How to make a square from a circle.

In The Murderous Maths of Everything we meet THE THREE UNSOLVED PROBLEMS OF ANTIQUITY.

The ancient Greeks tried to do all their drawings just using a straight edge and a pair of compasses, but one of the things they found impossible was drawing a square that has exactly the same area as a circle. The Murderous Maths Organisation likes a challenge, so we solved the unsolved problem.

1/ We need to start with a circle. If the diameter =1, then the circumference = p.

The formula for the area of any circle is pd2 / 4, so if d=1, the area of our circle = p/4.

Our first job will be to make a rectangle with this same area. If we have a rectangle measuring p along the bottom and 1/4 up the side then that'll do it!

2/ In the book we show how to convert the circumference of the circle into a straight line of length p. That's the tricky part that the Greeks couldn't do but we used a secret weapon - A TIN OF BEANS!

This is how we drew a line of length p using a tin of beans with a diameter of 1.

If the Greeks had managed to draw this diagram of the circle and the p line, then they could have taken measurements off it and made a square with the same area as the circle! We didn't have space in the book to show exactly how they would have done it, so that's why this webpage is here.

3/ To make our rectangle, we start by drawing a line of length p and then construct a right angle at one end.

4/ Now we need to get a line of length 1/4. We get this by bisecting the diameter of the circle twice. (Remember the diameter = 1) Once we've done this we can use it to set our compasses to a length of 1/4.

5/ Here's how to make the rectangle with area p/4. We continue with our line that has the right angle constructed on the end.

This rectangle has the same area as the circle.

6/ So far so good. Now we need to convert this rectangle into a square of the same area p/4. This is a pretty good trick on its own!

And that's how we solved one of the great UNSOLVED problems of antiquity using our SECRET WEAPON. It's such a brilliant weapon that we're surprised that the ancient Greeks didn't think of it first!

If you enjoyed this page, find out how we TRISECT AN ANGLE!

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