Try it on paper first. A hint, the key idea and the worked solution are here when you need them.
Three circles, each of radius , touch each other. Find the area of the curved triangle between them. Take and . Also write the exact form.
A window is a rectangle plus a semicircle. The inner edge of a track is and each lane further out is longer by . Three touching circles of radius enclose .
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Plan. Curved triangle the triangle of centres minus the three sectors inside it.
Step 1. The triangle of centres. Its side is . Its area is , which is .
Step 2. The sectors. Each angle of the triangle is , so each sector is of a circle. Three of them make half a circle: .
Step 3. The difference. .
Answer: the exact area is , which is with the given values.
Watch out. The answer is small, because it is a small gap between large shapes. Small errors in show up strongly here.
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