Question

The difference between circumference and diameter of a circle is 15 m. What is the area of the circle?
A. \[25\,{{m}^{2}}\]
B. \[65\,{{m}^{2}}\]
C. \[38.5\,{{m}^{2}}\]
D.None of these

Answer 1

Hint: The diameter of a circle is double the radius of a circle. As a result, diameter is written as 2r. Similarly, we know that the circumference of a circle is equal to \[2\pi r\] . Given that the difference between the perimeter and the diameter of the object is 15 m, we get \[2\pi r-2r=15\,m\] . Solve the above equation to get the value of r. then substitute in the formula of area of circle and get the answer.

Complete step-by-step answer:

Here in this figure, to find the area first of all we need to find the radius for that we need to use the \[d=2r\] and circumference is the perimeter of the circle that means a measure of the outer round length of the circle.
But it is given in the question that the difference between circumference and diameter of a circle is 15 cm then, we have to find the radius of the circle is r. then we know that the diameter of the circle is double the radius. So, the value diameter can be given as \[2\times radius\] .
We get the diameter of a circle \[=2r\]
Now, we know that the circumference of a circle is given by the formula \[2\pi r\] . Given the difference between circumference and diameter of a circle is 15 cm, which clearly means that the difference \[2\pi r\] and \[2r\] is 15 m. then we get:
 \[2\pi r-2r=15m----(1)\]
Take 2r common on LHS, we get:
 \[2r(\pi -1)=15\] , substituting the value of \[\pi \] as \[\dfrac{22}{7}\] , we get:
 \[\Rightarrow 2r\left( \dfrac{22}{7}-1 \right)=15\]
By simplifying further, we get:
 \[\Rightarrow 2r\left( \dfrac{22-7}{7} \right)=15\]
Multiply 7 on both side and further solving we get:
 \[\Rightarrow 2r\left( 15 \right)=15\times 7\]
By simplification we get:
 \[\Rightarrow 30r=105\]
Therefore, we get the value of r from above equation we get:
 \[\Rightarrow r=\dfrac{105}{30}\]
 \[\Rightarrow \therefore r=3.5\,m\]
As we know that Area of a circle is \[\pi \,{{r}^{2}}\] we have to substitute the value of r in this formula we get:
 \[A=\dfrac{22}{7}\,\times {{(3.5)}^{2}}\]
After simplifying we get
 \[A=\dfrac{22}{7}\,\times (12.5)\]
Further solving this we get:
 \[A=38.5\,{{m}^{2}}\]
So, the correct option is “option C”.
So, the correct answer is “Option C”.

Note: To answer the circle question, one needs to understand the radius, diameter, circumference, and area of a circle, as well as how to compute the area of a circle. When estimating the area of a circle, there are a few things to keep in mind. First, see which value of \[\pi \] , which is 3.14 and \[\dfrac{22}{7}\] will give you the simplest solution and help in the calculation.