Question

The_______ of a circle is called the circumference of a circle.
(a) Area
(b) Volume
(c) Perimeter
(d) Radius

Answer 1

Hint: For solving this problem we consider a circle of radius $${}^{\prime }{r}^{\prime }$$ as follows

We know that the formula circumference of a circle is given as $${}^{\prime }2\pi r^{\prime }$$. Then we check which of the options having the same formula as that of circumference.

Complete step by step answer:
Let us assume a circle of radius $${}^{\prime }{r}^{\prime }$$
We know that the circumference of circle having radius $${}^{\prime }{r}^{\prime }$$ is
$$\Rightarrow C=2\pi r$$
Now, let us go through each option.
(a) Area
We know that the area of a circle of radius $${}^{\prime }{r}^{\prime }$$ is given as
$$\Rightarrow A=\pi r^2$$
Here, we can say that the area is not equal to the area of the circle is not the circumference.
(b) Volume
We know that the circle is a 2D figure.
We know that a 2D figure has no volume.
Here, the volume of the circle is ‘0’ which is not equal to circumference.
(c) Perimeter
We know that the perimeter of a figure is nothing but the total length of the surface.
For finding the perimeter let us assume a small sector making an angle $${}^{\prime }d{\theta }^{\prime }$$ as shown below

For any small angle, we know that the arc length is given as
$$\Rightarrow AB=rd\theta$$
Now total length of surface is given by integrating ‘AB’ applying the limits of $${}^{\prime }{\theta }^{\prime }$$ from $$0\to 2\pi$$ as
$$\begin{array}{rl}& \Rightarrow P=\underset{0}{\overset{2\pi }{\int }}r.d\theta \\ & \Rightarrow P=2\pi r\end{array}$$
Here, we can say that the perimeter is equal to circumference.
So, the perimeter of a circle is called the circumference.

So, the correct answer is “Option c”.

Note: We can say this from the definition of circumference of the circle. The circumference of the circle is defined as the length covered by the circle in one revolution. This can be defined in another way as the total surface length of the circle which is nothing but the perimeter definition.
So, we can say that the perimeter is called the circumference of a circle.