Answer to Riddle #76: Area of an Annulus

76. The diagram shows two concentric annuli (two dimensional ring shapes,) given that the two red lines are the same length, prove that the shaded areas are the same too.

A bit of geometry for a change, you really only need to know Pythagoras theorem and that the area of a circle is given by πr2. I've studied a lot of maths over the years and I never covered this until I read a puzzle about carpeting a lighthouse:

two concentric annuli with equal chord lengthBefore reading the answer can I interest you in a clue?

Let's just crack in to the geometry. Essentially and obviously the area of an annulus (ring) is calculated by working out the area of the outer circle and subtracting from it the area of the inner circle.
diagram for calculating the area of an annulus
So the area of our annulus is the area of the big circle minus the area of the little

Area = πR2 - πr2
Area = π(R2-r2)

We're going to need to need to show that we can eliminate r and R and thus prove the area is only dependent on L. Lets take a look with Pythagoras and see what happens. rRL is a right angle triangle where R is the hypotenuse. Pythagoras gives:

R2 = r2 + L2
L2 = R2 - r2

We can substitute R2 - r2 back in to the equation for area we had above:

Area = πL2

Which is sufficient to prove that if the red line is the same length then in the question then the area is the same.

Let's phrase it like a puzzle

A mad professor lives at the top of a light house. He wants the ground floor covered with carpet tiles. He calls a flooring company and explains that the room is ring shaped. The company explains that their carpet tiles are 1m2 and they will need to send someone out to measure to see how many tiles they need to buy.

The worker visits the lighthouse to measure the sizes but finds the door locked and the following note pinned to the door.
Hand written diagram of the annulus ground floor of a light house
Satisfied the worker goes back to the flooring shop and gets a clip around the ear. His boss said he was an idiot and they needed to know the exact measurements to work out the area. 'I know the area' said the worker.

What is it?
the area is πL2 = π•502 = 7854m2

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