079i - Consolidation Circumference and Area of a Circle

Lesson 79 — Problem Solving and Consolidation: Circumference and Area of a Circle

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

Learning Intentions

• To consolidate fluency with the circumference and area formulas across a full range of circle, composite and sector problems.
• To reason flexibly between circumference, area, radius and diameter, moving confidently in any direction between them.

Success Criteria

I can:

1. Select correctly between , and depending on what is given and what is required.
2. Solve multi-step problems combining circles, composite shapes and sectors.
3. Work backwards fluently from a circumference or area to find a radius or diameter.
4. Justify each step of a solution and evaluate whether a final answer is reasonable.

Warmup

(6 minutes — “Always, sometimes, never”, pairs)

Decide whether each statement is always, sometimes, or never true. Give a supporting example or counterexample.

1. Doubling a circle’s radius doubles its circumference.
2. Doubling a circle’s radius doubles its area.
3. A sector’s perimeter is less than the circumference of the full circle it came from.
4. Two circles with the same circumference have the same area.

Answers: 1. Always — is directly proportional to . 2. Never — area is proportional to , so doubling quadruples the area. 3. Sometimes — true for most sectors, but a sector with a very large angle (close to ) has a perimeter (arc plus two radii) that can exceed the full circumference, since the two radii add extra length the full circle doesn’t have. 4. Always — equal circumference means equal radius (since is a one-to-one relationship), so the areas must also be equal.

Discussion of 3: Ask students to test a semicircle (): perimeter , compared to the full circumference . Still less. Now test angle : arc , plus , giving — more than the full circumference. A rich discussion point.

Activities

Activity 1 — Fluency Circuit: Mixed Formula Practice (12 min)

Individual, timed circuit of 8 short problems — students rotate or work down the list. Emphasis on quick, correct formula selection.

1. Find the circumference of a circle with radius cm ().
2. Find the area of a circle with diameter cm ().
3. A circle has circumference cm. Find its radius ().
4. A circle has area . Find its diameter ().
5. Find the area of a sector with radius cm and angle ().
6. Find the arc length of a sector with radius cm and angle ().
7. A semicircle has diameter cm. Find its area and its perimeter ().
8. Two circles have radii cm and cm. Find the ratio of their areas without fully calculating either area.

Activity 2 — Consolidation Problems: Two-step and Composite Reasoning (12 min)

Pairs. Every answer must carry correct units and a one-sentence justification.

Problem 1. A circular clock face has an area of . Find the length of the metal rim around its edge, using . (Two steps: area to radius, radius to circumference.)

Problem 2. A running track’s infield is a rectangle m by m with a semicircle attached to each short end (together one full circle of diameter m). Find the total infield area, using .

Problem 3. A sector has an area of and radius cm. Find the sector’s angle, using . (Working backwards through the sector formula.)

Answers:

Activity 3 — Rich Problem-solving Task: the Goat and the Shed (13 min)

Whole-class launch, then pairs work through Polya’s stages, sharing back.

A rectangular shed measures m by m and stands in the middle of a large open paddock. A goat is tied by a rope of length m to the midpoint of one of the shed’s m walls. The rope cannot pass through the shed.

Find the total area the goat can graze.

This is genuinely tricky: the rope wraps around the two corners nearest the tie-point once the goat reaches them, giving grazing regions of different radii on different sides of the shed.

Socratic scaffolding:

Prompt	Purpose
Understand: what shape does the goat’s grazing area actually form?	Not a full circle — the shed blocks part of it, and the rope can wrap around corners.
Draw a diagram. Where is the goat tied?	Midpoint of a m wall means m of wall on each side of the tie-point.
What happens directly in front of the wall?	The goat sweeps a semicircle of radius m (the full rope length), since the wall itself blocks the other half.
What happens when the goat reaches a corner?	The remaining rope, after using m to reach the corner, is m. The goat can then sweep a further quarter-circle of radius m around that corner, along the shed’s m side wall.
Does this happen on both sides?	Yes — by symmetry, the same m quarter-circle occurs at both corners of the tied wall.
Devise a plan	Total area = one semicircle (radius ) + two quarter-circles (radius ).
Carry out the plan	See working below.
Looking back	Does the m remaining rope reach past the shed’s m side wall? No — , so the goat’s sweep stays within a quarter turn and does not wrap a second corner. Confirm this assumption explicitly.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Find the area of a circle with radius cm, using .
2. A circle has area . Find its circumference, using (two steps: find , then ).
3. Find the arc length and area of a sector with radius cm and angle , using .
4. Reasoning. Two circles have circumferences of cm and cm respectively (exact form). Without converting to decimals, state the ratio of their radii and the ratio of their areas.

Answers: 1. ; 2. , cm, cm; 3. arc cm, area ; 4. Radii ratio (since ); area ratio (since ).

Common Misconceptions

Misconception	How to pre-empt it
Treating every “backwards” problem (area or circumference given) as one step, forgetting the intermediate radius.	Explicitly label each step: “Step 1: find . Step 2: find what’s actually asked.”
Assuming a sector’s perimeter is always less than the full circumference.	Revisit the warmup discussion — test a sector close to to see the perimeter exceed the circumference.
In wrap-around rope/goat problems, forgetting that the remaining rope length shortens after each corner.	Insist on subtracting the wall length used before calculating the next arc’s radius.
Losing track of units across a multi-step problem (e.g. mixing cm and m mid-solution).	Convert to one unit as the very first line of working, every time.
Rushing the Looking Back step and submitting a numerically correct but contextually implausible answer.	Require one sentence of justification (“is this reasonable, and why?”) on every non-trivial problem.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A circle of radius has the same area as a sector of radius and angle . Find .

Answer

E2 (AMC Junior style). A square of side cm has a circle inscribed inside it (touching all four sides) and, separately, a circle circumscribed around it (passing through all four corners). Find the ratio of the circumscribed circle’s area to the inscribed circle’s area, in exact form.

Answer

Inscribed radius cm. Circumscribed radius is half the square’s diagonal: cm.

The circumscribed circle has exactly double the area of the inscribed circle — true for any square, not just side .

E3 (Challenge — Socratic). A circular pizza of radius is cut by straight cuts, all passing through the centre, into equal sectors (). Find, in terms of and , the perimeter of one sector.

Answer

Prompt	Purpose
What angle does one sector have?	degrees, since equal sectors share the full circle.
What does the sector’s perimeter consist of?	Two straight radii plus one arc.
Write the arc length in terms of	.
Combine	Perimeter .

Check with (quarter circles): perimeter , matching the quarter-sector formula used earlier in this unit.

Homework

1. Find the circumference and area of a circle with radius cm, using .
2. A circle has area . Find its circumference, using (show both steps).
3. Find the area and perimeter of a sector with radius cm and angle , using where convenient, or .
4. A composite shape is a rectangle cm by cm with a semicircle of diameter cm removed from one short end. Find the remaining area, using .
5. Reasoning. A circle’s radius is tripled. Explain, using ratios, the effect on (a) its circumference (b) its area. Then explain why doubling a circle’s area does not double its radius.
6. Challenge. A goat is tied by a m rope to a corner of a square shed of side m, in an open paddock (rope cannot cross the shed). Find the total grazing area, using . (Hint: after the goat reaches an adjacent corner, how much rope remains, and what shape does it then sweep?)

Answers: Q1 — cm, . Q2 — , cm, cm. Q3 — area ; arc cm; perimeter cm. Q4 — rectangle minus semicircle , giving . Q5 — (a) circumference triples, since directly; (b) area increases by a factor of , since ; doubling area only requires the radius to increase by a factor of , not , because area depends on the square of the radius. Q6 — the goat sweeps a sector of radius m (three-quarters of a circle, since the shed itself blocks the remaining at the tied corner), plus, at each adjacent corner, the remaining rope is m, sweeping a further sector of radius m on each side: total .