1. Part A
Property 1: The perimeter of the arbelos
1. If the diameter of the largest semicircle is 1 and the diameter of one of
the smaller semi‐circles is a, what is the diameter of the other semi‐
circle? Sketch the arbelos and label it.
2. What is the length of the arc of the largest semi‐circle?
3. What is the sum of the length of the two smaller semi‐circles?
4. What is the perimeter of the arbelos?
Property 2: The area of the arbelos
5. Sketch another arbelos. This time, label the radii of the two smaller
semicircles a and b. What is the radius of the largest semi‐circle? Label
it.
6. What is the area of the arbelos? Write your answer in terms of a and b
as simply as possible.
Property 3: The area of circle CD
7. Construct a line segment CD as in figure 2. CD should be perpendicular
to the diameters of the semi‐circles.

2. Figure 2
2a 2b
8. Using the Pythagorean theorem, calculate the length of CD in terms of
a and b. (Hint 1: There are 3 right angled triangles in figure
2. You will need to use all 3 triangles. Still stuck? There is
another hint at the bottom of this worksheet. Try to work
it out on your own first though)
9. Show (prove) that the circle with diameter CD has the same area as
the arbelos.
In a paragraph or more of writing, summarize your results from Part A
Part B
Investigate other shapes
In Part A you discovered 3 properties of the arbelos. Now you will investigate the
relationship of the area and perimeter for other shapes that are constructed in
the same way ie. 3 half‐squares
Or 3 similar half‐rectangles

3. Make sure the three shapes you use are all mathematically similar – ie. The ratio
of all corresponding sides needs to be the same.
Write or type up your findings on a sheet of paper illustrating with drawings
where necessary.
Question 8 hints:
1. Using the Pythagorean theorem, write an equation linking
the sides of triangle ADB
2. Do the same for triangles ADC and DBC
3. Try to replace the e 2 and f 2 terms in your first equation
with expressions from the other two equations.
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