From Wikipedia, the free encyclopedia
Measurement of a Circle (Greek: Κύκλου μέτρησις, Kuklou
metrēsis) is a treatise that consists of three propositions
by Archimedes. The
treatise is only a fraction of what was a longer work.
The circle and the triangle are equal in area.
Proposition one states:
The area of any circle is equal to a right-angled triangle in
which one of the sides about the right angle is equal to the
radius, and the other to the circumference, of the circle.
Any circle with a circumference
c and a radius
r is equal in area with a
with the two legs being
c and r. This proposition is proved by the method of
Proposition two states:
The area of a circle is to the square on its diameter as 11 to
This proposition could not have been placed by Archimedes, for
it relies on the outcome of the third proposition.
Examples of how Archimedes calculated pi. Archimedes used a
96-sided polygon to find his estimate.
Proposition three states:
The ratio of the circumference of any circle to its diameter is
but less than .
This approximates the mathematical constant π. He found the upper and lower
limits to the value of π by inscribing and circumscribing a circle with two similar 96-sided regular
Approximation to square
This proposition also contains accurate approximations to the square root of
3 (one larger and one smaller) and other larger non-perfect square roots; however,
Archimedes gives no explanation as to how he found these
gives the upper and lower bounds to √3 as 
- ^ Heath, Thomas Little (1921), A History of Greek
Mathematics, Boston: Adamant Media Corporation, ISBN 0543968774, http://books.google.com/books?id=zGIYbEtzD-QC&printsec=frontcover, retrieved
- ^ a
Britannica. 2008. http://www.britannica.com/EBchecked/topic/32808/Archimedes. Retrieved
- ^ a
Heath, Thomas Little
(1897), The Works of
Archimedes, Cambridge University, pp. lxxvii
; 50, http://books.google.com/books?id=bTEPAAAAIAAJ&printsec=titlepage, retrieved
- ^ Heath, Thomas Little (1931), A Manual of Greek
Mathematics, Mineola, N.Y.: Dover Publications,
p. 146, ISBN 0486432319, http://books.google.com/books?id=_HZNr_mGFzQC&printsec=frontcover