Where does the area formula of the circle come from?



Math is so beautiful that everything is related.The area formula of the circle is from the circumference formula of the circle.
area of the circle
Now let's write the formulas;

  • Area formula of the circle; π r 2 

  • The formula of the perimeter of the circle; 2π r

  • Area formula of the rectangle; A*B
Let's first consider that we cut the circle into slices.Let's arrange the pieces like pizza slice side by side.And we get an image like the gif below.

circle gif


 Let's say we do this slicing process forever.As you can see in the gif, the number of slices is increasing gradually.As a result, when we combine the slices, we approach to obtain a smooth rectangle.

 What did we have ? 

 The perimeter formula and the area of the rectangular formula.The long edge of the rectangle we obtained is actually half the circumference of the circle.Let the long edge of the rectangle be A.The short edge of the rectangle equals the radius of the circle.Let B be the short edge of the rectangle.
A=π r                   B=r

We apply them to the area formula of the rectangle.


rectangle area



Consequently A * B = π r 2

 The result we get is actually the area formula of the circle.
Math is awesome.

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