Have you ever noticed that every manhole cover you see is round? Have you ever wondered how engineers came to this decision? Did you realize that the reasoning comes down to the geometry of shapes?
That’s right: manhole covers are round because circles are the only shapes that cannot fall through themselves.
Let’s examine some of the properties of shapes to see why this is true.
First, it’s important to recognize that the shapes will only fall through themselves when they are rotated to be vertical. If they were lying flat then they would be covering the hole successfully!
Next, let’s examine shapes with unequal sides beginning with a triangle. If the triangle had side lengths of 3, 3, and 2 feet then we could rotate it so that the side with length 2 feet was parallel to the ground. This would mean that the width of the manhole cover would now be 2 feet, with the width of the opening approaching 3 along the sides of the hole. Since 3 is greater than 2, the cover would be able to fall through the opening. Similarly for a 4-sided figure if one side is shorter or longer than the others then we will find the same result.
There is a pattern to notice from examining the previous shapes. The shortest width of each shape is compared to the longest length. But what if the covers were regular shapes with all sides and angles being the same?
Let’s look at a square with length of 1 foot on each . The shortest width of a square will be the length of a side. But how about the longest length? That can be found by measuring from the upper left-hand corner to the lower-right hand corner.
Using Pythagorean’s Theorem, we know that 1^2 + 1^2 = diagonal^2. So the diagonal is square root of 2, or approximately 1.414. This means that this square could still fall through itself.
So now our focus can shift to finding a shape whose longest length and shortest width are the same. But this would mean that we would need all widths and lengths to be the same because we cannot have the longest length be shorter than the shortest width. This leads us to a circle.
A circle will always have the width of its diameter no matter which way it is rotated, so this will be the shortest width. But the longest length will also be the diameter as well because any chord will be shorter than the diameter. So the circle is unique to all polygons and shapes in that it can never fall through itself.
Want to test geometry with physical objects? Since manhole covers are heavy you should explore this idea using Tupperware containers. See how many different lid shapes will fall into the container, and how many will not – then post your comments!
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This post was originally posted in the Everyday Explanations – Answers to Questions Posed on a Middle School Bus Ride by Sean Mittleman. We have his permission to re-post in the MSP2 blog.