how to find the area of the circle in the simplest method?

Question

area of circle = (pi)*r*r
where,
pi=3.14
r=radius of the circle

How to arrive at this formula using the simplest method?

Answer 1

Pi, by definition, is the ratio between the circumference and the diameter of a circle. That yields the formula C = 2rπ... but that is derived directly from definition.

How this was extended into the two dimensional analysis of area is beyond me. Sounds like integral calculus. This formula for area existed well before calculus reared its ugly head. But there is a calculus proof for it, if you care to know.

Answer 2

The only way I know is to use integration. I've done it before but I can't remember the details without doing it all again from scratch. Here's an outline of the proof.

Pretend you've sliced up your circle into a series of tiny vertical slivers. For -r < x < r, the height of each sliver is 2 sqrt(r^2 - x^2). That means you can calculate the area using the integral

2 sqrt(r^2 - x^2) dx

That's easier said than done. You need to make a trigonometric substitution. Not sure the best one, but let's try x = r sin t. Then dx/dt = r cos t. Integral becomes

2r (cos t)^2 dt

2 (cos t)^2 can be rewritten as 1 + cos 2t so you need to make another substitution u = 2t. From there I think you can work it out.

Answer 3

Imagine the circle cut into a very large number of segments. Now imagine that these segments are cut from the circle, and placed top to toe. Effectively, you end up with a rectangle, of breadth r, the radius of the circle, and length pi*r. The area of this rectangle is very very close to the area of the circle, and is length * breadth, which is:

r*pi*r = pi*r^2.

The calculus provides a more mathematical proof of this, but I`m assuming that you wanted a very simple proof.

Hope this helps, Twiggy.

Answer 4

find area circle simplest method

Answer 5

The proofs are not really that simple but a yahoo search gave two or three websites that describe integration and geometrical approximation techniques. Use the search words, "Are of circle proof"

Answer 6

Because