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Divide a circle up into pie slices

Proof for the area of a circle: Divide a circle up into sections like an orange

Euclid’s proof

We know it’s true that A=πr-squared is the area of a circle, but how can we prove that this formula is always true for every possible circle? Here’s how Euclid did it.

Who was Euclid?
Area of a circle
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Divide a circle into sections

Start by dividing a circle into sixteen sections like an orange. We know that together they add up to the area of the whole circle.

Line the sections up like teeth

Now take all the green sections and line them up next to each other like the bottom teeth of some wild animal. On top of them, line up all the orange pieces like the top teeth of the animal. Then fit the sections together as if the animal had closed his mouth.

A row of orange sections of a circle pointing down, fitted into a row of green sections of a circle pointing up.

Fitting the sections of the circle together to approximate a rectangle

Is it a rectangle?

All the teeth together look almost like a rectangle. The short side of this rectangle is the radius of the circle. The long side of the rectangle is half of the circumference of the circle, or 2πr. What if we multiply them together to get the area of the rectangle? Then we get r (πr) or πr-squared – the area of a circle.

What is circumference?
Proof for a circle’s circumference
Area of a rectangle

If we cut our circle into smaller sections, our rectangle will be straighter, but this is enough to see what it would be like.

More about the area of a circle
And more about circles
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Bibliography and further reading about circles:


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