Prove that the area of a circular path
Question:

Prove that the area of a circular path of uniform width h surrounding a circular region of radius r is πh (2r + h).

Solution:

We know that the area of a circle of radius $r$ is $A=\pi r^{2}$

It is given that a circular path of width h surrounds the circle of radius r.

So, radius of the outer circle $=r+h$

Using the value of radius in above formula,

Area of the outer circle $=\pi(r+h)^{2}$

Hence,

Area of the circular path = Area of outer circle-Area of inner circle

$=\pi(r+h)^{2}-\pi r^{2}$

$=\pi\left(r^{2}+2 r h+h^{2}\right)-\pi r^{2}$

$=2 \pi r h+\pi h^{2}$

$=\pi h(2 r+h)$

Area of the circular path $=\pi h(2 r+h)$