Question

The diameter of a circle is 10 cm. Find the length of the arc, if the corresponding central angle is 180 degrees.

Answer 1

Hint: From the given diameter find out the radius of the circle. Then find out the arc length by using the formula, arc length $=\dfrac{\theta }{{{360}^{\circ }}}\times 2\pi r$, where r is the radius, and $\theta $ is the central angle.

“Complete step-by-step answer:”
In the question it is given that the diameter of a circle is 10 cm.
The distance from one point of a circle through the center to another point on the circle is known as the diameter of a circle.
The distance from the center to the circumference of a circle is known as the radius of a circle.
We know that the diameter is twice of the radius.
So the radius of the given circle is,
$r=\dfrac{10}{2}=5$ cm.
The arc of a circle is a portion of the circumference of a circle.
Here we have to find out the length of the arc which makes an angle of 180 degrees with the center.
That means if we add the two end points of the arc with the center then the corresponding angle will be 180 degrees.
We know that the arc length, radius and the angle has a relation between them. That is,
Arc length $=\dfrac{\theta }{{{360}^{\circ }}}\times 2\pi r$
Here r is the radius of the circle which is 5 cm here.
$\theta $ is the corresponding angle, which is 180 degrees here.
Therefore the arc length is:
$=\dfrac{{{180}^{\circ }}}{{{360}^{\circ }}}\times 2\pi \times 5=\dfrac{1}{2}\times 10\pi =5\pi $
Hence the arc length is $5\pi $ cm or we can put the value of $\pi =\dfrac{22}{7}$
The arc length is $=5\times \dfrac{22}{7}=\dfrac{110}{7}$ cm.

Note: Alternatively we can say that, since the corresponding central angle of the arc is 180 degrees, which is half of 360 degrees, the arc is basically the semi circle without the diameter.
Therefore the arc length will be $\pi r$ cm.