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

To do:

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

Solution:

Radius of the inner circle $= r$

Width of the path $= h$

This implies,

Outer radius $R = r + h$

Therefore,

Area of the path $= \pi R^2 - \pi r^2$

$= \pi [(r + h)^2 - r^2]$

$= \pi (r^2 + h^2 + 2rh – r^2)$

$= \pi (2rh + h^2)$

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

Hence proved.

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Updated on: 10-Oct-2022

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