076i - Area of a Circle

Lesson 76 — Explicit Instruction: The Area of a Circle

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

Learning Intentions

• To understand where the formula for the area of a circle comes from.
• To use the formula to calculate the area of a circle given its radius or diameter.

Success Criteria

I can:

1. Recall the formula and explain why it uses the radius, not the diameter.
2. Calculate the area of a circle given its radius, using an appropriate approximation for .
3. Calculate the area of a circle given its diameter, by first halving to find the radius.
4. Calculate the radius of a circle given its area.
5. Express answers in exact form (in terms of ) or as a sensibly rounded decimal, with correct units.

Warmup

(5 minutes — mini whiteboards, rapid-fire)

Recall our work on circumference from last lesson.

1. State the formula for the circumference of a circle in terms of the radius.
2. A circle has radius cm. Find its circumference, using .
3. Circumference is measured in a length unit (cm, m). What kind of unit do you expect an area to be measured in?
4. Look at this square of side cm sitting around a circle of radius cm. Estimate — what fraction of the square do you think the circle covers?

Teacher note: Question 4 is the hook. Most students guess between and . Do not resolve it yet — return to it explicitly in Activity 1.

Activities

Activity 1 — Explicit Instruction: where Does come From? (10 min)

Show the classic “unrolled circle” demonstration: a circle is cut into many thin sectors (like pizza slices) and rearranged into a shape that approximates a rectangle/parallelogram.

• The rearranged shape has a height equal to the radius .
• The rearranged shape has a base equal to half the circumference, since the sectors alternate direction: .

Return to the warmup hook: the circle of radius cm sitting in the square. Square area . Circle area — about of the square, not or .

Critical distinction to display prominently:

Activity 2 — Explicit Instruction: Calculating Area (13 min)

I do: A circle has radius cm. Using :

I do (given diameter): A circle has diameter cm. First halve to find the radius, cm. Using :

Explicit warning: A very common error is to substitute the diameter directly into . Model checking aloud: “Have I been given the radius, or do I need to halve the diameter first?”

We do: Radius cm, using ; diameter m, using .

You do:

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

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

Teacher note: Questions 1 and 2 are deliberately matched (radius vs diameter ) so students discover they give the same area — a useful self-check.

Activity 3 — Working Backwards: Finding the Radius from the Area (10 min)

I do: A circle has area . Find its radius, using .

We do: A circle has area , using .

You do:

1. Area () — find .
2. Area () — find , then .
3. Area () — find .

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

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Find the area of a circle with radius cm, using .
2. Find the area of a circle with diameter cm, using .
3. A circle has area . Find its radius, using .
4. Reasoning. Explain the mistake in this working, then correct it: “A circle has diameter cm, so .”

Answers: 1. ; 2. radius cm, ; 3. , mm; 4. The diameter, not the radius, was substituted into the formula. Correct working: cm, .

Common Misconceptions

Misconception	How to pre-empt it
Substituting the diameter directly into .	Insist students write "" as an explicit first line whenever a diameter is given.
Confusing with , mixing the two formulas up.	Keep both formulas displayed side by side; ask “am I finding a length or a region?” before starting.
Treating as .	Model order of operations explicitly: square the radius first, then multiply by . Show the numerical difference with a worked contrast.
Forgetting to square-root when working backwards from area to radius.	Rehearse the inverse relationship explicitly: squaring and square-rooting undo each other.
Writing area answers with linear units (cm instead of cm²).	Insist every area answer carries squared units; contrast directly with circumference answers from Lesson 75.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A circle has the same numerical value for its area and circumference. Find its radius.

Answer

Since , .

E2 (AMC Junior style). A circular pizza has radius cm. If the radius is doubled, by what factor does the area increase?

Answer

The area increases by a factor of , not — a common trap.

E3 (Challenge). A circle is inscribed exactly inside a square of side cm (the circle touches all four sides). Find the area inside the square but outside the circle, using .

Answer

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

Homework

1. Find the area of each circle, using : (a) radius cm (b) diameter cm (c) radius m.
2. Find the area of each circle, using : (a) radius cm (b) diameter cm (c) radius mm.
3. A circle has area . Find its radius and diameter, using .
4. A circular garden bed has diameter m. Find its area correct to decimal place, using .
5. Reasoning. Two circles have radii in the ratio . Explain, using the formula , what the ratio of their areas must be.
6. Challenge. A washer (a flat ring) is made by cutting a circular hole of radius cm from the centre of a circular disc of radius cm. Find the area of the remaining washer, using .

Answers: Q1 — (a) (b) (c) . Q2 — (a) (b) (c) . Q3 — , cm, cm. Q4 — m, . Q5 — the areas are in the ratio , since area is proportional to the square of the radius. Q6 — .