Question

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

Answer 1

Let $h$ be the width of the path. Let the radius of inner circle be $r$.

$\therefore$ Radius of outer circle $=(r+h)$

$\Rightarrow$ Area of the path $=$ Area of outer circle $-$ Area of inner circle.

$\Rightarrow$ Area of the path $=\pi (r+h{)}^2-\pi r^2$

$\Rightarrow$ Area of the path $=\pi (r^2+h^2+2rh)-\pi r^2$

$\therefore$ Area of the path $=\pi (h^2+2rh)=\pi h(h+2r)$