Experimental Pi

March 14, 2021
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Welcome to International Day of Math

We all know that π is the ratio between the circumference and the diameter of any circle. But did you know this special number shows up in unexpected places too?

Here we suggest two experiments in which the number π shows up. After you perform them and see our celebrated number appear, think about why you think it shows up? What does the experiment have to do with anything you know about 3.14157…?

Beans Pi

Material:

  • Bag of beans
  • Mid-size container with a squared bottom and lid
  • Scrap paper
  • Pencil
  • Ruler
  • Compass
  • Tape or glue
  1. Draw the outline of the container on a piece of paper and cut it out. Notice that since the bottom of the container has to be a square, then all sides should be about the same length.
  2. Fold the square piece of paper in two. Unfold and fold back in the other direction. These folds should mark the center of the square.
  3. Place the point of the compass in the center and open it up to the width of the square.
  4. Draw a circle. It should fit exactly in the square.circle in a square
  5. Tape or glue the piece of paper to the bottom of the container. You can do it on the outside if you prefer, but make sure the circle is facing towards the inside of the container.
  6. Put 20 beans inside the container and close it. Counting the beans carefully is important.
  7. Shake the container, then carefully put it down, open it and take out the beans that landed outside the circle.
  8. Count those beans and calculate how many beans landed inside the circle.
  9. Repeat the experiment at least 5 times, more if possible, and keep track of your results. You might want to make a chart like the following:
Repetition Beans outside the circle Beans inside the circle
1
2
3
4
5
Total
  1. Add up all the numbers of beans that fell inside the circle.
  2. Calculate the quotient

Total number of beans inside the circle
Total number of beans thrown between all experiments

For the denominator, for example, if you repeat the experiment 5 times since you are throwing 20 beans each time, that would make a total of 100 beans thrown.

Times the result by 4 and what you get should be pretty close to Pi. The more times you repeat the experiment, the closest it will get!

You can also see a computer simulation of the experiment on this website, but the real experiment is more surprising! On the website, n represents the number of beans thrown (dots on the diagram). Move the slider to increase the number of virtual beans. They count the dots inside the circle for you, but you can verify the calculation by counting yourself.

The game of the needle

Material:

  • Sheet of paper
  • Triangle ruler
  • Pencil
  • 50 needles, sewing pins, toothpicks, or small matches (all of the same size)
  1. Measure carefully the length of the needle.
  2. Align the short edge of the triangle with the edge of the sheet, measure twice the length of the needle, and make a mark. Make another mark again measuring twice the length of the needle after the first mark. Keep going in this fashion.

  1. Now align the triangle with the other edge of the sheet and make lines passing through each one of the marks.
  2. After this process, your sheet should have a set of lines parallel to its edge, separated by twice the length of the needle.

Grid

  1. Take the needles and throw them on the sheet. Any needles that land outside the paper should be thrown again. (Click on the image to see an animation of the experiment.)

  1. Remove the needles that cross any line and count them.
  2. Repeat the experiment at least 4 times and keep track of your results.
  3. Add up all the numbers of needles that crossed any line.
  4. Calculate

Total number of needles thrown between all experiments
Total number of needles that crossed a line

This result should also be pretty close to π! Again, the more times you repeat the experiment, the better your estimation will be. Here’s an animation of a simulation of the experiment.

This post was contributed by Buckeye Aha! Math Moments, the outreach program from OSU’s Department of Mathematics. Check out their website for more fun math activities.

Sources

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