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

A circle is a closed two-dimensional figure with a center where all the points in the plane are equidistant. Some examples of circles are wheels, pizzas, circular ground, etc. The area of a circle formula is useful for measuring the space occupied by a circular field or a plot.

Suppose you have the plot to fence it, then the area formula will help you to check how much fencing is required. Or suppose you have to buy a table cloth, then how many portions of cloth are needed to cover it completely.

But why is the area of a circle pi times the square of the radius? The usual definition of pi is the ratio of the circumference of a circle to its diameter so that the circumference of a circle is pi times the diameter, or two pi times the radius.

This amazing DIY project shows that a circle can be cut and rearranged to closely resemble a parallelogram with height r and base pi times r by dividing the circle into slices. If you do more slices, the approximation obtained in this manner would be even better. By the way, there is also another beautiful way to understand the math behind the area of a circle! You can check it out here.

I got this idea first from Steven Strogatz’s beautiful book, Infinite Powers, about the history of calculus when I was at college.

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. ### Ali Kaya

This is Ali. Bespectacled and mustachioed father, math blogger, and soccer player. I also do consult for global math and science startups.

BBC Bitesize

30 min

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