075i - Circumference of a Circle

Lesson 75 — Explicit Instruction: The Circumference Formula

Strand: Measurement | Descriptor: AC9M8M03 | Duration: 45 minutes

Learning Intentions

• To understand that the circumference of any circle is proportional to its diameter, and that this constant ratio is .
• To use the formulas and to calculate the circumference of a circle.

Success Criteria

I can:

1. Correctly identify the radius, diameter and circumference of a circle.
2. Recall that is an irrational number, and use an appropriate approximation such as or .
3. Calculate the circumference of a circle given its radius or diameter.
4. Calculate the diameter or radius of a circle given its circumference.
5. Express answers in exact form (in terms of ) or as a sensibly rounded decimal.

Warmup

(5 minutes — mini whiteboards, rapid-fire)

Recall from our earlier work on irrational numbers:

1. Is a rational or irrational number? How do you know?
2. Write correct to decimal places.
3. If you measured the circumference and the diameter of ten different circular objects, then divided for each, what would you notice?
4. What is the relationship between the radius and the diameter of a circle?

Teacher note: Question 3 is the hook — every ratio should come out close to , regardless of the circle’s size. This is the entire basis of today’s formula.

Activities

Activity 1 — Explicit Instruction: Finding Circumference (12 min)

I do: A circle has diameter cm. Using :

We could also leave this in exact form: cm.

I do (using radius): A circle has radius cm. Using (a useful approximation when the radius is a multiple of ):

We do: Diameter cm (use ); radius cm (use ).

You do:

1. Diameter cm ().
2. Radius cm ().
3. Diameter m ().
4. Radius cm ().

(Answers: 1. cm 2. cm 3. m 4. cm)

Activity 2 — Explicit Instruction: Working backwards from Circumference (12 min)

I do: A circle has circumference cm. Find its diameter and radius, using .

We do: Circumference cm, using .

You do:

1. Circumference cm () — find and .
2. Circumference cm () — find and .
3. Circumference cm () — find .

(Answers: 1. cm, cm 2. cm, cm 3. cm)

Activity 3 — Applied Task: the Bicycle Wheel (10 min)

Pairs, then whole-class share.

A bicycle wheel has a diameter of cm. Use .

(a) Find the circumference of the wheel. (b) How far does the bike travel in one full rotation of the wheel? (c) How many full rotations are needed to travel km? Should you round up or down — and why?

Socratic scaffolding for part (c):

Prompt	Purpose
Understand the problem	We need whole rotations, and the wheel travels a fixed distance per rotation.
What units do we need to match?	Convert km into centimetres first: m cm.
Devise a plan	Divide the total distance by the circumference (distance per rotation).
Carry out the plan	rotations.
Looking back — round up or down?	full rotations would fall short of km; only rotations covers the full distance. Always round up when counting “enough” whole units to meet or exceed a target.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Find the circumference of a circle with diameter cm, using .
2. Find the circumference of a circle with radius cm, using .
3. A circle has circumference cm. Find its radius, using .
4. Reasoning. Explain why is described as irrational, and why we always use an approximation like or when calculating with it.

Answers: 1. cm; 2. cm; 3. cm, cm; 4. ‘s decimal expansion never terminates or repeats, so it cannot be written exactly as a decimal or fraction — any value we calculate with is therefore always a rounded approximation, never the exact answer.

Common Misconceptions

Misconception	How to pre-empt it
Using the diameter value in the formula without halving it first.	Always state which measurement (radius or diameter) is given before choosing a formula.
Treating as an exact value rather than an approximation.	Regularly show more decimal places of and remind students the “true” answer is only ever approached, never reached, by a decimal.
Forgetting units, or writing for a circumference (a length).	Circumference is always a length — reinforce that only area answers carry squared units.
Rounding too early in a multi-step problem (e.g. rounding the circumference before dividing to find rotations).	Carry extra decimal places through working, and only round the final answer.
Confusing radius and diameter when working backwards from a given circumference.	Always find diameter first (), then halve for radius — never the reverse.

Enrichment — Competition-Style Problems

E1 (Investigation). The Earth’s equatorial radius is approximately km. Estimate its circumference, correct to the nearest km.

Answer

Rounded to the nearest km, approximately km.

E2 (Kangaroo style). A circular running track has a circumference of m. Find its diameter, correct to decimal place.

Answer

E3 (Challenge). A wheel of diameter m rolls, without slipping, through complete rotations. How far does it travel, in kilometres?

Answer

Homework

1. Find the circumference of each circle, using : (a) diameter cm (b) radius cm (c) diameter m.
2. Find the circumference of each circle, using : (a) radius cm (b) diameter cm (c) radius m.
3. A circle has circumference cm. Find its diameter and radius, using .
4. A circular pond has diameter m. Find its circumference correct to decimal place.
5. Reasoning. Two circles have circumferences in the ratio . Explain, without calculating actual values, what the ratio of their diameters must be.
6. Challenge. A car’s wheel has diameter cm. Using , find how many complete rotations the wheel makes over a journey of km. Round appropriately and justify your rounding direction.

Answers: Q1 — (a) cm (b) cm (c) m. Q2 — (a) cm (b) cm (c) m. Q3 — cm, cm. Q4 — m. Q5 — the diameters must also be in the ratio , since means circumference and diameter are directly proportional. Q6 — circumference cm; km cm; rotations , so complete rotations are needed to travel at least km (rounding up, since rotations would fall short).