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Project Summary

Difficulty	3
Time required	Very Short (a day or less)
Prerequisites	None
Material Availability	Readily available
Cost	Very Low (under $20)
Safety	No issues

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Sponsored by a generous grant from Motorola

Abstract

Objective

In this experiment, you will see which formula is the most accurate for estimating the volume of an M&M candy.

Introduction

Geometry is the study of how to use math to describe and investigate different points, lines and shapes. The way that a shape is described in geometry is with a formula, which is simply a mathematical way to calculate different properties of a shape like size, area or volume. Volume is a unique property of three-dimensional shapes because three-dimensional shapes take up space in three different directions. Most real-world objects are three dimensional: balls, cars, food, etc.

The problem with geometric formulas is that they describe "perfect" or "ideal" shapes. A sphere is an "ideal" 3-dimensional shape that is perfectly circular in all directions. Even though a ball is spherical in shape, it is not a perfect sphere. If geometric formulas describe "ideal" shapes and not "real" shapes, then how are they useful in the "real" world?

Most real-world shapes are not simple shapes and use complex geometry to be calculated. The properties of real-world shapes can also be approximated, or estimated, to the best possible measure with a geometric formula. This is called making a geometric model, and the most important part of making a good geometric model is choosing the formula that best describes the object. Even the most irregular objects can be modeled by using geometry: cars, airplanes, electronics, plastics, food, etc. geometric modeling is very important for manufacturing because a product needs to have the same shape, made the same way, every time.

In this experiment we will use geometry to produce a mathematical model of an M&M candy. If you look closely, the volume of an M&M candy is a bit irregular, it is not quite perfectly round. It looks like a ball shape (sphere) that has been squished on one side. We will use three different formulas for a sphere, a cylinder and an ellipsoid, to see which formula makes the best geometric model of an M&M candy. We will test each model by calculating the volume with each formula and comparing it to the actual volume of a single piece of candy.

Terms, Concepts and Questions to Start Background Research

To do this type of experiment you should know what the following terms mean. Have an adult help you search the Internet, or take you to your local library to find out more!

• geometry
• model
• volume
• height
• radius
• diameter
• sphere
• cylinder
• ellipsoid

• Which formula will calculate the most accurate volume of an M&M?
• How are geometric formulas different from each other?
• What other ways can I use geometric formulas to measure real-world objects?

Bibliography

• This site is a really good site for kids to review math skills, solve puzzles and read stories about math. They also have a section on geometry:
  Karen, 2005. "Cool Math 4 Kids," Coolmath.com, Inc. [accessed: 3/3/06] http://www.coolmath4kids.com/
• To, S.D.F., 2006. "Agricultural and Biological Engineering: Tools - Volume Calculator Page," Dept. of Agricultural and Biological Engineering, Bagley College of Engineering, Mississippi State University. [accessed: 3/3/06] http://www.abe.msstate.edu/Tools/Volume/index.php
• NCES, 2006. "Create a Graph," National Center for Education Statistics (NCES) U.S. Dept. of Education. [accessed: 3/3/06] http://nces.ed.gov/nceskids/createagraph/
• Here is a more advanced project on the geometry of Thompson Seedless Grapes that is a nice example of a real-world application of geometry for estimating volumes for an industry:
  Cheung, M. and Yin, M., 1996. " Some Simple Methods for the Estimation of Surface Area and Volume of Thompson Seedless Grapes," CSU Fresno, The California Agricultural Technology Institute (CATI) Publication #960902. [accessed: 3/3/06] http://cati.csufresno.edu/verc/rese/96/960902/index.html

Materials and Equipment

• one package of M&M's
• metric ruler that measures in centimeters (cm)
• metric measuring glass that measures in milliliters (mL)
• water
• clay
• computer with Internet connection

Experimental Procedure

1. Before you can find out the volume of an M&M, you need to make some careful measurements. Make sure you do your background research and know what the terms radius, diameter, height, sphere, cylinder and ellipsoid mean. You should also make a data table to record your measurements:

   	Long Radius (cm)	Short Radius (cm)
   Diameter of 10 M&M's
   Diameter of 1 M&M (divide by 10)
   Radius of 1 M&M (divide by 2)
   	Actual Volume (mL)
   Starting Volume (mL)	20mL
   Final Volume (mL)
   Actual Volume of 10 M&M's (mL)
   Actual Volume of 1 M&M (mL)

2. Now you are ready to make some measurements of the M&M's.
3. Open the package and dump out the M&M's.
4. Measure the M&M's using this neat little trick. Line up 10 M&M's on their flat side end-to-end. You can poke them into some clay to keep them in a neat row with each M&M touching the next, like this:

   How to measure M&M's

5. Measure the whole group of M&M's together from end-to-end. Divide your answer by 10. This is the long diameter of a single M&M candy. Write the data in your table.
6. Divide your answer by 2, this is the long radius of a single M&M candy. Write the data in your table.
7. Do the same thing, but this time line up the M&M's on their side so that you are measuring across the short side. Again, use some clay to hold them in place in a neat row with each M&M touching the next.
8. Measure all of the M&M's from end-to-end. Divide your answer by 10. This is the short diameter of a single M&M candy. Write the data in your table.
9. Divide your answer by 2, this is the short radius of a single M&M candy. Write the data in your table.
10. Now you are ready to measure the actual volume of the M&M's with a water displacement test. Fill a metric measuring glass with 20 milliliters (mL) of water. Dump in 10 M&M's. Record the new volume of water.
11. Subtract the beginning volume of water (20 mL) from the new volume of water (that you just measured) to calculate the actual volume of the 10 M&M's.
12. Divide your answer by 10. This is the actual volume of a single M&M in milliliters (mL).
13. Next you will be making some calculations of volume using different formulas to see which one best calculates the volume of the M&M. Make a data table for your calculations:

    	Calculated Volume
    Actual Volume
    Sphere - Long Radius
    Sphere - Short Radius
    Cylinder
    Ellipsoid

14. You will need to use some Internet resources to perform the calculations. Go to the Volume Calculator Page at Mississippi State University. There you will see several different shapes. Each shape is a link to an online volume calculator that you can use.
15. First we will calculate the volume of a sphere . Click on the Full Sphere link and you will see the following calculator:

    The Sphere Volume Calculator (from Dept. of A&BE at MSU, 2006)

16. We will be making our calculation two ways, with the long radius and the short radius.
17. Type the long radius into the box under "Radius" and click "CALCULATE." Write the answer in your data table next to "Sphere - Long Radius."
18. Type the short radius into the box under "Radius" and click "CALCULATE." Write the answer in your data table next to "Sphere - Short Radius."
19. Go back to the Volume Calculator Page at Mississippi State University by clicking this link or by using the back button of your browser.
20. Next we will calculate the volume of a cylinder. Click on the Cylinder link and you will see the following calculator:

    The Cylinder Volume Calculator (from Dept. of A&BE at MSU, 2006)

21. Type the long radius into the box under "Outer Radius", type a zero into the box under "Inner Radius" (since the M&M is not hollow), and type the short diameter (short radius x2) into the box under "Height" then click "CALCULATE." Write the answer in your data table next to "Cylinder."
22. Go back to the Volume Calculator Page at Mississippi State University by clicking this link or by using the back button of your browser.
23. Next we will calculate the volume of an ellipsoid. Click on the Ellipsoid link and you will see the following calculator:

    The Ellipsoid Volume Calculator (from Dept. of A&BE at MSU, 2006)

24. Type the long diameter into the box under "Major Axis" and into the box under "Minor Axis" because in the case on an M&M they are actually the same. Then type the short diameter into the box under "Vertical Axis" and click "CALCULATE." Write the answer in your data table next to "Ellipsoid."
25. Now you are ready to make a bar graph of your data. You can make one by hand or you can try using the Create a Graph website for kids from the National Center for Education Statistics. Make one bar for each type of volume calculation and for the actual volume measurement.
26. How do each of the different calculated volumes compare to the actual volume that you measured? Which ones were more and which ones were less? Which calculation came the closest? Which formula do you think is the best one to use for an M&M?

Variations

• Another way to look at your data is to calculate the difference between each calculation and the actual volume measurement. You can do this by subtracting the actual volume from the calculated volume for each formula. A bigger number is more different from the actual volume than a smaller number. You can also calculate something called the percent difference by dividing your answer by the actual volume. If you make another graph comparing the percent difference of each method, what does it show?
• You can use this same experiment to find the best formula to calculate any other volume. Try using it for an egg, a football, an apple, a bar of soap, or any other irregular shaped object. Just make sure that you choose an object that can safely be submerged in water! Which formula is the best?

• For more science project ideas in this area of science, see Pure Mathematics Project Ideas.

Credits

Sara Agee, Ph.D., Science Buddies

Last edit date: 2006-04-20 01:01:54

Career Focus

If you like this project, you might enjoy exploring careers in Pure Mathematics.

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