Question

If the perimeter of a sector of a circle is 27.2 m. And the radius of the circle is 5.7 m. Then, find the area of the sector.

Answer 1

Hint:- Let us find the length of the arc made by the sector using radius and the given value of the perimeter of the circle. We use the formula of perimeter for getting the length of arc. After getting the length of arc use the formula of area of sector.

Complete step-by-step solution -
As, we can see that the radius(r) of the given circle is 5.7 m.
And, the perimeter(p) of the sector of the circle is 27.2 m.
As we all know that perimeter of the complete circle is 2\[{\text{\pi }}\]r, where ( \[{\text{\pi }}\] = 3.14 ).
A perimeter of any shape is the sum of the length of all its sides (boundaries).
And as we can see from the above figure that the length of two sides (OA and OB) of the sector of the circle is equal to the radius of the circle and the length of the third side (AB) will be the length of the arc made by the sector.
So, the perimeter of the sector of any circle with radius r will be 2r + length of the arc made by sector.
Let the length of the arc (AB) made by the given sector will be x metres.
So, perimeter of the sector (OAB) = (2r + x) metres
So, 27.2 = 2*(5.7) + x
27.2 = 11.4 + x
x = 15.8 metres
So, the length of arc (AB) made by the sector will be 15.8 m.
As we know that the area of any sector of circle having radius r and length of arc l is given as \[\dfrac{{\text{1}}}{{\text{2}}}{\text{*l*r}}\].
So, the area of sector OAB will be \[\dfrac{{\text{1}}}{{\text{2}}}{\text{*x*r}}\].
So, area of sector OAB = \[\dfrac{{\text{1}}}{{\text{2}}}{\text{*(15}}{\text{.8)*(5}}{\text{.7)}}\]\[{{\text{m}}^{\text{2}}}\] = (7.9)*(5.7) = 45.03 \[{{\text{m}}^{\text{2}}}\]
Hence, the required area of the sector will be 45.03 \[{{\text{m}}^{\text{2}}}\].

Note:- Whenever we come up with this type of problem then first, we will use a formula for the perimeter of the sector and find the length of the arc. After that we can easily apply formulas to find the area of the sector of the circle if radius and length of arc is known.