Car Talk Annulus Puzzler


RAY: This puzzler is from a fellow named Jerry Olsen. Ed and his two sons, Biff and Skip, have been hired to paint the floor of a merry-go-round. They want to make sure they measure the floor area exactly, because they don’t want to buy any extra paint. The carousel, of course, is a circle. Here’s the catch: In the middle of the carousel is a smaller circle, which contains all the machinery for the carousel.

TOM: An annulus, in other words.

RAY: Exactly. It’s a ring we’re dealing with. Ed tells Biff, “We need to know the area of the carousel, including the area of the big outer circle that we’re going to paint and the area of the inner circle where there’s nothing but the machinery. “Once we have the areas of both circles we can subtract the inner circle from the other circle and we’ll know how much paint we need.”

Biff goes to the carousel and says to himself, “I can’t do this. All the machinery is in the middle. I can’t get to the center to measure the diameter.” He thinks, “I’ll cheat. The old man will never know!” Biff measures a straight line from one edge of the carousel to the other edge, not going through the center.

TOM: In other words he’s going to make what’s called a chord of the big circle.

RAY: Right. Any line that goes from one edge of the circle to the other that isn’t a diameter is a chord. As luck would have it, the tape measure touches the inner circle, or in geometric terms, is tangent to the inner circle at one point.

Biff returns to his dad and says, “I couldn’t do what you wanted me to do. I got this measurement and it’s 70 feet.” The old man administers a swift dope slap. He says, “How the heck are we going to figure this out. We don’t know either diameter.”

The other brother Skip says “I think I can figure it out. ”

Can he or can’t he?

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