Question

The radius of a certain circle is 30 cm. Find the approximate length of an arc of this circle; if the length of the chord of the arc be 30 cm.

Answer 1

Hint:
Here we need to find the approximate length of an arc of this circle. For that, we will draw a chord in a circle and we will join the end points of this chord to the center of this circle which will form a triangle, and the triangle formed will be an equilateral triangle. From there, we will get the angle made by the arc at the center and then we will use the formula to find the length of the arc.

Complete step by step solution:
First we will draw a chord in a circle and we will join the end points of this chord to the center of this circle which will form a triangle.

We can see from the figure,
$AO=B0=30cm$ as these are the radii of the same circle.
The triangle formed is an equilateral triangle as all the sides are equal and thus the each angle of a triangle is equal to ${60}^{\circ }$.
Thus, $\mathrm{\angle }AOB={60}^{\circ }$
Now, we will find the length of the arc formed corresponding to that chord.
We know the formula to find the length of arc is given by
$\Rightarrow \text{length of arc=}{\displaystyle \frac{\theta }{{360}^{\circ }}}\times 2\pi r$
Here $\theta$ is the angle made by the arc at the center and $r$ is the radius of the circle.
Substituting the value of angle and radius of the circle, we get
$\Rightarrow \text{length of arc=}{\displaystyle \frac{{60}^{\circ }}{{360}^{\circ }}}\times 2\times {\displaystyle \frac{22}{7}}\times 30$
On multiplying the terms, we get
$\Rightarrow \text{length of arc=}31.42cm$

Thus, the required length of the arc is equal to 31.42 cm.

Note:
Here we have calculated the length of an arc formed. An arc of a circle is defined as the part or portion of the circumference of the circle. The formula to find the length of arc is equal to the product of circumference of circle and the ratio of angle made by the arc to the total angle i.e.
$\Rightarrow \text{length of arc=}{\displaystyle \frac{\theta }{{360}^{\circ }}}\times 2\pi r$.