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📐 Mathematics - Class 7
Lesson 7 · Perimeter and Area

Circumference and Area of a Circle

MathematicsSubject
14 minEstimated read
The shape that refuses scissors

Every area formula in the last topic came from cutting a shape up and sliding the pieces into a rectangle. A circle has no straight edges, so that trick seems to fail. It does not fail. It just needs more cuts.

Turning a circle into a rectangle

Cut a paper circle into eight equal sectors and arrange them alternately, one up and the next down, in a row. The result already looks a bit like a parallelogram. With sixteen it looks much more like one, and with thirty two the edges are almost straight.

Now measure that rectangle

The height of the rectangle is one radius. The length is made of the curved edges, half pointing up and half pointing down, so it is exactly half the circumference, which is pi r. Area = length times height = pi r times r.

\[ A = \pi r^2 \]

Square the radius, not everything else

Pi r squared means pi times r times r. Squaring the radius first and then multiplying by pi is the safest order, because it keeps the two operations separate.

Circumference or area?

Circumference Area
FormulaC = 2πr = πdA = πr²
Unitscm, mcm², m²
It answersHow far round the edge?How much space inside?
Real questionsWheel distance, fencing a pond, edging a tableclothPainting a lid, watering a field, the size of a pizza
Doubling does different things to each

Double the radius and the circumference exactly doubles, but the area becomes four times bigger. This is the rule from Unit 3: lengths scale by k, areas scale by k squared. It explains why a pizza of twice the diameter feeds four people rather than two.

Rings and paths

A ring is a big circle with a smaller one removed. Work out both areas and subtract. Be careful when a question gives the inner radius and the width of the path: the outer radius is the inner radius plus the width.

Radius, again

The formula uses the radius. Questions nearly always give the diameter instead. Halve it first. A diameter of 14 cm put straight into pi r squared gives 616 instead of 154, which is four times too big.

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Worked Example 1

Find the area of a circle of radius 21 cm. Take pi as 22/7.

Step 1: square the radius firstr² = 21 × 21 = 441
Step 2: multiply by piA = 22/7 × 441 = 22 × 63 = 1386 cm²
Why the question chose 21Radii of 7, 14, 21 and 28 are chosen so that the 7 in 22/7 cancels neatly.
Worked Example 2

A circular table top is 1.4 m across. Find the cost of polishing it at Rs 450 per square metre.

Across means diameterd = 1.4 m, so r = 0.7 m.
AreaA = 22/7 × 0.49 = 1.54 m²
Cost1.54 × 450 = Rs 693
Sense checkThe table would just fit inside a 1.4 m square of area 1.96 m². Our 1.54 is a bit less, which is right: a circle fills a little over three quarters of the square around it.
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Activity 7.2

Cut a circle into a rectangle and prove the formula yourself

What you needTwo paper circles of the same size, a compass, a protractor, scissors and glue.
MethodCut one circle into 8 sectors of 45 degrees, lay them alternately up and down and glue them. Then do the same with 16 sectors of 22.5 degrees underneath.
Then measureMeasure the height and length of each shape. Compare the height with the radius, and the length with half the circumference.
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1

Area of a circle: A = pi r squared, meaning pi times r times r.

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1Explain where the formula for the area of a circle comes from.
2What is the difference between the circumference and the area of a circle, and how do the units help?
3Why does doubling the radius of a circle multiply its area by four rather than by two?
4A circular pond of radius 14 m has a path 3.5 m wide all the way round it. Find the area of the path, showing every step.
5Why is it useful to know that a circle fills a little over three quarters of the square drawn around it?

Question 1 of 10

1The area of a circle is
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Slide 1 of 4
Unit 7 · Area of a Circle

Area of a Circle

Mathematics · Grade 7

What You Will Be Able To Do

✂️

Prove the area formula yourself, with scissors and a paper circle

π

Use A = pi r squared confidently, forwards and backwards

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Find the area of a ring or a path without any new formula

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Explain why a pizza of twice the width feeds four people, not two

“

A circle has no straight edges to cut along, so the scissors trick seems to fail completely. It does not fail. It just needs more cuts.

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Presenter notes: Open by reminding the class what worked last lesson and then breaking it. Every shape gave in to a pair of scissors: cut it, slide the piece round, and it became a rectangle. Hold up a paper circle and ask where they would cut. There are no straight edges, so there is nowhere obvious to start. Let the difficulty sit for a moment before you say that the trick still works, it just needs a lot more cuts, and that today they will cut a circle into a rectangle with their own hands.