Mathematics 7–10 · Years 7–8
Measuring pi: circumference divided by diameter for real round objects
Measurement and space
The idea
For every circle the circumference is the same multiple of the diameter, and that multiple, measured with string and a ruler, comes out a little over 3.
What you need
- 6 round objects of different sizes per group (tin can, jar lid, roll of tape, bucket, plate, coin)
- 1.5 m of non-stretch string per group (the 300 mm bucket has a circumference of 942 mm, so 1 m is too short to wrap and hold at both ends)
- 30 cm ruler and 1 m tape measure (the ruler reads diameters up to 250 mm, the tape the wider diameters and every circumference)
- Recording table with columns object, diameter d (mm), circumference C (mm), C ÷ d
How to do it
- Measure the diameter of each object across its widest part to the nearest millimetre, with the ruler for objects up to 250 mm across and with the tape held taut across the rim for wider ones such as the bucket; take the largest of three attempts, since a chord shorter than the diameter is the common error.
- Wrap the string once around the object, mark where it meets, then lay it flat along the tape to read the circumference to the nearest millimetre.
- Compute C ÷ d for each object to two decimal places.
- Plot C against d for the six objects and draw the line through the origin that fits best; read its gradient.
- Compare the group's mean ratio with the class mean and with 3.14159.
What you should see
Each C ÷ d value lies close to 3.14. A 1 mm error in the diameter shifts the ratio by about π ÷ d: 0.13 on a 25 mm $1 coin, 0.05 on a 60 mm lid and 0.01 on a 300 mm bucket, and a 1 mm error in the circumference shifts it by 1 ÷ d, about a third as much, so the larger objects give the closer values. The two common slips, a chord measured instead of the diameter and a string wrapped loosely, both push the ratio above π, so class values tend to sit slightly high rather than scatter evenly about 3.14. The plot of C against d is a straight line through the origin with gradient near 3.14. The learner knows it worked when the ratio is close to the same for the smallest and largest objects despite their sizes differing by a factor of 10 or more.
What changes
- What you change
- diameter of the object
- What you measure
- circumference
- What you keep the same
- string pulled snug but not stretched
- measurements read at eye level
- same ruler and tape
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- Pi is a property of a particular circle rather than of all circles.
- Pi equals 22/7 or 3.14 exactly.
- A bigger circle has a bigger ratio of circumference to diameter.
Safety card
Hazards
- sharp rim on an opened tin can
Controls
- use unopened cans or file the rim
- check objects before handing out
Note
No chemicals or heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; ordinary classroom supervision.
Curriculum references
The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.
- Mathematics K–10 Syllabus (2022), Stage 4 Length; page read 2026-09-22MA4-LEN-C-01
- Australian Curriculum v9AC9M7M03AC9M8N01AC9M8M03
Sources
The pages the author read to write this activity.