Key concepts
Geometry
Mathematics
Shapes
Dimensions
Volume

Introduction
Has an adult ever caught you munching on candy and asked, "How much candy have you eaten?" Instead of saying, "I don't know," and possibly receiving a scolding, wouldn't you rather respond, "I ate precisely 10.7 cubic centimeters of candy"? In this activity, you will investigate which mathematical formula is most accurate for estimating the volume of an M&M. Figure this out and the next time you are discovered while snacking on sweets, you might make a better impression.

Background
Geometry uses math to describe and investigate different points, lines and shapes. A shape is described in geometry using a formula, which is simply a mathematical way to calculate different properties, such as its size, area or volume. Volume is a property of three-dimensional shapes—such as cubes and spheres—and takes into account the space the shapes take up in each of the three different directions.

The challenge of using geometric formulas in the real world is that these mathematical formulas often describe "perfect" or "ideal" shapes. A sphere is an "ideal" three-dimensional shape that is perfectly circular in all three directions. Even though a tennis ball, for example, is spherical in shape, it is not a perfect sphere (think of the lines that mark its surface). Most real-world objects are not simple shapes and require complex geometry to be calculated. The properties of real-world shapes can, however, be approximated, or estimated with a geometric formula. This is called "making a geometric model." The most important part of making a good geometric model is choosing the formula that best describes the object. Using that formula, you can geometrically model all kinds of irregular objects, such as cars, airplanes, toys and M&Ms.

Materials
•     Two pieces of paper
•     Table or countertop
•     Clay or Play-Doh. Use a small amount that you do not mind ruining.
•     Metric measuring glass with milliliter markings
•     Water
•     110 M&Ms. One seven-ounce (198-gram) bag holds about 210 M&Ms.
•     Ruler with centimeter markings
•     A pen or pencil
•     A computer with an Internet connection

Preparation
•     Place a sheet of paper on a clean table or countertop.
•     On top of the paper, place a small amount of clay or Play-Doh. Flatten it and stretch it out into a line about 20 centimeters (cm) long by three cm wide. The line should be only a couple cm high. You will be measuring the M&Ms on this.
•     The clay or Play-Doh may be ruined by the M&Ms' dyed coating, so use a portion that you will not mind ruining.
•     Fill the metric measuring glass with 100 milliliters (ml) of water. If you intend to eat some of the wet M&Ms after this activity is completed, remember to clean the measuring glass carefully before you fill it with water.

Procedure
•     Line up 10 M&Ms on their flat side, end to end, on the clay. Make sure the line is straight and flat and that each M&M is touching the next, with no gaps in between. You can poke them into the clay to keep them in a neat row.
•     Measure the line of M&Ms in cm and divide this number by 10. This gives you the long diameter of a single M&M candy. What is the long diameter of an M&M candy? Write this measurement down.
•     Divide the long diameter by two. This gives you the long radius of a single M&M candy. What is the long radius of an M&M candy? Write this down.
•     Remove the M&Ms from the clay.
•     Now line up the 10 M&Ms on their side so that you are measuring across the short side. Again, use the clay to hold them in a neat row in which each M&M touches the next.
•     Measure the line of M&Ms and divide this number by 10. This gives you the short diameter of a single M&M candy. What is the short diameter of an M&M candy? Write this down.
•     Divide the short diameter by two. This gives you the short radius of a single M&M candy. What is the short radius of an M&M candy? Write this down.
•     Now you are ready to measure the actual volume of the M&M with a water-displacement test. Make sure that the metric measuring glass has exactly 100 ml of water in it (check by looking at the bottom of the meniscus).
•     Dump 100 M&Ms into the glass of water. (If you want to eat soggy M&Ms later, use fresh M&Ms during this stage since the ones you used previously may have clay on them.) What is the new water level of the glass? Write this down.
•     Subtract 100 ml from the new water level. Divide this number by 100. This is the actual volume of a single M&M in ml (the same as cubic centimeters). What is the volume of one M&M? Write this down.
•     You will now be making some volume calculations using different formulas to see which one best calculates the volume of an M&M. Go to the Unit-Free Volume Calculators page at Mississippi State University.
•     Click on the "Full Sphere" link. For the "Radius" box, enter the long radius you measured for a single M&M and click "Calculate." What is the M&M volume using its long radius? Write this down.
•     Now for the "Radius" box, enter the short radius instead. What is the M&M volume using its short radius? Write this down.
•     Go back to the Unit-Free Volume Calculators page and click on the "Cylinder" link. For the "Outer Radius" box, enter the long radius; for the "Inner Radius" box, enter zero; and for the "Height" box, enter the short diameter. What is the M&M volume using the cylinder formula? Write this down.
•     Go back and click on the "Ellipsoid" link. For the "Major Axis" and "Minor Axis" boxes, enter the long diameter (for both). And for the "Vertical Axis" box, enter the short diameter. What is the M&M volume using the ellipsoid formula?
•     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 candy?
•     Extra: Another way to look at your data is to calculate the percent difference between each calculation and the actual volume measurement. You can do this by dividing your answer using each formula by the actual volume. Which formulas give you an answer that is closest to, or most different from, the actual volume based on percent difference?
•     Extra: 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, other types of candies or any other irregularly shaped object. Just make sure that you choose an object that can be safely submerged in water! Which formula is the best?
•     Extra: The shape of a candy can affect how well many of those candies pack together. Use the water displacement test on a couple differently shaped candies to determine the actual volume of a single candy. Then fill a measuring glass with a certain amount of each type of candy, one type at a time (without water). Divide the volume the candy took up by the number of candies to determine how much space one candy took up, on average, when taking packing into account. How much space does each type of candy take up in the measuring glass (when packing is taken into account) compared with the actual volume of one candy? In other words, which types of candies pack together the best? How do you think their shape affects this?

Observations and results
Did you find the ellipsoid formula to give the closest answer to the actual volume you measured for one M&M candy?

Using the water displacement test, you should have found the actual volume of a single M&M candy to be about 0.60 to 0.65 cubic centimeter (milliliter). (Adding 100 M&Ms to 100 ml of water should have caused the water level to rise to about 160 to 165 ml.) When using the sphere formula with the long radius, the calculation gives a volume (about 1.3 cubic centimeters) that is a little bigger than the actual volume of the M&M candy. If you look closely, you will see that the volume of an M&M is not quite perfectly round but is shaped like a sphere that has been squished on one side. If it were not "squished," the sphere formula with the long radius would fit. The cylinder formula also gives a volume (about one cubic centimeter) that is too big because it assumes that the entire length of the M&M is as wide as its short diameter is, but it actually tapers around the edges. The sphere formula with the short radius gives a volume (about 0.2 cubic centimeter) that is much smaller than the actual volume of the M&M. The ellipsoid formula should give a volume (about 0.6 cubic centimeter) that is very close to the actual volume of the candy. An M&M indeed has an ellipsoid shape, specifically, a type called an oblate spheroid.

Cleanup
•     If portions of the clay or Play-Doh have been dyed by the M&Ms' coating, you can try to pinch these parts out and throw them in the trash.
•     If you want to eat the soggy M&Ms when you are done with this activity, you may do so, but you should not eat the M&Ms that were in contact with the clay or Play-Doh.

More to explore
Agricultural and Biological Engineering: Tools: Unit-Free Volume Calculators from Mississippi State University
Unique Shape of M&M's Interests Scientists from NPR's Talk of the Nation's Science Friday
Math Tables: Areas, Volumes, Surface Areas from Math2.org
Geometry from MathIsFun.com
M&M Geometry from Science Buddies

This activity brought to you in partnership with Science Buddies