How to Show the Area of a Circle is πr^2?

An amazing do it yourself project to prove the area of a circle is π x r².


Step 1

Gather the ingredients.

Step 2

CUT OUT CIRCLE: Use the compass to draw a circle on the paper. Cut it out, fold it in half and use the ruler to draw a line along the fold.

Step 3

CUT INTO SEGMENTS: Fold it in half again and use a ruler to draw a line along the fold. Repeat twice more until you have 8 equal sectors marked in the circle…

Step 4

Then colour in half of the sectors before cutting them all out. You should now have 8 sectors.

Step 5

MAKE A PARALLELOGRAM: Rearrange the sectors with four on the bottom and four on top, to form a shape that looks a bit like a parallelogram.

Step 6

RECTANGLE IS πr x r : Fold the end sector in half and cut along the fold to make two half sectors. Put one at each end to form a shape close to a rectangle…

Step 7

.the rectangle’s height is just the radius of the circle, or ‘r’ for short. Its base is almost exactly the curved outsides of the coloured sectors, or half the circumference of the circle. So if the circumference is 2πr., the base must be πr.

Step 8

The area of a rectangle is base x height. So if the base is πr and the height is r, the area of the rectangle is πr x r. Another way of writing this is π r², because r x r = r². TIP: π is just over 3, so the area of a circle is roughly 3 x r².

Step 9

 A quick way to remember the area of a circle is to imagine an r x r square. Its area is r x r = r². Now if you imagine slicing this square diagonally you can see that the area of the circle is more than 2 r². But less than 4 r². So roughly 3 r².

DIY Details


BBC Bitesize

Avg. Duration

30 min

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