Measuring Pi Experiment: String and Five Round Things

Measure one mug and you will get 3.2, decide pi is roughly three-ish, and learn almost nothing. Measure a coin, a mug, a tin, a saucepan lid and a bicycle wheel, and something much better happens: the answers get closer to 3.14 as the objects get bigger, in a clean trend your kid can see in their own table. That trend is the real content of a measuring pi experiment. It is not a fact about circles. It is a fact about measurement, and it explains why Archimedes did not measure anything round at all.

Key Takeaways

• You’ll measure circumference and diameter of five round objects, divide, tabulate, and compare the results against each other.
• It teaches what pi is (a ratio that is the same for every circle) and why measurement alone cannot pin it down past about two decimal places.
• Best from 8, once dividing with a calculator and reading a ruler to the millimetre are both comfortable.
• Diameter is the hard measurement, not circumference. Nearly every bad result traces to it.
• Use objects spanning a 20-to-1 size range. Without the big one, the main lesson does not appear.

What You’ll Need

Item	How much	Where to get it	If you don’t have it
Round objects	5, sizes from 2 cm to 60 cm	A coin, a mug, a tin of beans, a saucepan lid, a bicycle wheel	A bucket, a dinner plate, a hula hoop, a globe
Non-stretch string	About 3 m	Kitchen string or cotton twine	Dental floss or a strip of paper; never wool or elastic
Ruler and tape measure	1 each	School supplies and a sewing box	A metre of string marked in centimetres
Two books of equal thickness	2	Shelf	Two identical boxes, or two blocks of wood
Calculator	1	Phone	Long division, which takes four times as long
Paper and pencil	2 sheets	Anywhere	Squared paper is better for the table

Time: gathering 10 min, measuring 30 min, calculating 10 min Cost: nothing Mess level: low

Do the Experiment: A Measuring Pi Experiment

Step 1 – Gather objects that differ wildly in size

This is the step that decides whether the activity works. You want something tiny, something enormous, and three in between: a coin around 2 cm, a mug around 8 cm, a tin around 7 cm, a saucepan lid around 25 cm, and a bicycle wheel around 60 cm. If everything you measure is mug-sized, the pattern will not show up.

Step 2 – Draw the table before measuring anything

Four columns: object, circumference, diameter, circumference divided by diameter. Add a fifth for “how sure are you, in millimetres”. That last column looks like fuss and is the most important one here.

Step 3 – Measure circumference with string

Wrap the string once around the widest part, pinch where it meets, mark the spot with a pen, then lay the string flat along the ruler. Keep the string snug but not stretched. Wool and elastic both lie to you by several millimetres; cotton twine does not.

Step 4 – Measure diameter with two books, not by eye

Stand two books of equal thickness upright on the table, slide the object snugly between them, and measure the gap between the book spines. That gap is the diameter. Measuring across the middle with a ruler and hoping you found the centre is where most of the error in this experiment comes from, and this trick removes it.

Step 5 – Divide, and record all the digits

For each object, divide circumference by diameter on the calculator and write down three decimal places even though they are not all meaningful. You want to see the spread. Do not round to 3.1.

Step 6 – Sort the table by object size and look

Rewrite the rows from smallest object to largest. The ratios from the small objects scatter; the ratios from the large objects cluster tightly near 3.14. Ask your kid why, before telling them.

Step 7 – Average the five and compare with 3.14159

Add the five ratios and divide by five. Then write 3.14159 underneath. Then do something better than averaging: throw away the coin result and average the remaining four, and see whether it improves. Deciding which data to trust is a real scientific judgement.

What you should see: Individual ratios between about 3.0 and 3.3. The coin is reliably the worst, often 2.9 or 3.4, because one millimetre of error on a 20 mm diameter is a 5% mistake. The bicycle wheel is reliably the best, usually 3.12 to 3.16. The average of five lands within about 0.05 of pi, which looks like a success but is partly luck: errors cancelled. The trend down the sorted table is the result that always appears.

What’s Actually Going On

Pi is defined as the ratio of a circle’s circumference to its diameter, and the whole point is that this ratio is identical for every circle, from a coin to a planet. The reason is similarity: all circles are scale copies of one another, so doubling the diameter doubles the circumference exactly, and the ratio cannot change. Your child’s table is five independent attempts to measure the same number, which is why the scatter is informative.

Now the part that matters more than the number. The ratio you compute inherits the errors of both measurements, and relative error is what counts. Suppose you can read a length to within 1 mm. On the bicycle wheel, 1 mm out of 600 mm is 0.17%, so your ratio is good to about ±0.005: that is 3.14 give or take a thin margin. On the coin, 1 mm out of 20 mm is 5%, so your ratio is good to about ±0.15, which is 3.14 give or take almost nothing useful. The same ruler gives you two decimal places on a wheel and none on a coin. This is the single most transferable idea in the activity, and it is why precision engineers measure long things when they can.

It also means something uncomfortable: you cannot discover pi’s digits this way. String and rulers stop at two or three decimal places no matter how patient you are. Which makes what Archimedes did around 250 BC genuinely remarkable. He never wrapped string round anything. He inscribed and circumscribed regular polygons inside and outside a circle, doubling the number of sides until he reached 96-gons, and used their perimeters as bounds. The result he proved was 223/71 < pi < 22/7, and the average of those two bounds is 3.1418, which is off by about 0.0002. He got five times the precision of your bicycle wheel by not measuring at all.

That history also kills the most persistent myth in this topic. Pi is not 22/7. Archimedes knew it was not: 22/7 is his upper bound, and he made no claim to have found the exact value. Pi is irrational, which was proved in the eighteenth century, so it has no exact fractional form and no repeating decimal. It is worth saying this to a child who has been told “pi equals 22 over 7” by a worksheet. For comparison, the Egyptian Rhind Papyrus of about 1650 BC uses a value equivalent to 3.16, and the Chinese mathematician Zu Chongzhi, in the fifth century, had 355/113, which is accurate to six decimal places and is still the best fraction of its size.

How to Explain It to Your Kid

Ages 5–8: three and a bit, every time

Skip the division. Cut a piece of string the length of the diameter, then see how many of those lengths fit around the edge. Three fit, and a little bit is left over. Do it on a mug, then a plate, then a bucket. Three and a bit, every single time, no matter the size. That sentence is a true and complete version of pi for a six-year-old, and laying the string pieces end to end around the rim makes it physical.

Ages 9–12: the ratio and the error column

Introduce the word ratio and the division. Then make the error column do work: ask how wrong their diameter measurement could be, in millimetres, and work out what percentage of the object that is. Do it for the coin and the wheel. When they see 5% against 0.17%, the sorted table stops being mysterious. This is their first encounter with relative error, and it will come back in every science class they ever take.

Ages 13+: bound it instead of measuring it

Give them Archimedes’ method. Draw a circle of radius 10 cm, inscribe a regular hexagon, and compute its perimeter exactly: six sides each equal to the radius, so the perimeter is 60 cm against a circumference of about 62.8 cm, giving pi > 3. Then do a 12-gon with trigonometry, then a 24-gon. Watch the bound tighten. They are reproducing a 2,000-year-old argument and getting better accuracy than their ruler can deliver, which is the whole difference between measuring and proving.

The question to ask: “If I had a perfectly round hoop the size of a football pitch and a ruler marked in millimetres, how many decimal places of pi could I get? What if the hoop were the size of a coin?”

Why Teaching This Early Pays Off

Pi arrives in school as a button on a calculator and a number to memorise. A child who has measured it badly five times understands something different and more durable: that pi is a ratio, that ratios can be measured, and that measurement has a resolution.

The relative-error idea is the real prize. It explains why lab equipment is expensive, why you weigh a hundred rice grains instead of one, why opinion polls need sample sizes, and why engineers specify tolerances rather than numbers. The Framework for K-12 Science Education lists using mathematics and computational thinking among its eight core practices, and the error column is exactly that practice in a form a nine-year-old can carry out.

Be honest about the limits. One afternoon of string does not change anybody’s grade, and the research on hands-on work does not claim it will. Deslauriers and colleagues found something more useful to a parent: students in active learning often feel they learned less while actually learning more. Expect your kid to say this was slow and fiddly. Fiddly is the lesson. The experiment that produces a clean answer with no scatter is the one that has hidden the interesting part.

Before You Start: What Parents Should Know

Best age	8 and up. The floor is set by comfort with calculator division and reading millimetres, not by the concept. A six-year-old can do the string-length version in the Ages 5–8 section.
Supervision	Light, except for the tin. You hold the bicycle wheel steady; a wheel that rolls away mid-measurement loses the measurement.
Real hazards	The cut edge of an opened tin is genuinely sharp. Use an unopened tin or a plastic tub instead. If you use a bicycle wheel, lay it flat on the floor rather than measuring it on the bike, where the spokes and chain can pinch fingers.
Setup	A table for the small objects and floor space for the big one. The whole activity needs about 2 m of clear floor at one point.
Frustration point	Step 4, the diameter. Without the two-books method, children measure a chord instead of a diameter, get a ratio over 3.3, and conclude they are bad at measuring. Show the trick before they struggle.
Don’t	Don’t substitute elastic or wool for the string, and don’t let them round the ratios to one decimal place. Both destroy the result you came for.

When It Doesn’t Work

• Every ratio is above 3.3. The diameters are being measured too small, because the ruler is crossing a chord and not the centre. Switch to the two-books method and remeasure.
• Every ratio is below 3.0. The string is stretching, or it is not sitting in one plane around the object. Swap to cotton twine, and check that the string is not spiralling on a mug.
• The results do not improve with object size. The big object was measured sloppily, which is common with bicycle wheels because they move. Lay it flat, weight it, and redo it.
• The average is suspiciously perfect. Sometimes errors cancel and you get 3.141. Do not celebrate: ask whether any single measurement was that good. Usually none were, and that is a useful conversation about averaging.
• They lose interest after two objects. Give them the biggest object second rather than last. The bicycle wheel is the fun one and it pulls them through the rest.

Take It Further

Plot ratio against size and fit a trend. Put object diameter on one axis and the measured ratio on the other, with error bars from the millimetre column. The scatter visibly narrows as diameter grows. This is a proper uncertainty analysis done with kitchen equipment, and it is a strong science fair project because the result is a method finding, not a number.

Do Archimedes’ polygons. Start with the inscribed hexagon, which needs no trigonometry at all, then double to a 12-gon. Compare each bound with the best measurement from the string experiment. A 12-gon beats the mug; a 24-gon beats the bicycle wheel.

Measure something that is not quite a circle. An oval mirror, a slightly dented tin, a ring of rope. The ratio comes out wrong, which is the point: pi belongs to circles, and this is a way to test whether something really is one.

What to Watch For Over the Next 3 Months

• Week 4: They ask how accurate a measurement is, unprompted, usually while weighing or measuring something for cooking.
• Month 2: The phrase “that’s only a few millimetres out of a metre” or something like it shows up in an argument about whether an error matters.
• Month 3: The question worth encouraging is why pi never ends. The honest answer is that it is irrational, which was proved in the 1760s, and that no fraction equals it exactly.

Frequently Asked Questions

Is pi exactly 22/7?

No. 22/7 is about 3.1429, while pi is about 3.14159. The fraction appears in the historical record because it is the upper bound Archimedes proved, not because it is a value for pi. Pi is irrational, so no fraction equals it exactly.

Why does the coin give such a bad answer?

Because relative error dominates. A one-millimetre mistake on a 20 mm diameter is 5% of the measurement, so the computed ratio can be off by about 0.15. The same one-millimetre mistake on a 600 mm wheel is 0.17%, which is why the big object is so much better.

What is the best household object to measure?

A bicycle wheel, a hula hoop or a large bucket, because they are big and rigid. Anything flexible gives a circumference you cannot trust, and anything under about 10 cm across cannot support two reliable decimal places.

Can we get more decimal places by measuring more carefully?

Barely. Household measurement tops out at two or three decimal places because the ruler and the string set a floor. Getting further requires calculation rather than measurement, which is exactly the route Archimedes took with his polygons.

My kid got 3.1 and was unimpressed. Now what?

Give them the error column and the size comparison, which turn a dull number into a question with a real answer. Then hand them the inscribed hexagon, where ten minutes of arithmetic proves pi is greater than 3 without measuring anything.

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About the author

Ricky Flores is the founder of HiWave Makers and an electrical engineer with 15+ years of experience building consumer technology at Apple, Samsung, and Texas Instruments. He writes about how kids learn to build, think, and create in a tech-saturated world. Read more at hiwavemakers.com.

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Sources

1. O’Connor, J. J., & Robertson, E. F. “Pi through the ages.” MacTutor History of Mathematics Archive, University of St Andrews. https://mathshistory.st-andrews.ac.uk/HistTopics/Pi_through_the_ages/
2. National Institute of Standards and Technology. “SI Units” and measurement guidance, Office of Weights and Measures. https://www.nist.gov/pml/owm/metric-si/si-units
3. National Research Council. (2012). A Framework for K-12 Science Education. National Academies Press. https://doi.org/10.17226/13165
4. Deslauriers, L., McCarty, L. S., Miller, K., Callaghan, K., & Kestin, G. (2019). “Measuring actual learning versus feeling of learning in response to being actively engaged in the classroom.” PNAS, 116(39), 19251–19257. https://doi.org/10.1073/pnas.1821936116
5. Chi, M. T. H., & Wylie, R. (2014). “The ICAP Framework: Linking Cognitive Engagement to Active Learning Outcomes.” Educational Psychologist, 49(4), 219–243. https://doi.org/10.1080/00461520.2014.965823
6. NRICH, University of Cambridge. “Painted Cube” and related measurement investigations. https://nrich.maths.org/problems/painted-cube

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