Question

Write the formula to find the area of a sector and length of an arc for the following:
(a) Semicircle
(b) Quarter circle

Answer 1

Hint: We are given a question asking us to write the formula for the area of a sector and length of an arc for a semicircle and a quarter circle. A sector refers to a part of a circle with two radii and an arc. In order to find the area of a sector, we have to take in account the angle formed by the two radii and divide it by the whole angle of a circle and multiply this ratio to the area of a circle. Similarly, an arc is a part of the circumference of a circle. So, in order to find the length of the arc, we divide the angle subtended by the arc at the center by the complete angle of a circle and multiply the ratio by the circumference of the circle. Hence, we will have the required formulae.

Complete step-by-step solution:
According to the given question, we are asked to write the formulae of the area of the sector and the length of the arc for the given parts of a circle.
A sector refers to a part of a circle with two radii and an arc. In order to find the area of a sector, we have to take in account the angle formed by the two radii and divide it by the whole angle of a circle and multiply this ratio to the area of a circle
an arc is a part of the circumference of a circle. So, in order to find the length of the arc, we divide the angle subtended by the arc at the center by the complete angle of a circle and multiply the ratio by the circumference of the circle.
First, we have,
(a) Semicircle
A semicircle is the half of a circle. Based on the above explanation, we have the area of the sector.
The angle formed between the radii is \[{{180}^{\circ }}\],so we have,
Area of the sector = \[\dfrac{{{180}^{\circ }}}{{{360}^{\circ }}}\times \pi {{r}^{2}}\]
\[\Rightarrow \] Area of the sector = \[\dfrac{1}{2}\times \pi {{r}^{2}}\]
where ‘r’ is the radius of the circle and \[\pi {{r}^{2}}\] is the area of a circle
Also, based on above explanation, we have the length of the arc as,
Length of the arc = \[\dfrac{{{180}^{\circ }}}{{{360}^{\circ }}}\times 2\pi r\]
\[\Rightarrow \] Length of the arc = \[\dfrac{1}{2}\times 2\pi r\]
where \[2\pi r\] is the circumference of a circle with radius ‘r’
(b) Quarter circle
A quarter circle is one when you divide the circle in four equal parts. Based on the above explanation, we have the area of the sector.
The angle formed between the radii is \[{{90}^{\circ }}\], so we have,
Area of the sector = \[\dfrac{{{90}^{\circ }}}{{{360}^{\circ }}}\times \pi {{r}^{2}}\]
\[\Rightarrow \] Area of the sector = \[\dfrac{1}{4}\times \pi {{r}^{2}}\]
where ‘r’ is the radius of the circle and \[\pi {{r}^{2}}\] is the area of a circle
Also, based on above explanation, we have the length of the arc as,
Length of the arc = \[\dfrac{{{90}^{\circ }}}{{{360}^{\circ }}}\times 2\pi r\]
\[\Rightarrow \] Length of the arc = \[\dfrac{1}{4}\times 2\pi r\]
where \[2\pi r\] is the circumference of a circle with radius ‘r’

Note: The area of the circle and the circumference of a circle must be known clearly and correctly as well. When talking about area, we consider the entire region specified within a boundary and so we use \[\pi {{r}^{2}}\]. In case of length of arc, we cannot measure the length using a ruler since an arc is a curved line, so we will use here, the circumference of the circle, which is, \[2\pi r\].