AREA OF A CIRCLE-In SAVAN's approach


Here is a way to find the formula for the area of a circle:


Cut a circle into equal sectors (12 in this example) circle 12 sectors

Divide just one of the sectors into two equal parts. We now have thirteen sectors – number them 1 to 13:

circle 13 including 2 half slices

Here Area of sector1=Area of sector13=1/2*Area of sector2.

Rearrange the 13 sectors like this:

sectors laid out like rectangle

Which resembles a rectangle:

sectors with rectangle on top

What are the (approximate) height and width of the rectangle?

The height is the circle's radius: just look at sectors 1 and 13 above. When they were in the circle they were "radius" high.

     sectors laid out like rectangle 

Note that Here length of the approximated rectangle is bumpy edges of 6 sectors. And initially, we divided the circumference of the circle( 2 × π × radius)equally into 12 sectors.

The width (actually one "bumpy" edge) is half of the curved parts around the circle ;which are shared equally at both sides(length sides)... in other words, it is about half the circumference of the circle.

We know that:

                       Circumference = 2 × π × radius:

And so the width is about:

                        Half the Circumference = π × radius

And so we have (approximately):

rectangle is (pi x radius) by radiusradius
π × radius 

Now we just multiply the width by the height to find the area of the rectangle:

Area = (π × radius) × (radius)

= π × radius2

Note: The rectangle and the "bumpy edged shape" made by the sectors are not an exact match.

But we could get a better result if we divided the circle into 25 sectors (23 with an angle of 15° and 2 with an angle of 7.5°).

And the more we divided the circle up, the closer we get to be exactly right.

Conclusion:

                                    Area of Circle = π r2 


 

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