How to make a square from a circle - Euclid

How to put a square peg in a round hole

"I love right angles," said Euclid. "They're just so neat and tidy."

"Remember how we made a right angle by rotating a line around a point until it was right at the top? Until it was neither an acute angle or an obtuse angle?"

"We remember!" said Euclid's friends, who were really getting into this geometry thing now,

"Well, look what happens if we rotate another line up to a right angle from the point at the other end of the first line!" said Euclid.

"ooooh!" said Euclid's friends, oohing and ahhhing.

"And look what happens if we join the two points at the top with another line... we get four equal lines, four right angles and two pairs of parallel lines."

"That is so cool!" said Euclid's friends. It looks sort of...

"Square is the word you are looking for," said Euclid. "When all the lines are of equal length and at ninety degrees to each other, we call it a square!"

"All these angles and right angles," said Euclid's friends. "Is there some way we can measure them?"

"Ahh!" Euclid beamed. "That's the really interesting bit. Look at this circle. If you put all the angles inside it together, they add up to 360 measuring thingy-bits that we will call Degrees." (Euclid really was good at thinking up names for things.)

Euclid's friends looked baffled. "You just made that up!" they grumbled.

"No, look!" said Euclid. "Here is Earth at the centre of the Universe. Every year the sun goes round the Earth 360 times. That's where the number of degrees comes from." (In those days, they didn't know that it's the other way around and the Earth really goes around the Sun!)

"But there are 365 days in the year!" Euclid's friends protested.

"Well, strictly speaking, you're right," said Euclid, "but 360 is a much a better number - look... 360 can be divided by a whole bunch of numbers which makes it really easy to work with.

360 divided by 2 equals 180, divided by 3 equals 120, by 4 - 90, by 5 - 72, by 6 - 60, by 8 - 45, by 9 - 40, by 10 - 36, by 12 - 30, by 15 - 24, by 18 - 20 and by 20 - 18. Don't you see how useful that is?"

Euclid's friends looked baffled. "So how many angles are there in a square?" they asked.

"360!" said Euclid. "Look - watch this... this is a circle."

"Right..." said Euclid's friends.

"But see how you can turn the circle into a square..."

"Oh!" said Euclid's friends.

Euclid smiled. "Each right angle is 90 degrees. 4 times 90 equals 360."

"That is amazing!" said Euclid's friends. They were genuinely impressed and wanted to know more.

"So how many degrees are there in a triangle?" They asked.

"I think we'll do that next time," said Euclid. "We've done quite enough for today!"

Well, I hope you enjoyed that. I like to think that's Euclid's version of String Theory.