The Area of Circle

The basic terms

The locus of point traversing at a fixed distance with respect to a given point gives us a circle. The given point is often known as the centre of a circle or the fixed point. The distance between the fixed and the moving point is constant, tracing a circular path. Thus, the distance from one another is represented as the radius of a circle. The direct distance between a fixed point and any random point on the boundary is the radius of the circle.

There are infinite lines passing through the circle and cutting it into different sections. The line passing through the centre and intersecting on the boundary is the longest possible line inside the circle. This line is known as the diameter of the circle. Eventually, there can be many diameters. The radius and the diameter of the circle are linked as –

Diameter = 2 x Radius

The locus of a point with respect to the fixed point determines the boundary. Now, the length along the boundary or the perimeter of the circle can be given as:

C=2πR or C=ΠD

Here, R is the radius, 

D is the diameter, and 

π is the mathematical constant used whose value is 22/7 or approximated at 3.14. 

Now, the circumference of the circle can be used to evaluate the area of the circle. 

Area of the Circle

The locus of a point with respect to the fixed point makes the boundary of the circle making it a closed figure. The area can be evaluated for the closed figure and thus, we can say that the area of the circle is the space enclosed by the boundary of the circle. 

The radius of the circle can be used to compute the area. Hence, we have the area of circle as—

Area = πR2

Thus, the area of the circle can be computed. 

Also, if the measurements of the diameter and/or circumference is given, then the area of a circle can be evaluated as—

Area =πD2 / 4=C2 / 4π

Derivation of the Area of the Circle

The basic definition of area of the rectangle can be used to derive the formula for area of the circle. Thus, we can find the dimensions for the rectangle from the given circle and compute the area. 

Now, assume that we have a circle with the given radius R. We can find the circumference of the given circle using the formula. Now, we cut the circle into small sectors or sections. These sections can be rearranged and placed side by side and alternatively, forming a rectangle like shape. If the sections are smaller, after arranging the section, we can see a rectangle. Thus, if we make infinitely many sections, then the circle looks or tends to look like a rectangle.

Now, we can see that the one side of the rectangle is described by the radius of the circle. And, as the sections are placed alternatively, the other side of the rectangle formed can be said to be half the circumference of the circle. 

Hence, we can say length=2πR2=ΠR and breadth=R

Thus, the circle is now equivalent to that of a rectangle. Hence, evaluating the area of rectangle with corresponding length and breadth, we get the area of a circle,

Area of circle=Area of rectangle=lb=πR × R=ΠR2

If the radius or the constant distance is given, the area of the circle can be computed. Also, we can evaluate the area if we know the length of diameter.

Application of Area of Circle

The area of the circle can be used infinitely in many sectors. This can be used in different bakeries, to make a pizza or cake with a definite radius and area covered on the pan. Also, the area covered by the football stadium or cricket stadium can be evaluated using the area of a circle. Different ferris wheels, or the wheel of cars, can be used to find the area of the circle and the distance it covers all over. 

Conclusion

When it comes to the concept of the formula for area, importance should be to the following:

1. The area of the circle can be computed if radius or the circumference is known or given. 

2. The ratio is the mathematical constant , for the diameter and the circumference for any circle.

3. The sector in a circle is the area between any two radius of the circle. And the circumference consumed by the sector formed is known as the arc. 

4. The circle is a point circle if it does not enclose any area and the radius of such a circle is 0.