Question

The radius of a circular ground is 56 m. Inside it, runs a road of width 7 m all along its boundary. Find the area of this road.

Answer 1

Hint: We will first find the radius of the inner circle formed using the given condition. Then, calculate the area of both circles. To find the area of the road, subtract the area of the inner circle from the area of the outer circle.

Complete step by step solution: We are given that the radius of the inner circle is 56m and there is the path of 7m width inside the circle.
The path divides the circle into two parts, the inner circle and the outer circle.
The radius of the outer circle is 56m
And we can calculate the radius of the inner circle by subtracting the width of the path from the radius of the outer circle.
That is, $56m - 7m = 49m$ is the radius of the inner circle.

We have to find the area of the road.
We can calculate the area of the road by subtracting the area of the inner circle from the area of the outer circle.
Also, the area of the circle is given by $\pi {r^2}$, where $r$ is the radius of the circle.
If radius of outer circle is 56m, then the area of the outer circle is $\pi {\left( {56} \right)^2}{{\text{m}}^{\text{2}}}$.
If the radius of inner circle is 49m, then the area of the inner circle is $\pi {\left( {49} \right)^2}{{\text{m}}^{\text{2}}}$
Then, the area of the road is,
$\left( {\pi {{\left( {56} \right)}^2} - \pi {{\left( {49} \right)}^2}} \right){{\text{m}}^{\text{2}}}$
Simplify the expression using the identity ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$ and solve for the area of road,
$
  A = \pi \left( {{{\left( {56} \right)}^2} - {{\left( {49} \right)}^2}} \right) \\
   \Rightarrow A = \pi \left( {56 + 49} \right)\left( {56 - 49} \right) \\
   \Rightarrow A = \pi \left( {105} \right)\left( 7 \right) \\
$
On substituting the value, $\pi = \dfrac{{22}}{7}$, we get,
$
  A = \left( {\dfrac{{22}}{7}} \right)\left( {105} \right)\left( 7 \right) \\
   \Rightarrow A = 2,310{m^2} \\
$

Hence, the area of the road is $2,310{m^2}$

Note: Area of the required region is of the shape of a circular ring. The area of the circular ring is the difference of the area of the outer circle and the area of the inner circle. Also, use $\pi = \dfrac{{22}}{7}$ to avoid tricky calculations. The area is always measured in square units.