008i - Pi and Circle Measurement

Lesson 8 — Exploring as an Irrational Number in Applied Contexts

Strand: Number | Descriptor: AC9M8N01 | Duration: 45 minutes

Learning Intentions

• To recognise as an irrational number and understand why approximations are used in practice.
• To apply to calculate the circumference and area of circles in applied, real-world contexts.

Success Criteria

I can:

1. Explain why is irrational and why we use approximations such as or .
2. Calculate the circumference of a circle using or .
3. Calculate the area of a circle using .
4. Solve applied measurement problems involving , choosing an appropriate level of accuracy.

Warmup

(5 minutes — mini whiteboards, discussion)

1. If you measured the distance around a circular plate (circumference) and divided it by the distance across (diameter), what value would you expect to get, roughly?
2. Is that value the same for a small plate and a large plate?
3. Estimate without a calculator. Compare with your answer to Question 1.
4. Why might only be an approximation for this ratio, rather than the exact value?

Teacher note: Question 4 previews the lesson’s core idea — is a fixed, irrational constant, and is a convenient rational approximation, not an exact equivalent (this connects directly to Lesson 6).

Activities

Activity 1 — Explicit Instruction: , Circumference and Area (10 min)

I do: Establish as the constant ratio of circumference to diameter for every circle, and state its irrational decimal expansion.

’s decimal never terminates and never recurs — it is irrational — so we round it for practical use: or .

State the formulas:

Model — circumference: a circular pond has diameter m.

Model — area: the same pond, radius m.

We do: Together find the circumference and area of a circle with radius cm, using .

You do: Find the circumference and area of a circle with (a) radius m (b) diameter cm. Use .

(Answers: (a) m, ; (b) radius cm, cm, .)

Activity 2 — Guided Practice: Choosing Appropriate Accuracy (10 min)

Discuss: different situations call for different levels of rounding — sometimes an exact value in terms of is wanted (e.g. m), sometimes a rounded decimal is wanted, and the number of decimal places should suit the context (e.g. a tape measure reads to the nearest mm, not the nearest micrometre).

I do: A circular running track has radius m. Give the circumference (a) exactly, in terms of (b) rounded to the nearest metre.

We do: Together give the area of a circle of radius cm, exactly in terms of , then rounded to decimal place.

You do: A circular garden bed has diameter m. Give the area (a) exactly in terms of (b) rounded to decimal places.

(Answers: radius m; (a) (b) .)

Activity 3 — Applied Task: in Real Contexts (14 min)

Pairs.

Problem 1. A bicycle wheel has a diameter of cm. How far does the bicycle travel in one full rotation of the wheel? Give your answer to the nearest cm.

Problem 2. A pizza has a diameter of cm and costs the same as a smaller pizza of diameter cm plus a 41.5$ times the area of the small one?

Problem 3 (harder — applied, multi-step). A circular athletics track has an inner radius of m. Lane is m further out from the centre than Lane (i.e. radius m). A runner completing one full lap in Lane runs further than a runner in Lane . In a real race, Lane runners start slightly ahead of Lane runners to make the race fair — this is called a “staggered start.” How much further does one lap of Lane cover than one lap of Lane ?

Socratic scaffolding for Problem 3:

Prompt	Purpose
Understand the problem	We need the difference in circumference between two circles with radii m apart, not either circumference alone.
Devise a plan	Calculate each circumference using , then subtract — or find a shortcut using the formula directly.
Carry out the plan — direct method	m.
Is there a shortcut?	Since , increasing by always increases by exactly m — regardless of the starting radius.
Looking back	Does it make sense that the difference doesn’t depend on the actual size of the track? Test with very different radii, e.g. and : difference is still m. The pattern holds because circumference grows linearly with radius.

Answers: Problem 1 — cm. Problem 2 — small pizza area ; large pizza area ; ratio , so yes, the large pizza has more than times the area (in fact more than double). Problem 3 — m further.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. A circle has radius cm. Find its circumference, using .
2. A circle has diameter m. Find its area, rounded to decimal place ().
3. Explain why the circumference of a circle can never be written as an exact terminating decimal, even though we can measure it with a ruler.
4. Two circles have radii and . Find, in exact form using , how much greater the second circle’s circumference is than the first’s.

Answers: 1. cm. 2. radius m; . 3. Circumference , and is irrational, so unless , the product with an irrational number is (in general) irrational too — its exact decimal never terminates; any ruler measurement is necessarily a rounded approximation. 4. .

Common Misconceptions

Misconception	How to pre-empt it
Using the diameter instead of the radius in the area formula: .	Always require students to write “radius ___” as an explicit first line before substituting into .
Treating (or ) as the exact value of .	Reinforce, as in Lesson 6, that these are rounded approximations; distinguish “exact answer in terms of ” from “approximate decimal answer” in every worked example.
Forgetting squared units for area (writing cm instead of cm), or using squared units for a length answer.	Insist on units being written at every step, and sanity-check: “is this a length or an area?”
Assuming doubling the radius doubles the area.	Contrast directly: doubling multiplies circumference by (linear), but multiplies area by (quadratic) — revisit Problem 2 in Activity 3 as evidence.
Rounding intermediate working (e.g. rounding radius or an intermediate value) before the final step, causing accumulated error.	Model keeping full calculator precision throughout, and rounding only the final answer.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). A circle has circumference cm. Using , find its radius.

Answer

E2 (Kangaroo style). A square has the same perimeter as a circle’s circumference. If the circle has radius cm (use ), find the side length of the square.

Answer

Circumference cm. Square side cm.

E3 (Challenge). A circle is inscribed exactly inside a square of side cm (the circle touches all four sides). Find the area of the square not covered by the circle, to decimal place.

Answer

The circle’s diameter equals the square’s side, so radius cm.

E4 (Investigation). Which grows faster as increases: the circumference () or the area ()? Explain, using as evidence.

Answer

Area grows faster. Circumference is proportional to (linear growth — doubling doubles ), while area is proportional to (quadratic growth — doubling quadruples ). Evidence: at , gives ratios , but gives ratios — area increases far more steeply.

Homework

1. A circle has radius cm. Find, using : (a) its circumference, to decimal place (b) its area, to decimal place.
2. A circular swimming pool has diameter m. Find its area exactly in terms of , then rounded to the nearest whole .
3. A wheel of diameter cm rolls without slipping. How far does it travel in full rotations? Round to the nearest cm.
4. Explain, in one or two sentences, why is classified as irrational rather than rational.
5. Reasoning. A circular table has its radius increased by cm. Show, using the formula , that the circumference always increases by exactly cm, no matter the original radius.
6. Challenge. Two circles have areas in the ratio . What is the ratio of their radii? What is the ratio of their circumferences?

Answers: 1(a) cm. (b) . 2. radius m; exact area ; . 3. one rotation cm; rotations cm. 4. ‘s decimal expansion never terminates and never recurs, and it cannot be written exactly as a fraction of two integers, so by definition it is irrational. 5. cm, independent of the starting radius , because circumference is a linear function of radius. 6. Since area is proportional to , a ratio of areas gives a ratio of radii (since ); circumference is proportional to , so the ratio of circumferences is also .