How do we arrive at the formula for area of a circle as #pir^2#?

Answer 1

Please see below.

#pi# is defined as the ratio of circumference #C# of a circle to its diameter, which is double the radius #r#, hence by definition

#pi=C/(2r# and #C=2pir#

There are various ways of deriving formula for area of circle, including complicated one using calculus, however the simplest one found extensively on internet involves splitting a circle into smaller and smaller sectors and then rearranging them as shown below

It is apparent that rearrangement forms roughly a rectangle, which becomes more and more accurate, as we divide a circle in to fine sectors, whose width is #r# and length is half the circumference i.e. #pir#

Hence, area of circle is #pirxxr=pir^2#

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Answer 2

The formula for the area of a circle, (A = \pi r^2), can be derived from the definition of a circle and basic geometry principles.

A circle is defined as the set of all points in a plane that are equidistant from a fixed point called the center. The distance from the center to any point on the circle is called the radius (r).

To find the area of a circle, we can divide the circle into infinitely many infinitesimal sectors and rearrange them to form a shape that resembles a parallelogram or a rectangle. When we rearrange these sectors, they form a shape that closely approximates a rectangle with height equal to the radius (r) and width equal to half the circumference of the circle.

The circumference of a circle is given by the formula (C = 2\pi r), where (r) is the radius.

So, the width of the rectangle is (C/2 = (2\pi r)/2 = \pi r), and the height is (r).

The area of a rectangle is given by the formula (A = \text{width} \times \text{height}). Substituting the width and height we found earlier, we get (A = (\pi r) \times r = \pi r^2).

Therefore, the formula for the area of a circle is (A = \pi r^2).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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