Question

How do you find the perimeter of a $\dfrac{3}{4}$ circle?

Answer 1

Hint:
Here we need to know that the perimeter of the complete circle is given by the formula $2\pi r$ where $r$ is the radius of the given circle. But when we will see the figure we will come to know that including the $\dfrac{3}{4}$ of the complete perimeter we also need to add two times the radius as we can see in the figure.

Complete step by step solution:
Perimeter is actually the total length of the circumference of the circle which is given by the formula $2\pi r$ where $r$ is the radius of the given circle.
Now if we draw the diagram of the three-fourth of the circle we will come to know that the length of the total boundary will be equal to $\dfrac{3}{4}$ of the complete perimeter plus the two times radius as now it is included in the boundary of that $\dfrac{3}{4}$ of the complete circle.
So we know that it will be shown by the figure as:

Now we know the perimeter of the complete circle$ = 2\pi r$
So perimeter of $\dfrac{3}{4}$the complete circle arc$ = \dfrac{3}{4}\left( {2\pi r} \right)$
So we can simplify it and write it as$ = \dfrac{3}{2}\pi r$
Now we can see from the figure clearly that in the boundary of this circle we have two times the radius also which is also to be counted so as to find the perimeter of $\dfrac{3}{4}$the complete circle

So we can say perimeter of $\dfrac{3}{4}$the complete circle$ = \dfrac{3}{2}\pi r + 2r$.

Note:
Here the student can make the mistake by just multiplying the whole perimeter with $\dfrac{3}{4}$.
So he must keep in mind that we need to draw the figure and notice it carefully so that we come to know all the boundary lengths we need to count.