KS3 · Maths

Area of a circle

Draw a square on a circle's radius. How many of those squares fit inside the circle? More than 2, fewer than 4, and the exact answer is a famous number.

Before any formula

How many radius squares fit in a circle?

Take any circle and draw a square on its radius. Call it a radius square: its area is r × r = r². How many radius squares does the circle hold? The line is measured in radius squares. Place the four clues first, then commit to where the circle itself goes.

Around — The square around the circle (each side is the diameter, 2r)Inside — The square inside the circle, corners on the circle (exactly half of the square around it)Whole — Grid clue: a circle of radius 5 cm, centred where grid lines cross on a 1 cm grid, has 60 squares wholly inside it (one radius square is 5 × 5 = 25 cm²)Cover — Grid clue: 88 squares on the same grid cover the circle (every square with any part of the circle in it)Circle — The circle itself: your best estimate

Why it's exactly π r²: slice the circle

8 → 16 → 32 slices

Cut the circle into equal slices, like a pizza. Each slice is a sector, and they're all congruent (identical). Now lay them in a row, alternately pointing up and down, so they interlock like teeth.

1 / 5

Which would you do?

Sam's rug

A circular rug is 6 m across. That 6 m is its diameter. Sam wants to know the rug's area.

Which is closest to what you'd do?

Use 2πr with r = 3. That gives 6π m².Put the 6 straight in: π × 6² = 36π m².Halve 6 to get r = 3, then π × 3² = 9π m².r = 3, and 3² is 6, so the area is 6π m².

How sure are you?

I'm sureI think soI'm guessing

Use 2πr with r = 3. That gives 6π m².

This is a common misconception

2πr is the circumference: the distance round the edge. It's a single length, so it's measured in metres, not square metres. Area needs two lengths multiplied together: π × 3 × 3.

Put the 6 straight in: π × 6² = 36π m².

This is a common misconception

6 m is the diameter, and A = πr² needs the radius. Halve first, so r = 3. Using 6 gives 36π, which is four times too big: doubling the length you square makes the answer four times as big.

Halve 6 to get r = 3, then π × 3² = 9π m².

This matches the evidence

Spot on. Radius or diameter? Diameter, so halve it. Square the radius: 3 × 3 = 9. Multiply by π: 9π m², which is about 28.3 m².

r = 3, and 3² is 6, so the area is 6π m².

This is a common misconception

Squaring means multiplying a number by itself: 3² = 3 × 3 = 9, not 3 × 2. Sneaky, because it lands on the same wrong answer as using 2πr, so it can look as if two methods agree.

Radius first, then square it, then multiply by π. 2πr goes round the edge; πr² fills the inside.

Worked example: diameter to area

Problem

A circle has a diameter of 10 cm. Find its area (a) exactly, in terms of π, and (b) correct to 1 decimal place.

Now go backwards

From area to radius: estimate first

A circle has an area of 200 cm². About how long is its radius? Use the fact that a circle holds a little over 3 radius squares.

Your estimate

35cm

A circle has an area of 200 cm². About how long is its radius? Use the fact that a circle holds a little over 3 radius squares. Estimate between 0cm and 70cm.

0cm70cm

Maths · Algebra

From area to diameter, line by line

Step forward — the next line lights up, the rest dim back.

GoalA circle has an area of 40 cm². Find its diameter, correct to 1 decimal place.

40 = πr²

Put the area into A = πr². Now it's an equation to solve for r.

Step 1 of 5

Put the area into A = πr². Now it's an equation to solve for r.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

How many squares fit inside a circle? The answer is π — and that's the whole formula.

What you need to know

A radius square is a square whose side is the radius, so its area is r².
A circle holds π radius squares, about 3.14 of them: more than 2r², less than 4r².
On a grid, a circle's area lies between the number of squares wholly inside it and the number of squares that cover it.
Slice a circle into thin sectors and rearrange them: you get a near-parallelogram with base πr and height r, so A = πr².

The big picture

A circle's area is π lots of the square on its radius: A = πr². You can see why two ways: squeeze the area between squares you can count, or slice the circle into a near-parallelogram with base πr and height r. Then use the formula forwards and backwards.

Key points

1A = πr², where A is the area and r is the radius.

2Area multiplies two lengths, so the radius is squared. The circumference, 2πr, is a single length round the edge.

3Check whether you've been given the radius or the diameter. Halve a diameter before doing anything else.

4The order is: radius, square it, multiply by π.

5For an exact answer, leave it in terms of π (like 25π cm²). Otherwise use the π key and round to a sensible accuracy.

6Working backwards from the area: divide by π, then square root to get r. Double r for the diameter, or use it in 2πr for the circumference. Round only at the end.

7The radius can be read from a grid, or from the coordinates of the centre and a point on the circle.

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Worked example

Problem

A circle has its centre at (2, 1) and passes through the point (2, 7). Find its area in terms of π, and correct to 1 decimal place.

⚠ Watch out

Using 2πr (the circumference) for the area, or putting the diameter in as if it were the radius. Area needs the radius, squared: halve a diameter first, then work out r × r, then multiply by π.

🧠

Memory hook

Round the edge: 2πr. Fill the inside: πr², squared because area is two lengths multiplied. And a circle is always π radius squares: a bit more than 3.

✓

Check yourself

A circle has a diameter of 4 cm. Without a calculator, write its area in terms of π. Is it more or less than 12 cm²? (r = 2, so 4π cm²: just over 12.)

Flashcards

(13)

What is a radius square?

A square whose side is the circle's radius. Its area is r².

Between which two squares does a circle's area lie?

More than the square inside it (2r²), less than the square around it (4r²).

How can you estimate a circle's area on a grid?

Count the squares wholly inside (a lower bound) and the squares that cover any part of it (an upper bound). The area lies between them.

What is the formula for the area of a circle?

A = πr², where A is the area and r is the radius: π lots of the radius square.

Why is the radius squared in the area formula?

Area multiplies two lengths. In the sliced-up circle they are πr and r, so πr × r = πr².

Which circle formula gives a length, and which gives an area?

2πr is the circumference, a length round the edge. πr² is the area, the space inside, in square units.

What happens when you cut a circle into thinner and thinner sectors and rearrange them?

The shape looks more and more like a parallelogram, with the same area as the circle.

In the rearranged circle, what are the base and the height?

Base = half the circumference = πr. Height = the radius, r.

You're given a circle's diameter. What's your first step?

Halve it to get the radius.

What does 'give the area in terms of π' mean?

Leave π as a symbol, like 49π cm². Nothing has been rounded, so it's exact.

How do you find the radius from the area?

Divide the area by π to get r², then take the square root.

In a problem with several steps, when should you round?

Only at the final answer. Keep full calculator values until then.

Nobody has told you the radius. Where else can it come from?

Read it off a grid, or count from the centre's coordinates out to a point on the edge.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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How many radius squares fit in a circle?

Sam's rug

From area to radius: estimate first

From area to diameter, line by line

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