Washer

College/University

Definition

A washer, or an annulus (in latin, this means little ring), is a ring shaped object. Its area is bounded by two concentric circles that has different radii. Both of the circles has the same center point. You are able to find the area of the annulus by subtracting the hole in the middle from the total area of the bigger circle.

Worked examples

\(A = \pi R^2 - \pi r^2 = \pi(5^2) - \pi(3^2) = 25\pi - 9\pi = 16\pi\)

Outer radius 5, inner radius 3: subtract the hole's area from the big circle's area.

\(A = \pi(R^2 - r^2) = \pi(7^2 - 4^2) = \pi(49 - 16) = 33\pi\)

Factor out π first, then subtract the squared radii—same result, fewer steps.

Common mistakes

\(A = \pi(R - r)^2\) → \(A = \pi(R^2 - r^2)\) You must square each radius separately before subtracting, not subtract then square.
\(A = \pi R^2 - r^2\) → \(A = \pi R^2 - \pi r^2\) Both circle areas need π; factor it out or include it in both terms.

Where you'll use it next

Washers are the foundation of the washer method in calculus, where you revolve a region around an axis to find volumes of solids of revolution with hollow centers.

Found in 1 StudyPug lesson

Mastering the Disc Method for Volumes of Revolution

Calculus 2

Unlock the power of the disc method to calculate volumes of solids of revolution. Perfect your calculus skills with our comprehensive guide, from basic concepts to advanced applications.

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