Question

The shape of the top of a table in a restaurant is that of a sector of a circle with the centre O and $\angle BOD=90{}^\circ ,BO=OD=60cm$. Find the perimeter of the table.

Answer 1

Hint: We have to find the length of the curve first using the formula, $\dfrac{\theta }{360}\times 2\pi r$, where r is the radius and $\theta $ is the angle at the centre. Then we will add it with twice the radius of the circle to get the required answer.

Complete step by step answer:
In the question, we have been given a sector of a circle which represents the shape of a table. It has a centre O and we can represent it by the figure given below.

We have been given that the length of BO and OD are equal, which measures as 60 cm and the angle BOD is $90{}^\circ $. We have been asked to find its perimeter. The given shape is a sector of a circle, where B is lying on its circumference and O is the centre. Thus, the distance between B and O can be considered as the length of the radius, whose value is 60 cm. So, we can say that BO or OD is the radius of the circle and that is equal to 60 cm.
Now, in order to find the perimeter of the table, we have to find the sum of the length of arc BD and the sum of the lengths of BO and OD or twice the radius.
We generally use the formula, $\dfrac{\theta }{360}\times 2\pi r$ to find the length of the arc BD, where r is the radius and $\theta $ is the angle at the centre. Now, here the value of $\theta $ is $\left( 360{}^\circ -90{}^\circ \right)=270{}^\circ $ and the radius is 60 cm. So, we get the length of the arc BD as,
$\dfrac{270}{360}\times 2\pi \times 60=90\pi $
The value of $\pi =3.14$, so the length of the arc BD will be,
$90\times 3.14=282.6cm$
And now, we will find the perimeter, which will be the length of the arc BD + twice the radius. So, we get the perimeter as,
$\begin{align}
  & 282.6cm+60cm\times 2 \\
 & \Rightarrow 282.6cm+120cm \\
 & \Rightarrow 402.6cm \\
\end{align}$

Hence, the perimeter of the table is 402.6 cm.

Note: We can find the length of the arc BD by using another method also. We can first find the perimeter of the whole circle first by using the formula $2\pi r$ and also find the perimeter of the quadrant, which is given by $\dfrac{1}{4}\times 2\pi r$, where r is the radius. Then we can subtract the perimeter of the quadrant form the perimeter of the circle to get the perimeter of the table.