Circular Reasoning: Finding Pi

A science activity from Science Buddies that measures up

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Can you uncover a useful mathematical formula with just a few household objects and some ribbon? It's easier than you think. In fact, we would wager a slice of pi that you can!

Key conceptsCircumferenceMathematical formula

Introduction

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

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
• Scissors (An adult should help you use them.)
• 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.
• To measure the diameter you need the length of a which states that the diameter create a piece of twine with the longest length a 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.
• Now you have everything you need to start exploring.
• If so, this would mean the longer piece is twice as long as the shorter piece.
• Try the activity again with a different-size circular object.
• Repeat the circumference- and diameter-finding until you have explored a tiny, medium, big and very large circle.
• 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.
• Extra: 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. )
• 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.
• Extra:

Observations and results

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 explorePrehistoric Calculus: Discovering Pi, from Better ExplainedDescribing Nature with Math, from Talking Pi and Pie for Pi Day, from Science Buddies

Science Buddies

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