Why is the formula for circle's area pi times r squared?

Answer 1

Here is one way of approaching this formula.

We need to know two things:

* the area of a triangle is half the base times the altitude; * the circumference of a circle is 2 pi r. Draw a circle and inscribe a hexagon inside the circle. Then draw the radii from the centre of the circle to each of the six vertices of the hexagon. (Sorry, I don't have a diagram.)

The hexagon has been divided into six triangles. Look at one of these triangles: it has base one side of the hexagon and altitude a bit less than the radius of the circle.

The area of all six triangles is

6 times (1/2) times (side of hexagon) times (altitude of triangle).

Shuffle this slightly to get

(1/2) times 6 times (side of hexagon) times (altitude of triangle).

Now 6 times (side of hexagon) is the perimeter of the hexagon. So

area of hexagon = (1/2) times (perimeter of hexagon) times (altitude of triangle).

Do this again with a 12-sided figure instead of a hexagon, then a 24-sided figure, and so on. We get

area = (1/2) times (perimeter of many-sided figure) times (altitude of triangle).

If we take a figure with a lot of sides, its area will be very close to that of the whole circle, its perimeter will be very close to the circumference of the circle, and the altitude of one of the (very thin) triangles will be very close to the radius.

So (waving my hands a bit here),

area of circle = (1/2) times (perimeter of circle) times (radius of circle).

If we know that the perimeter of the circle is 2 pi r, we get

area of circle = (1/2) time 2 time pi times r times r = pi times r squared.

This isn't quite a precise proof, because of the hand-waving bit. But it could be made into one. See Archimedes' proof in http://en.wikipedia.org/wiki/Area_of_a_disk.