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Application of an Arbelos in Real Life

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

The arbelos shape is an extremely interesting one in both mathematics and the real world. In this lesson, we will look at the definition of this shape, some of its properties, and applications of an arbelos in real life.

Arbelos

We are all fairly familiar with common shapes such as squares, rectangles, triangles, and circles, but some shapes are less well-known. One such shape is the arbelos.

An arbelos is a shape that is made up of three half-circles. The largest half-circle contains the smaller two half-circles inside it in such a way that the diameters of the smaller two half-circles sum up to the diameter of the largest half-circle. Well, that was a mouthful! Take a look at the image of an arbelos to help clarify your understanding.


arbreal1


Pretty interesting, huh? Let's make it even more interesting by looking at the area and perimeter of this shape!

Area and Perimeter of an Arbelos

Because of how it is set up and defined, an arbelos satisfies some pretty neat properties when it comes to its perimeter and area. For starters, because the sum of the diameters of the smaller half-circles is equal to the diameter of the larger half-circle, it turns out that the perimeter of the entire arbelos is the same as the perimeter of the largest circle:

  • Perimeter = 2πR = Dπ, where R and D are the radius and diameter of the largest half-circle, respectively.

In the same way, the area of an arbelos also is equal to the area of another part of the arbelos. That is, the area of an arbelos is equal to the area of the circle that has diameter PQ, where P is the point at which the two smaller half-circles touch, and Q is the point directly above P on the larger half-circle.

  • Area = π(|PQ| / 2)2 = (π / 4)|PQ|2


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It is because of properties like this that the arbelos has fascinated mathematicians for some time!

You may be thinking that since this shape isn't very common, it probably doesn't really have any applications in the real world, but it does! Let's explore!

Applications in the Real World

Art

Two very common areas that the arbelos shape shows up in is art and in shoemaking. There is a giant arbelos sculpture on the roadside in the Netherlands.

Arbelos sculpture in the Netherlands
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When this sculpture was built, they would have had to know how much metal material would be needed to build it. In other words, the perimeter of the arbelos. The diameters of the half-circles of this sculpture are 24.2 meters, 19.3 meters, and 4.9 meters. All the builders would have needed to know was the diameter of the largest half-circle (24.2 meters) to find the perimeter of the arbelos.

  • Perimeter = 24.2π ≈ 76.0265

By simply knowing this, the builders could have determined that they would need approximately 76 meters of metal to make the sculpture.

Shoemaking

You probably wouldn't think of an arbelos being used in shoemaking, but this is actually how the shape got its name. In Greek, the word arbelos means 'shoemaker's knife', and this is because a knife that is used in shoemaking is the shape of an arbelos.

arbreal4

When a knife is in the shape of an arbelos, its blade is curved, and it has sharp points on each end. These characteristics make it is great for cutting and trimming leather at very precise angles.

Suppose a factory that produces these knives is programming a machine to produce a batch of knives that have the following measurements:

  • Diameter of the largest half-circle = 8 inches
  • Diameters of the smaller half-circles = 4 inches each

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