of the areas?
  2. What will the ratio of the areas of two rectangles if they are similar with a scale factor of 5? Explain your reasoning. (A picture is really helpful.)
  3. Consider these two similar rectangles below.
    1. What is the scale factor?
    2. What is the area of the blue rectangle? The red?
    3. What is the area of the red divided by the area of the blue? (That is, what is the ratio of the areas in decimal form?)

  4. Look at these two triangles.
    1. Explain why these two triangles are similar.
    2. What is the scale factor?
    3. What is the ratio of their areas in decimal form? (Divide the larger by the smaller.)
    4. Add your data to the following table to help see the relationship between scale factor and areas.
      (The results you should have gotten from #1-3 are already inserted.)
      Scale Factor
      Ratio of Areas
      2
      4
      5
      25
      1.5
      2.25
         
    5. What relationship do you think there is between scale factor and the ratio of areas?
  5. What about circles? Consider one circle with a radius of 9 and another circle with a radius of 90.
    1. Explain why all circles are similar to each other.
    2. What is the scale factor for these two circles?
    3. What is the area of each circle? (Remember A = πr2.)
    4. What is the ratio of their areas in decimal form? (Divide the larger by the smaller.)
    5. Add your data to the table above and see if your conjecture still holds true.

  6. If two figures are similar with a scale factor of 8, what do you think will be the ratio of their areas?
    Draw a picture and calculate the ratio to test your guess.

  7. Hopefully you saw that for similar figures, the ratio of the areas is the square of the scale factor. 22 = 4, 52 = 25, 1.52 = 2.25, etc.
    1. Explain why you think the ratio of the areas should be the square of the scale factor.
    2. What would ratio of the areas of two similar figures if the scale factor is 8/13?
    3. Draw a picture and show two similar figures with a scale factor is 8/13. Show the areas are in the ratio of the square of 8/13.
    4. What scale factor would give a ratio of areas between two similar figures of 2?

  8. Now in 3D we see a different ratio. Consider the following rectangular prism.
    1. What is the volume of the prism? (Google search if you need help.)
    2. Consider a larger and similar prism with a scale factor of 2.
      1. What would the dimensions be of the larger prism?
      2. What would the volume be?
      3. What is the ratio of the volumes?
  9. Draw a cylinder with any dimensions (height and radius) you would like.
    1. What is the volume of your cylinder? (Google search if you need help.)
    2. Consider a larger and similar cylinder with a scale factor of 5.
      1. What would the dimensions be of the larger cylinder?
      2. What would the volume be?
      3. What is the ratio of the volumes?
  10. Consider the square based pyramid to the right whose height is 12m.
    1. Show why the volume of the pyramid is 900m3. (Google search if you need help.)
    2. Consider a larger and similar pyramid with a scale factor of 5.
      1. What would the dimensions be of the larger pyramid?
      2. What would the volume be?
      3. What is the ratio of the volumes?
    3. Consider a larger and similar pyramid with a scale factor of 7/3.
      1. What would the dimensions be of the larger pyramid?
      2. What would the volume be?
      3. What is the ratio of the volumes?
  11. Create a table to show the relationship we've seen so far between scale factor of similar figures and the ratio of their volumes.
    1. What relationship do you think there is? (Hint: in 2D it was a square relationship. Now we're in 3D.)
    2. Test your conjecture by drawing two similar 3D figures (rectangular prisms are easy) with a scale factor of 2.5 and calculating their volumes.
    3. Did you see that the ratio of their volumes is 2.53? If not, then carefully recalculate their dimensions and volumes.
    4. Explain why you think the ratio of the volumes of similar shapes is the cube of the scale factor.

  12. Now to find out how much Robert Wadlow would have weighed if he had the same "physique" as Andre the Giant, the following equation should be used:

    Simplifying, we have the equation

    Now solving for x:

    Given
    Simplifying
    Rewriting
    Cross multiplying
    Dividing by 0.555

    So Robert Wadlow would have weighed about 900 pounds if he'd had the "physique" of Andre the Giant.
    Did you catch that? Growing from 7.333 to 8.925 feet changed the weight from 500 to 900 pounds! How could that be?
    Remember, to keep a 3D object similar the height, width, and depth all change. That means a change in one dimension affects all three dimensions if that 3D object is to remain similar.

 

That ends investigation 3. Save your work.