Key concepts
Circles
Circumference
Diameter
Mathematical formula
Pi

Introduction
Mathematicians get excited about the discovery of mathematical relationships. They look at the world around them in terms of numbers, formulas and equations. Mathematics is fun and practical, too. It comes in handy when calculating how long you need to save your allowance before you can buy a new game. You use math to double or triple a recipe or to calculate how late you can leave your house and still get to school on time, too.

You can also use it to predict the size of things—on a page or in the real world. In this science activity you will examine circular objects and see what relationships you can discover about their sizes. You will explore whether the circumference of a circle relates in a particular way to its diameter. As you explore the relationship, you might be surprised at how useful the results can be; maybe it will inspire you to save up for a bigger bike!

Background
Did you know people have been studying mathematics for thousands of years? Many mathematical terms we use today originate from Greek and Latin, revealing the origin of some of the ancient scholars who worked on these problems. The word circumference, for example, comes from the Latin words circum (which means around) and ferre (to carry). It can refer to the line bordering the circle as well as to the length of that line. The word diameter is derived from the Greek words dia (across) and metron (measure). It refers to a straight line that starts at a point on the circumference, goes through the center of the circle and ends at the other side of the circle. It can also refer to the length of this line. Mathematicians have proved that the diameter is also the longest distance across the circle.

Enough terminology! It’s time to start exploring.

Materials

• At least four circular objects of different sizes (For instance, you can use a large coin, round container lid, large container lid and a bicycle wheel.)
• A large roll of twine or ribbon that you can cut into small pieces
• Tape, such as masking tape (optional)

Preparation

• Assemble all of your objects in one area so that they are in easy reach.
• Start your exploration with a medium-size circle, such as a yogurt container lid. In the next couple of steps you will cut pieces of twine (or ribbon) that have the length of the circumference and the diameter of this circle. Once you have those pieces ready, you can start exploring if these relate in a particular way. You will repeat the procedure for different-size circles in the hope of discovering that the diameter and circumference of all your examples relate to each other in the same way.

Procedure

• To create a piece of twine the length of the circumference (the line bordering the circle) of your first circular object, hold the end of a piece of twine, with your thumb, on a point on the edge of the circular object.
• Wrap the twine exactly one time around the object and cut the twine where the wrapped-around twine meets its starting point. To make this a little easier, you can temporarily attach the beginning of the twine to the circular object with tape, then wrap and cut the twine to that piece of tape. Can you see that the length of your piece of twine is exactly the length of the circumference of your circular object?
• To measure the diameter you need the length of a straight line that starts at a point on the circumference, goes through the center of the circle and ends at its other side. Because it is not easy to find a circle’s center, you will use a mathematical fact about circles, which states that the diameter is also the longest distance across a circle. To create a piece of twine with the longest length across the circle use your thumb to hold the end of a new piece of twine (or ribbon) on a point on the edge of the medium-size circular object.
• Span a straight line of this twine across the circle to another point on the circumference of the circle. Now move the second point along the circumference—to the left and right. Do this until you find the longest straight spanned piece of twine possible. When you move the end of the twine away from this point, the spanned piece of twine gets shorter again. Cut off the piece of twine where it was longest to get a measure of the diameter of this circle. Did you observe that your spanned twine went through the center of the circle?
• Now you have everything you need to start exploring. Which distance is longer—the diameter or the circumference? Is it longer by a lot or a little?
• If you fold the longer piece of twine in half, does it fit the length of the other piece? If so, this would mean the longer piece is twice as long as the shorter piece. If you did not find a good fit by folding the longer piece in two, what about if you fold it in threes, fours or fives? Do you get an exact or an approximate fit? How would you translate your findings in words like "twice as long" or "three times as long"?
• Try the activity again with a different-size circular object. Do you expect the same relationship between circumference and diameter length to be valid for the different-size circle?
• Repeat the circumference- and diameter-finding until you have explored a tiny, medium, big and very large circle. Can you find a relationship that is valid for all the tested circles? Is it an exact or an approximate relationship? If you found a relationship, do you think you have enough data to conclude that your relationship is valid for all circles?
• Extra: Look around the house to locate some circular objects and make an estimate of the length of the diameter and the circumference of these objects. Which one was easier to estimate for you, the diameter or the circumference? Which one would be easier to measure with a ruler?
• Extra: If you found an approximate relationship between circumference and diameter, how could you make it more exact? Hint: you can use a ruler to measure the length of your twine pieces and do a little math. (For example, try dividing a circumference by its corresponding diameter; try again with each circle. Do you get a similar number each time—regardless of whether the circle size was the same?)
• Extra: One real-world application of this principle is when calculating the distance different-size wheels travel. To explore the relationship between the distance traveled on the ground and wheel size, mark a spot on the circumference of a wheel (such as a bicycle wheel) with tape. Place that spot on the ground and indicate this location on the ground with tape or chalk. Roll the wheel along a straight line until the same spot on the circumference touches the ground again. Mark this location on the ground with tape or chalk. Now compare the distance between the two marked locations on the ground with the length of the diameter and circumference of the wheel. Can you find a relationship? What does this mean for the number of rotations a bigger or smaller wheel might need to travel the same distance?
• Extra: Now that you know the relationship of the diameter to the circumference—and maybe even the distance traveled—can you see practical ways where this relationship can be useful?

Observations and results
If all went well, you should have discovered that the circumference was a little more than three times the diameter of the circle for each circle, no matter how small or big the circle was.

If you were able to work more exactly, you might have found that it was not exactly three times, but rather three and one seventh the diameter. And even that is not exact.

Mathematicians found that the ratio of the circumference to the diameter of a circle is a constant, meaning it is the same for all circles, no matter how large or small the circles are. They also discovered, however, that this ratio is a number that can never be determined precisely. Since the mid-1800s, this ratio has been referred to with the Greek letter π (pi), which is a remarkably interesting number. It appears not only in geometry but also in other mathematics such as probability theory. It also shows up in the natural world, such as in the description of waves—from the visible ripples on the water to the invisible waves of light and sound.

More to explore
Prehistoric Calculus: Discovering Pi, from Better Explained
Describing Nature with Math, from NOVA
Talking Pi and Pie for Pi Day, from Science Buddies

This activity brought to you in partnership with Science Buddies 