IntroductionHas 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.
This activity is not appropriate for use as a science fair project. Good science fair projects have a stronger focus on controlling variables, taking accurate measurements, and analyzing data. To find a science fair project that is just right for you, browse our library of over 1,200 Science Fair Project Ideas or use the Topic Selection Wizard to get a personalized project recommendation.
BackgroundGeometry 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.
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 ResultsDid 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.
More to Explore
Agricultural and Biological Engineering: Tools: Unit-Free Volume Calculators from Mississippi State University
Teisha Rowland, PhD, Science Buddies
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