ARBELOS OF ARCHIMEDES
EXPOSITORY PAPER
Micki McConnell
In partial fulfillment of the requirements for the Master of Arts in Teaching with a
Specialization in the Teaching of Middle Level Mathematics in the Department of
Mathematics
Jim Lewis, Advisor
July 2008
Arbelos of Archimedes
1
The term “arbelos” is a Greek word meaning “shoemaker’s knife”. Archimedes is
credited with naming the geometric figure that he was first to study as the arbelos. It is stated
that he played with this figure for fun and identified a number of interesting properties for the
figure. In this paper, some information about Archimedes is given and a few of the arbelos’
properties and their proofs are examined.
Archimedes was a Greek mathematician, physicist, engineer, inventor and astronomer.
His greatest contribution to mathematics was in geometry with his work on the area of plane
figures and volume and area of curved surfaces. He is considered to be one of the greatest
mathematicians of antiquity and of all time.
Archimedes was born around 287 BC in the seaport city of Syracuse, Sicily. At the time
of his birth, Syracuse was a colony of Magna Graecia. His father was an astronomer named
Phidias, and it was believed that Archimedes was related to King Hiero II, the ruler of Syracuse.
There is no record to show if he was ever married or had children. Archimedes died around 211
or 212 BC when the city of Syracuse was captured during the Second Punic War. It is said that
Archimedes was working on a mathematical diagram when a Roman soldier commanded him to
come and meet General Marcellus. Archimedes refused to leave until he was done with the
problem; this enraged the soldier, who then killed Archimedes with his sword. Archimedes’ last
words were, “Do not disturb my circles.” He was referring to the circles in the mathematical
drawing that he was working on at the time.
It is believed that Archimedes studied in Alexandria, Egypt under Euclid. Some of his
discoveries and inventions changed how people viewed and/or did things. A commonly cited
anecdote regarding Archimedes explains how he invented a method for measuring the volume of
an object with an irregular shape when the king asked him to find out if his new crown was solid
Arbelos of Archimedes
2
gold. This discovery is known as Archimedes’ Principle and states that a body immersed in a
fluid experiences a buoyant force equal to the weight of the displaced fluid. He also proved that
the area of a circle was equal to  multiplied by the square of the radius of the circle. Some of
Archimedes’ writings include:

The Measurement of a Circle, where he gave the value of 3 .

The Quadrature of the Parabola, where he proved that the area enclosed by a parabola
and a straight line is 4/3 multiplied by the area of a triangle with equal base and height.

The Sand Reckoner, where he set out to calculate the number of grains of sand that the
universe could contain.

The Sphere and the Cylinder, where he obtained the result of which he was most proud –
namely, the relationship between a sphere and a circumscribed cylinder of the same
height and diameter.

Book of Lemmas or Liber Assumptorum, which was a treatise with fifteen propositions on
the nature of circles. It is believed that this is actually based on an earlier work by
Archimedes that is now lost.
One idea that caught the attention of Archimedes was the arbelos. It is said that Archimedes was
the first to study this figure. Even though the surviving works of Archimedes do not mention the
arbelos, it is claimed that he studied and named the arbelos in his Book of Lemmas. An arbelos is
a plane region bounded by a semicircle of diameter 1; connected to semicircles of diameters r
and (1-r), all oriented the same way and sharing a common baseline. (The baseline of a
semicircle is a straight line forming the diameter which connects the ends of an arc.) The figure
has a number of interesting properties; many first identified by Archimedes. Archimedes’ idea
entailed two smaller circles outside of each other but inside a third larger circle. Each circle was
Arbelos of Archimedes
3
tangent to the others and their centers were along the same straight line. He wanted to find the
area that was inside the larger circle, but outside the two smaller circles. Archimedes named the
arbelos using one-half of the circle, either the top half or the bottom half. It is important to note
that the arbelos is not represented by the circle being cut in half vertically, only horizontally (see
Figure 1).
Figure 1: the arbelos
The geometric properties of the arbelos have a wide range, one being that the length of
the lower boundary of the arbelos equals the length of the upper boundary. The proof for this
comes from the knowledge that the circumference of a circle is proportional to its diameter. A
more formal statement and proof occurs in the work of Behnaz Rouhani (2002):
Arc length along the enclosing semicircles is the same as the arc length along the two
smaller semicircles, i.e. Arc length AKB = Arc length AEC + Arc length CFB.
To prove this, let us look at the figure below.
Arbelos of Archimedes
4
As noted in the previous figure, point M is the center of the semicircle AKB. Point N is the
center of the semicircle AEC and point P is the center of semicircle CFB. From this figure, since
AM = a, and AN = b then we can say NM = a – b. Next, using semicircle CFB, we let CP = d so
we can conclude that MC = a – 2d. We know that the circumference of a semicircle is given by
the formula C =   r . Therefore, we can now make some equations dealing with arc length that
we will number for use in the sequel.
Arc length AKB =   a
(1)
Arc length AEC =   b
(2)
Arc length CFB =   d
(3)
Also from the previous figure we know that:
NM + MC = AN
(a-b) + (a-2d) = b
2a – 2d = 2b
OR
2a = 2d + 2b which simplifies to a = d + b
Now, using the information from (2) and (3):
AEC + CFB =   b    d   (b  d )    a  AKB
A more sophisticated property deals with the area of the arbelos. This property states that the
area of the arbelos equals the area of the circle whose diameter, CD, is the portion inside the
arbelos of the common tangent line to the two smaller semicircles at their point of tangency, C
(see Figure 2). In his Book of Lemmas, Archimedes devoted three of his fifteen Propositions to
the study of the arbelos. This property is Proposition 4. The other two are Propositions 5 and 6.
Arbelos of Archimedes
5
.
A
C
B
Figure 2: An area property of the arbelos
In one proof of this proposition we reflect across the line through points A and B and
observe that twice the area of the arbelos is what remains when the areas of the two smaller
circles with diameters AC and CB are subtracted from the area of the larger circle with diameter
AB. Since the area of a circle is proportional to the square of the diameter, the problem reduces
to show that 2(CD)2 = (AB) 2 – (AC) 2 – (CB) 2. Since the length of segment AB equals the sum
of the lengths of segment AC and segment CB, the equation can be simplified algebraically to
(CD) 2 = (AC)(CB).
The claim that the length of the segment CD is the geometric mean of the lengths of the
segments AC and CB can be made. Now, if we imagine the triangle ADB being inscribed in a
semicircle, it would have a right angle at point D and that consequently segment CD is “a mean
proportional” between segments AC and CB. This proof may be found given as a “proof without
words” by Roger B. Nelson (see Appendix A).
Here is another statement and proof that can be found in the work of Behnaz Rouhani
(2002):
The area of the circle with diameter DC is the same as the area of the arbelos.
For this proof, we use the diagram below. Draw the perpendicular segment DC from the
tangent of the two semicircles to the edge of the largest semicircle. We claim that the area of the
circle with diameter DC is the same as the area of the arbelos.
Arbelos of Archimedes
6
C
1-r
B
For this diagram, let AB = 1, and AC = r. Using the Pythagorean Theorem, we can write the
following equations that are numbered for future use.
In triangle ADC:
r2 + h2 = x2
(4)
In triangle BDC:
(1-r) 2 + h2 = y2
(5)
In triangle ADB:
x2 + y2 = 1
(6)
When solving equations (4) and (5) simultaneously, and after much simplification, we arrive at
the following equation.
(1-r) 2 + x2 – r2 = y2
OR
1 – 2r + x2 = y2
(7)
Now substituting equation (7) into equation (6), we get:
x2 + 1 – 2r + x2 = 1
2x2 = 2r
x= r
Next, we substitute the above expression for x into both equations (7) and (4), and find
corresponding expressions for y and h.
y = 1 r
and
h=
r  r2
Arbelos of Archimedes
7
We know that the radius of the circle with center at point O is
1
r  r 2 . Therefore, the area of
2
the circle with center at point O is given by equation (8).
1
( r  r 2 )2
4
 r  r2
A

4
4
A
(8)
From the above figure, we can tell that the area of the arbelos is the area of the semicircle with
diameter AB minus the sum of the areas of the semicircles with diameters AC and BC. Thus, we
can write, the following equation.
 1  r  2  1  (1  r )  2 
A          
 
8  2  2   2  2  

A

 (1  r ) 2

8
8
2
    r  (  2  r    r 2 )
A
8
2
2  r  2  r
A
8
 r  r2
 r  r2
A
OR
A

4
4
4
8

 r2
Therefore, the area of the arbelos is the same as the area of the circle with diameter DC as shown
in equation (8).
With modern mathematical knowledge and notation, we are able to give proofs that
would not have been available in Archimedes’ time. For our first identity, we know that all
circles are similar and that the circumference of a circle is 2  times the radius, so the
circumference of a semicircle would be  times the radius. Also, we are able to establish the
relationship using a circle of radius one.
Arbelos of Archimedes
8
1
x
[
1+x
1-x
]
The circumference for the larger semicircle would be  times 1 or  . The identity states that
the two smaller semicircle lengths will equal the arclength of the larger semicircle.
Thus, we need:  = 
( 1 2 x ) +  ( 1 2 x ).
Factor out  , we want 1 =
1 x 1 x
+
2
2
The x’s cancel out and so our desired identity is equivalent to, 1 = ½ + ½ .
The second identity is that the area of the circle with diameter DC (see page 6) is the
same as the area of the arbelos. We will use the following figure for our proof.
.
Arbelos of Archimedes
9
(x,y)
.C
A1
A2
(x,0)
Again, we will assume that the larger circle has radius 1. Thus, we know that x2 + y2 = 1 hence
y = 1  x 2 or y2 = (1-x2) . Also, since the area of a circle of radius r is  r2, the area of a
semicircle is (  r2 )/2. In our picture, A1 and A2 are semicircles and C is a circle with center at (x,
y/2) and radius y/2. In order to show that the area of C equals the area of the Arbelos, we need to
establish that 1/2  = A(A1 ) + A(A2 ) + A(C) or
1 x 2
1/2  =  1/2 (
)
2
(
)
1 x 2
+  1/2 (
)
2
Factoring out  , we need to establish that ½ =
(
)+  (
1  ( x) 2
4
)
(1  x) 2
(1  x) 2 1  ( x) 2
+
+
.
8
8
4
1  2x  x 2 1  2x  x 2
1 x2
The right hand side equals
+
+
.
4
8
8
If we get a common denominator, 2x will cancel out and we will get:
1  x 2  1  x 2  2  2x 2 4 1
  as desired.
8
8 2
Even though Archimedes was believed to be the first to study the arbelos, others after
him investigated its properties as well. Pappus of Alexandria discusses the arbelos in Book IV of
Arbelos of Archimedes
10
the Collection. Pappus does not cite Archimedes, but he also does not hold any claim to the most
famous theorem about the arbelos. The theorem states that a chain of inscribed circles, Cn, in an
arbelos has the following property. The distance from the diameter of the largest semicircle to
the center of the nth circle in the chain, Cn, is exactly equal to dn, where dn is the diameter of Cn.
Another person associated with the arbelos is Otto Mohr, a German civil engineer and professor
of mechanics. Since the arbelos is a classical example of what is known as pure mathematics, it
is well known in textbooks on solid mechanics under the name “Mohr’s circles.”
The arbelos was and is studied by mathematicians throughout the world. The arbelos
satisfies a number of unexpected identities of which the author only chose to evaluate two.
Properties of the arbelos can be examined and proven in many different ways, and it ought to be
mentioned that Archimedes enjoyed playing with the arbelos for fun. Certainly, for one to
completely understand all the known properties of the arbelos would take a vast amount of time.
However, even the introduction given here seems both valuable and extremely interesting.
Arbelos of Archimedes
11
REFERENCES
Archimedes. (n.d.) Retrieved June 19, 2008, from http://library.thinkquest.org
Archimedes. (n.d.) Retrieved June 19, 2008, from http://www.cs.drexel.edu
Arbelos-The Shoemaker’s Knife. (n.d.) Retrieved June 19, 2008, from http://www.cut-theknot.org
Proof Without Words: The Area of an Arbelos, Nelsen, R Mathematics Magazine, 2002, vol. 75,
no. 2, pp. 144
“Reflections on the Arbelos” HP Boas, American Mathematical Monthly, 2006, vol. 113, no. 3,
pp. 236-249
“The Arbelos” B Rouhani, (2002) Retrieved June 19, 2008, from http://jwilson.coe.uga.edu
Weisstein, Eric W. “Arbelos.” From MathWorld-A Wolfram Web Resource.
http://mathworld.wolfram.com/Arbelos.html
Arbelos of Archimedes
12
Appendix A
Proof Without Words: The Area of an Arbelos
-ROGER B. NELSEN LEWIS & CLARK COLLEGE
THEOREM. Let P, Q, and R be three pints on a line, with Q lying between P and R.
Semicircles are drawn on the same side of the line with diameters PQ, QR, and PR. An arbelos
is the figure bounded by these three semicircles. Draw the perpendicular to PR at Q, meeting the
largest semicircle at 5. Then the area A of the arbelos equals the area C of the circle with
diameter QS [Archimedes, Liber Assumptorum, Proposition 4].
A=C
Q
Proof.
A  A1  A2  B1  B2
Bi  Ai  Ci
A  A1  A2  A!Ci  A2  C2
C2