How to Find the Perimeter of an Arbelos

Instructor: Laura Pennington

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

This lesson will explain what an arbelos is and look at examples of this shape. We will also look at a neat property of an arbelos and use it to see an easy way to find the perimeter of an arbelos.

Arbelos

Suppose you are walking through an art show, and there is a sculpture that touches the ground at three points, such that those three points form diameters of half-circles within the sculpture as shown in the image.


Image of sculpture
arbelosper1


Did you know that the shape formed by these circles is actually a special shape itself? If not, read on and then you will!

The shape of this sculpture is called an arbelos. An arbelos is a two-dimensional shape that is formed by three half-circles. The half-circles intersect so that the two smaller half-circles are within the larger half-circle. The diameters of the two smaller circles make up the diameter of the larger circle. Phew! That's a mouthful! Basically, an arbelos is the shaded area in the image shown (The point at which the smaller and medium circle intersect the diameter of the larger circle is arbitrary.).


Arbelos
arbelosper2


Now suppose you decide that you like the look of the sculpture and that the sculpture would go perfectly with your office décor. You need to find the perimeter of, or the distance around, the sculpture in order to know if it will fit properly in your office. Let's take a look at how to do this!

Perimeter of an Arbelos

You don't have any measuring tools, but the display tag says that the diameter of the smallest circle is 2 inches, and the diameter of the medium circle is 4 inches. Okay, well, we know that the perimeter of a circle, or its circumference, can be found using the following formula:

  • Perimeter of a circle = 2πr or πd, where r is the radius and d is the diameter of the circle

Looking at the sculpture again, we see that the perimeter of the arbelos is made up of halves of the perimeters of the circles that form the arbelos. Therefore, to find it's perimeter, we would just need to find half of the perimeters of each of the three circles and add them together.


Perimeter of Sculpture
arbelosper3


Great! We can do this. We first calculate half of the perimeters of the small circle and the medium circle.


arbelosper4


We get that half of the perimeter of the small circle is π, and that half of the perimeter of the medium circle is 2π.

We just need to find half of the perimeter of the largest circle and then add them all up.

We know that the diameters of the small and medium circle make up the diameter of the largest circle, so the diameter of the largest circle is 6 inches. Therefore, the perimeter of the largest circle is 6π, and half of this is 3π.

Now, we just add them all up:

  • π + 2π + 3π = 6π

We get that the perimeter of the sculpture is 6π inches, or approximately 18.85 inches. Awesome! It will fit perfectly in your office!

An Easier Way

Now let me ask you a question. Did you notice that the perimeter of the arbelos sculpture was the same as the perimeter of the largest circle of the sculpture? They were both 6π! That's not a coincidence!

This is a property of an arbelos that makes it so interesting! You see, the perimeter of an arbelos is equal to the perimeter, or circumference, of the largest circle of the arbelos, so the perimeter of an arbelos is as follows:

  • Perimeter of an arbelos = dπ, where d is the diameter of the largest circle.

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