Question

Prove that the area of a circular path of uniform width h surrounding a circular region of radius r is $\pi h\left( 2r+h \right)$.

Answer 1

- Hint: We will be using the concept of area of circle. We know that the area of the circle is $\pi {{r}^{2}}$. Also, we will be using algebraic identities like ${{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right)$.

Complete step-by-step solution -

We have to prove that the area of a circular path of width h surrounding a circular region of radius r is $\pi h\left( 2r+h \right)$.
Now, we will draw the diagram according to the question.

Now, we have to find the area of the shaded region.
Now, area of the shaded region can be founded as = Area of outer circle – Area of inner circle.
Radius of outer circle = r + h
$\begin{align}
  & =\pi {{\left( r+h \right)}^{2}}=\pi {{r}^{2}} \\
 & =\pi \left( {{\left( r+h \right)}^{2}}-{{r}^{2}} \right) \\
\end{align}$
Now, we will use ${{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right)$.
$\begin{align}
  & =\pi \left( r+h+r \right)\left( r+h-r \right) \\
 & =\pi \left( 2r+h \right)h \\
 & =\pi h\left( 2r+h \right) \\
\end{align}$
Hence, prove that the area of the circular path is $\pi h\left( 2r+h \right)$.

Note: To solve these type of questions drawing a diagram which clearly represents the situation helps to simplify the problem