Attributes of a circle

What is common between these shapes?

They all have a distinct shape in which the distance between the point in the center and the periphery is constant. Such a shape is known as a circle. There are certain attributes associated with a circle. 

Here the center of the circle is O. Its distance from all the points in the periphery is constant and the constant distance is known as the radius. OA, OB, OC, and OD all are radii. If a straight line joins any two points on the circle boundary while passing through the center, then the line is called the diameter. The distance traveled along that periphery of the circle is called the circumference and represents the perimeter of the circle. In the circle, BD is the diameter.

Circumference and π

Let's observe a triangle circumscribed in a circle and the same circle inscribed in a square. From observation, we can see that the three vertices are at ‘r’ distance from the center where r is the radius of the circle. 

The triangle inequality law shows that the longest side must always be more than the sum of the other two sides. So one of the sides of the triangle joined with two radii must be at least greater than the r+r or 2r. That means the perimeter of the triangle must be greater than 6r. Since the center of the circle and square are the same, the side of the square is equal to 2r. That means the perimeter of the square is 2r+2r+2r+2r=8r. Since the path of the circle is in between the triangle and the square, the distance traveled around the circle will be more than 6r and less than 8r. The distance traveled around the circle can be measured by using a thread to make a circle and measuring the diameter. From experiments, it is found that the circumference is around 6.2832 times the radius. If the radius is converted into diameter, then the circumference is 3.1416 times the diameter. This experiment was found to be true for circles of any size. The constant of proportionality that relates the circumference “C” and diameter “d” is denoted by a Greek letter π. 

C=πd

or, C=π*2r

or, C=2πr