095i - Discovering Pi

Lesson 95 — Investigating the Relationship Between Circumference, Diameter and Π

Strand: Measurement | Descriptor: AC9M7M03 | Duration: 45 minutes

Equipment: circular objects (tins, lids, cups, plates — at least five per group, of clearly different sizes), string or tape measures, rulers, calculators.

Learning Intentions

• To discover that the ratio of circumference to diameter is the same for every circle.
• To understand π as that constant ratio.

Success Criteria

I can:

1. Name the radius, diameter and circumference of a circle.
2. Measure a circumference and a diameter accurately.
3. Calculate the ratio and recognise it is always about .
4. Explain what π is, in my own words.

Warmup

(6 minutes — circle vocabulary, board and mini whiteboards)

Draw a large circle with centre marked.

1. Name the distance from centre to edge. (Radius.)
2. Name the distance right across through the centre. (Diameter.)
3. Name the distance around the edge. (Circumference — not “perimeter”, though it is one.)
4. How are radius and diameter related? (.)
5. Estimate: is the circumference about , or times the diameter? Vote and record the class tally.

Do not resolve Q5. The vote is the hypothesis; the lesson is the experiment. Record the tally on the board to revisit at the end.

Activities

Activity 1 — The Measuring Investigation (18 min)

Groups of three. The heart of the lesson — do not rush it.

For each of your five circular objects:

1. Measure the diameter with a ruler, to the nearest millimetre. (Tip: the widest measurement across is the diameter.)
2. Measure the circumference by wrapping string around it, marking the overlap, then measuring the string.
3. Record both in a table, in the same units.
4. Calculate for each, to two decimal places.

Object	(cm)	(cm)

Circulating prompts:

Prompt	Purpose
Is the string taut and following the rim exactly?	The main source of error — slack string inflates .
Did you measure the widest way across?	Off-centre measurements shrink and inflate the ratio.
Your ratios: are they all the same number, or all in the same range?	Prepares the measurement-error discussion.
The biggest and smallest object — compare their ratios.	The punchline: size is irrelevant.

Class data pooling. Collect every group’s ratios on the board — expect a spread from about to . Compute the class mean.

The reveal, staged carefully:

1. “Every ratio is close to the same number — about .”
2. “The object’s size made no difference. A saucer and a dinner plate give the same ratio.”
3. “That number has a name: π (pi), and its exact value is , continuing forever without repeating.”
4. Return to the warmup tally: how did the class vote do?

The definition to record:

π is not a mystery symbol — it is the answer to a division. Every circle, divided the same way, gives it.

Why measurements varied: string stretches, rims are imperfect, rulers have limits. The mathematics gives one exact value; the measuring approximates it. This is Lesson 46’s distinction between measurement error and mathematical truth, now met a second time.

Activity 2 — Consequences of the Discovery (12 min)

Rearranging the relationship — three forms of one fact:

(Multi-variable formula work from Lesson 91 — any variable can be the target.)

I do — forward. A bicycle wheel has diameter cm:

Estimate first, as always: about cm — the answer should be a little more ✓

I do — backward. A tree’s circumference is cm:

We do: Find given cm, mm, m. Find given cm, m.

(Answers: cm; mm; m; cm; m.)

Convention for this course: use unless a calculator’s π button is available; state which was used.

Activity 3 — Inquiry: the Rope around the Earth (7 min)

Whole class — the classic result, and it lands hard.

A rope is tied tightly around the Earth’s equator (about km). Now imagine adding just one metre to the rope’s length and lifting it evenly all the way round.

1. Predict: could a sheet of paper slide under? An ant? A cat? A person?
2. Now compute the gap.

Socratic scaffolding:

Prompt	Purpose
Take the class vote first, and record it.	The prediction is nearly always “barely a hair”.
What does the rope’s length equal?	The circumference: .
So how does change if grows by m?	, so grows by m.
And the gap is the change in the radius, not the diameter.	Radius grows by half that: about m.
So the gap is…?	About cm — a cat walks under, comfortably.
Does the Earth’s size appear anywhere in that calculation?	No. The same cm results for a tennis ball.
Looking back	Because is proportional, the change depends only on the added length — never on the starting size.

Checks for Understanding

(5 minutes — exit ticket)

1. Define π in one sentence, without saying "".
2. A circle has diameter cm. Estimate, then calculate, its circumference.
3. A circle’s circumference is cm. Find its diameter.
4. A group measured cm and cm. Calculate their ratio and comment.
5. Reasoning. Two circles have very different sizes. What is the same about them?

Answers: 1. The number you get when any circle’s circumference is divided by its diameter; 2. About ; cm; 3. cm; 4. — close to π, slightly high, likely from slack string or an off-centre diameter; 5. The ratio of circumference to diameter — always π.

Common Misconceptions

Misconception	How to pre-empt it
π is “just a button” or a mysterious constant.	The class measures it into existence before it is named.
π depends on the circle’s size.	The pooled data covers tiny to large objects with the same ratio.
π equals exactly or exactly .	Both are approximations; state the exact value continues forever.
Confusing radius with diameter in .	Always identify which is given before substituting; noted as the same fact.
Treating measurement spread as failure.	Named explicitly as measurement error, as in Lesson 46.
Believing the rope result depends on the Earth’s size.	The inquiry’s final prompt makes the independence explicit.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A circle has radius cm. Find its circumference, using .

Answer

; cm. (This is why survives: it makes sevens vanish.)

E2 (AMC Junior style). A wheel of diameter cm rolls without slipping. How far does it travel in turns?

Answer

One turn covers cm; turns cm m.

E3 (Challenge). A circular pond has circumference m. Find its diameter and radius.

Answer

m; m.

E4 (Challenge). Two circles: the second’s diameter is three times the first’s. How do their circumferences compare? Justify without any numbers.

Answer

is a pure product — tripling triples . (Lesson 93’s clean scaling: no constant term to dilute the effect.)

E5 (Challenge). A rope around a tennis ball (circumference cm) is lengthened by m. How big is the gap now?

Answer

m — the same cm as the Earth. The starting size is irrelevant.

Homework

1. Define, in your own words: radius, diameter, circumference, π.
2. Find the circumference (use ): (a) cm (b) m (c) cm (d) m.
3. Find the diameter: (a) cm (b) m (c) km.
4. Measure three circular objects at home. Record , and in a table, and comment on how close your ratios are to π.
5. A bicycle wheel has diameter cm. (a) How far does the bike travel in one wheel turn? (b) How many turns to travel km? (Round sensibly and say why.)
6. A circular garden bed has radius m. Edging costs 14$ per metre. Estimate first, then find the cost.
7. Reasoning. A student says “bigger circles have a bigger π.” Explain the error, referring to your data.
8. Reasoning. Why do measured values of vary between groups, even though π is exact?
9. Challenge. A circular running track has circumference m. Find its diameter, and how much longer an outer lane is if it runs m further out all the way round. (Hint: the rope problem.)

Answers: Q2 — (a) cm (b) m (c) : cm (d) : m. Q3 — (a) cm (b) m (c) km. Q5 — (a) cm (b) turns — round to a whole number; a part-turn does not complete the distance, so turns. Q6 — ; m; cost 105.507.5 \times 14 = $105d = 400/\pi \approx 127.31.22\pi(1.2) \approx 7.54$ m longer — this is why staggered starts exist.