078i - Problem Solving with Circle Measurements

Lesson 78 — Problem Solving with Circle Measurements

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

Learning Intentions

• To apply the circumference and area formulas to solve multi-step, real-world problems.
• To select and justify the correct formula (or combination of formulas) for a given context.

Success Criteria

I can:

1. Decide whether a problem requires circumference, area, or both, and justify my choice.
2. Solve multi-step problems involving circles in practical contexts (cost, material, distance, capacity).
3. Compare circular measurements to make a reasoned recommendation.
4. Communicate my working clearly, using correct units throughout.

Warmup

(5 minutes — “Which formula?”, pairs)

For each scenario, decide whether you need circumference, area, or both. Do not calculate — just justify your choice.

1. How much fencing is needed to enclose a circular paddock?
2. How much turf is needed to cover a circular lawn?
3. How far does a wheel travel in one rotation?
4. How much pizza (as a fraction of the whole) does one slice represent?
5. How much icing is needed to cover the top of a circular cake, and how much ribbon is needed to go around its side?

Answers: 1. Circumference (a length, enclosing a boundary). 2. Area (covering a region). 3. Circumference (distance per rotation). 4. Both, potentially — area for the slice’s surface, but often expressed as a simple fraction of . 5. Both — icing needs area, ribbon needs circumference.

Teacher note: This warmup is the entire lesson’s thinking skill in miniature. Return to “which formula, and why” explicitly before every problem today.

Activities

Activity 1 — Worked Multi-step Problem: Cost and Material (12 min)

Explicit modelling of the full problem-solving process, using Polya’s four stages.

Problem: A circular swimming pool has a diameter of m. The pool owner wants to (a) buy a fitted pool cover to cover the entire surface, sold at $ per square metre, and (b) install a safety rope around the edge, sold at $ per metre (rope must be bought in whole metres, rounded up). Find the total cost.

I do — modelling Polya’s process aloud:

Understand: Two separate costs — a cover (area, in ) and a rope (circumference, in m, rounded up to whole metres).

Devise a plan: Find the radius from the diameter. Calculate area for the cover cost. Calculate circumference, round up, for the rope cost. Add the two costs.

Carry out the plan:

Looking back: Does $732.88 seem reasonable for an 8 m pool? Yes — check the cover cost alone is roughly $ for , about $ as stated. Also confirm rope was rounded up, since m of rope would leave a cm gap.

Activity 2 — Guided Practice: Applying the Process (13 min)

Pairs. For each problem, students must explicitly identify Understand → Plan → Carry out → Look back before the teacher reveals the worked solution.

Problem 1. A circular garden of radius m is to be surrounded by a m wide gravel path (so the path forms a ring around the garden, with outer radius m). Find the area of the gravel path, using .

Problem 2. A car has wheels of diameter cm. Find how many complete rotations each wheel makes over a journey of km, using .

Problem 3. A circular table seats people spaced cm apart around its edge. If the table has a diameter of m, estimate the maximum number of people who can be seated, using .

Answers:

Activity 3 — Inquiry Task: the Best Value Pizza (10 min)

Pairs, then whole-class share.

A pizza shop sells two options:

• A small pizza: diameter cm, price $.
• A large pizza: diameter cm, price $.

Which pizza gives better value for money, measured by area per dollar? Justify your answer with calculations, then decide: is area-per-dollar the only thing that matters when choosing a pizza? What else might a customer consider?

Socratic scaffolding:

Prompt	Purpose
Understand: what does “better value” mean here?	Area of pizza obtained per dollar spent — more per $ is better value.
What do you need to calculate first?	The area of each pizza, from its radius.
Devise a plan	Find each area, divide by price, compare the two rates.
Carry out the plan	Small: , , rate r=17.5A = 3.14 \times 17.5^2 = 961.625\ \text{cm}^2= 961.625 \div 24 \approx 40.07\ \text{cm}^2/$$.
Looking back	The large pizza gives more area per dollar — better value, since doubling the diameter roughly quadruples the area but the price didn’t quadruple.
Beyond the maths	A customer might also consider: can they finish a large pizza, storage of leftovers, variety from buying two smalls, dietary needs. Value isn’t the only factor in a real decision.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. A circular rug of diameter m needs a fringe sewn around its edge. How many metres of fringe are needed, using ?
2. A circular table of radius cm needs a tablecloth cut to exactly cover its top. Find the area of fabric required, using .
3. A bicycle wheel has diameter cm. How many complete rotations does it make travelling km, using ?
4. Reasoning. A student says, “To find how much paint covers a circular sign, I should use the circumference formula, because paint goes ‘around’ the sign.” Explain why this reasoning is incorrect.

Answers: 1. m; 2. ; 3. cm, km cm, rotations (round up to ensure the full km is covered); 4. Paint covers the flat surface of the sign, which is a region, not a boundary length — this requires the area formula, not circumference. “Around” refers to the sign’s edge, but painting a sign means covering its face.

Common Misconceptions

Misconception	How to pre-empt it
Automatically reaching for area whenever a problem “feels” circular, without checking what is actually being measured.	Always restate: “is this a boundary/length question, or a surface/region question?” before selecting a formula.
Rounding in the wrong direction (e.g. rounding down when “enough” material is needed).	Explicitly ask at the Looking Back stage: does this context need rounding up, down, or to the nearest unit? Justify each time.
Forgetting a second step in a two-part problem (e.g. finding area but never using it to find cost).	Model reading the full question again before writing a final answer — is every part of the question addressed?
Mixing units mid-problem (e.g. cm and m without converting).	Insist on converting to a single unit as an explicit first step whenever mixed units appear.
Treating “value for money” or similar comparisons as requiring only one calculation rather than a ratio/rate.	Reinforce that comparing two options requires a common measure (e.g. per dollar), not just two separate totals.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). A circular running track has an inner radius of m. A runner on the innermost lane runs one full lap. A second runner, on a lane m further out, also runs one full lap. How much further does the second runner travel, using ?

Answer

Notably, this extra distance does not depend on the original radius m at all — only on the m gap between lanes.

E2 (Kangaroo style). A goat is tied by a rope of length m to a corner post of a large rectangular paddock (the rope allows the goat to graze in a quarter-circle, since two fences meet at right angles at the post). Find the grazing area, using .

Answer

E3 (Challenge). A square sheet of metal has side cm. The largest possible circle is cut from it. Find the area of metal wasted, using . What percentage of the original sheet is wasted, correct to decimal place?

Answer

The largest circle has diameter cm, so cm.

Homework

1. A circular clock face has diameter cm. Find (a) the length of trim needed around its edge (b) the area of the clock face, using .
2. A circular flower bed of radius m is to be edged with bricks costing $ per metre (bricks are bought in whole metres, rounded up). Find the total cost, using .
3. A car’s wheel has diameter cm. Find how many complete rotations it makes over a km journey, using .
4. Two circular pizzas are on offer: a medium (diameter cm) for $, and a large (diameter cm) for $. Which is better value per ? Show full working.
5. Reasoning. A circular pool has its radius increased by . Explain, without fully recalculating, what happens to (a) its circumference and (b) its area. Use ratios, not raw numbers.
6. Challenge. A circular running track has an inner edge of radius m and is m wide (outer radius m). Find the area of the track surface itself (not the infield), using .

Answers: Q1 — (a) cm (b) . Q2 — m, rounded up to m, cost 128C=3.14\times50=1575=500,000=500,000\div157\approx3184.7\to3185r=15A=3.14\times225=706.5\ \text{cm}^2\approx44.16\ \text{cm}^2/$r=20A=3.14\times400=1256\ \text{cm}^2\approx48.31\ \text{cm}^2/$50%1:1.51.5^2=2.25125%A=3.14(54^2-50^2)=3.14(2916-2500)=3.14\times416=1306.24\ \text{m}^2$.