Area and Circumference of a Circle

Area of a Circle

The area of a circle is the number of square units inside that circle. If each square in the circle to the left has an area of 1 cm2, you could count the total number of squares to get the area of this circle. Thus, if there were a total of 28.26 squares, the area of this circle would be 28.26 cm2.

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

Cut a circle into equal sectors (16 in this example)

Rearrange the 16 sectors like this:

These sectors look like a rectangular region but not exactly so. The length of this rectangle will be equal to half of the circumference and breadth equal to the radius of the circle.

We know that:

Circumference = 2 × π × radius

And so the width is about:

Half the Circumference = π × radius

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

Area = (π × radius) × (radius)

= π × radius2

Hence, we get

Area of the circle = $\frac{1}{2}$ circumference x radius

= $\frac{1}{2}$ x 2$\pi$r x r²

= $\pi$r²

Circumference of a Circle

Verification:

Draw three circles of different radii. Measure the diameter of each one of them with the help of scale andfill teh table given below:

The ratio is denoted bycalled pi ($\pi$)

Here,

$\pi$ = 3.14(nearly)

= $\frac{22}{7}$ (nearly)

Thus,

$\frac{Circumference}{Diameter}$ = $\pi$

or, $\frac{c}{2r}$ = $\pi$

or, c = 2$\pi$r

$\therefore$ Circumference of a circle(c) = 2$\pi$r