In the beginning of the twelfth century CE, an interesting
geometry book appeared:
The Book of Mensuration of the Earth and its Division,
by Rabbi Abraham Bar Hiya (acronym RABH), a Jewish philosopher and scientist.
This book is interesting both historically and mathematically.
The second part of the book contains a beautiful mechanical derivation
of the area of the disk.
Roughly speaking, the argument goes as follows (see illustration above):
The disk is viewed as the collection of all the
concentric circles it contains.
If we cut the circles along the radius of the disk,
and let them fan out to become straight lines,
we get a triangle
(because the ratio of the circumference of the circles
to their diameters is constant).
The base length of the resulting triangle is equal to the
circumference of the original circle, and its height is equal to
the radius of this circle.
Thus, the area of a circle is equal to half of the
product of the radius and the circumference. Using modern terms,
this means that the area of the disk with radius R is equal to

2p R x R/2 = p
R^{2}

The first complete mathematical justification of this proof appeared
in [1]. This method was generalized to find the area of the
sphere and its horizontal sections in [2].

References.

[1]Every [Circle] whose Circumference, Higayon 3 (1995),
Department of Mathematics, Bar-Ilan University, Ramat-Gan, pp. 103--131. [in Hebrew] [2]
David Garber and Boaz Tsaban,
A mechanical derivation of the area of the sphere,
The American Mathematical Monthly108 (2001), 10--15. [Available here; for
related articles click here.]