096i - Relating Radius Diameter and Circumference

Lesson 96 — Relating Radius, Diameter and Circumference

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

Learning Intentions

• To move fluently between radius, diameter and circumference.
• To choose the appropriate form of the relationship for a given problem.

Success Criteria

I can:

1. Convert between radius and diameter without hesitation.
2. Use and correctly, choosing the convenient one.
3. Find the radius or diameter from a given circumference.
4. Estimate before calculating, and check that my answer is sensible.

Warmup

(6 minutes — the conversion drill, mini whiteboards)

1. cm. Find .
2. cm. Find .
3. m. Find .
4. mm. Find .
5. A circle’s radius is cm. Roughly how big is its circumference — , or cm?

Answers: 1. ; 2. ; 3. m; 4. mm; 5. , so cm.

Q5 is the lesson’s whole trap in miniature: given a radius, the doubling step must come first. Students who reach straight for get — the "" distractor.

Activities

Activity 1 — Explicit Instruction: Two Forms, One Relationship (14 min)

The relationship, three ways:

Why ? Substitute into . It is not a new fact — it is the same fact with the diameter unpacked.

The choosing rule:

Given	Use
Diameter
Radius	(or double first, then )
Circumference, want
Circumference, want

I do — from the radius. cm.

Name the estimate shortcut: , or . Every answer gets checked against it.

I do — backwards to the radius. cm.

(Multi-variable solving from Lesson 91 — substitute what is known, solve for the rest.)

We do:

1. cm — find .
2. m — find .
3. cm — find , then .
4. m — find to one decimal place.

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

Activity 2 — Practice Circuit (14 min)

Pairs. Estimate first on every item — written down before the calculation.

Set A — forwards.

1. cm
2. mm
3. m
4. km

Set B — backwards.

5. cm — find
6. m — find
7. km — find
8. cm — find (1 d.p.)

Set C — in context.

9. A round table has radius cm. What length of ribbon edges it exactly?
10. A tin’s label wraps once around a tin of diameter cm, with a cm overlap. How long is the label?
11. A circular pond of circumference m is to have a fence m outside it, all the way round. How long is the fence?
12. A trundle wheel has circumference exactly m. What is its diameter, to the nearest centimetre?

Socratic scaffolding for Q11:

Prompt	Purpose
What is the pond’s radius?	m.
What is the fence’s radius?	m further out: m.
Its circumference?	m.
How much longer is the fence than the pond edge?	m.
Where have you seen that number before?	— the rope problem again ( per metre of gap).
Looking back	The extra length depends only on the gap, not the pond’s size.

(Answers: 1. cm; 2. mm; 3. m; 4. km; 5. cm; 6. m; 7. km; 8. cm; 9. cm; 10. cm; 11. m; 12. cm.)

Activity 3 — Inquiry: the Rolling Wheel (8 min)

Pairs, with a trundle wheel or a marked circular lid if available.

A wheel rolls along the ground without slipping.

1. Mark a point on the rim. Roll the wheel exactly one full turn. How far did the wheel travel?
2. Explain the connection between “one turn” and “the circumference”.
3. A wheel of diameter cm rolls m. How many turns?
4. Why is a trundle wheel usually built with a circumference of exactly m?

Socratic scaffolding:

Prompt	Purpose
As the wheel turns, what unrolls onto the ground?	The rim — one turn lays down exactly one circumference.
So distance ?	Turns circumference.
For Q3: circumference first.	cm m.
Turns?	turns.
Interpret.	About full turns and a bit — the “bit” matters if you are counting clicks.
Q4: why m?	Each click is exactly one metre — no calculation needed in the field. Design chosen to make the maths disappear.

Checks for Understanding

(5 minutes — exit ticket)

1. cm. Estimate, then find .
2. m. Find .
3. A wheel of diameter cm makes turns. How far does it roll?
4. Reasoning. A student found for cm as cm. Diagnose the error.
5. Reasoning. Two circles’ radii differ by m. By how much do their circumferences differ, and does it depend on their sizes?

Answers: 1. Estimate ; cm; 2. m; 3. cm; turns cm m; 4. They used instead of — treated the radius as the diameter; correct answer cm; 5. By m; no — it depends only on the difference.

Common Misconceptions

Misconception	How to pre-empt it
Using instead of .	The warmup’s Q5 distractor; every radius problem states “double first” aloud.
Halving when they should double (and vice versa).	written on the board all lesson; check against the estimate.
Skipping the estimate.	Estimates are written before calculating and marked.
Forgetting units, or mixing them mid-problem.	Convert first; every answer carries a unit.
Rounding turns to the nearest when a whole rotation is required.	Q3 of the inquiry — interpret in context (L61’s rule).
Believing the fence result depends on the pond’s size.	Q11’s scaffolding ties it back to the rope problem.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A circle’s radius is cm. Using , find its circumference.

Answer

cm.

E2 (AMC Junior style). A wheel travels m in turns. Find its diameter.

Answer

Per turn: m ; so m.

E3 (Challenge). A running track’s inner lane has circumference m. Each lane is m wide. How much longer is lane than lane , measured round the full circle?

Answer

Lane ‘s radius is m greater; extra length m.

E4 (Challenge). Two pulleys have radii cm and cm. If the small one turns times, how many turns does the large one make? (The belt does not slip.)

Answer

Belt length moved: cm. Large pulley: turns. (Three times the radius, one third the turns.)

E5 (Challenge). A circle’s circumference and diameter are both whole numbers of centimetres. Explain why this is impossible.

Answer

, and is irrational — it cannot be written as a fraction. If and were both whole numbers, would be a fraction. So at most one of them is a whole number. (A genuine appearance of irrationality at Year 7 level.)

Homework

1. Find (estimate first): (a) cm (b) m (c) km (d) mm.
2. Find the requested value: (a) cm — find (b) m — find (c) mm — find .
3. A bicycle wheel has radius cm. (a) Distance per turn? (b) Turns to cover m?
4. A circular tablecloth has diameter cm. Trim costs 6.50$ per metre and is sold by the whole metre. Find the cost.
5. A garden hose is wound times around a circular reel of diameter cm. Estimate the hose’s length.
6. A circular pool of radius m gets a path m wide around it. Find the length of the path’s outer edge.
7. A wheel of circumference m rolls m. How many turns?
8. Reasoning. Explain why and are the same rule.
9. Reasoning. A student says “if I double the radius, the circumference goes up by .” Correct this precisely.
10. Challenge. Two circles have circumferences cm and cm. Find the difference in their radii, and check it against .

Answers: Q1 — (a) cm (b) m (c) km (d) mm. Q2 — (a) cm (b) m (c) mm. Q3 — (a) cm (b) turns to cover it. Q4 — m m bought; cost 39\approx 125.612\approx 15.075.5C \approx 34.5488 \div 2.2 = 40d = 2rC = \pi dC = 2\pi rrC257.52.515.7 \div 6.28 = 2.5$ ✓