What other way did Archimedesuse
to find the area of a circle?
The Greek mathematician Archimedes (c.
287–212 BCE; Hellenic) also found a way to
determine the area of a circle similar to the
Chinese comb: a method he first recorded
in his work, Measurement of a Circle(c.
225 BCE). He also used a sequence of
wedges to determine the area of a circle; as
the number of wedges (or triangles) inside
the circle increased toward infinity, they
became infinitely thin. By giving each
small triangle a base (b,a line connecting
the points where the wedges touched the
circle’s circumference), he determined that
the area was 1/2 times the radius (r) times
the base, summed over all the infinitesimal
triangles (or sum (1/2) rb). Because they all
had the same height, that was factored out.
Thus, the area became (1/2)r(sum (b))
1/2rc,with cbeing the circumference, or
the sum of the bases (b) of all the triangles (since the bases make up what is perceived as
the circle’s circumference). This is interpreted as one half times the radius times the cir-
cumference (c 2 πr), which is the same as saying πr^2. (For more about Archimedes and
his wedges, see “Mathematical Analysis.”)How are the surface areaand volumeof a three-dimensional geometric
figurecalculated?
The surface area(often abbreviated S.A.) of a three-dimensional geometric figure is
the total surfaces of the solid; it actually has units of distance or length squared. For
example, the surface area of a cube is 6a^2 , in which ais the length of the sides. To
translate, a cube has sides of equal lengths (a); the area of a cube is the sum of the
areas of the six squares (a^2 ) that cover it.
For more “diverse” figures, the surface area is actually equal to the lateral area
plus the area of each base. For example, the surface area of a prism or cylinder is the
lateral area plus the area of each base. (Because the bases for a prism or cylinder are
congruent, this is often expressed as twice the area of the base.) The surface area of a
pyramid or cone is the lateral area plus the area of the single base.
The volumeof a three-dimensional geometric figure is the total amount of space
190 the object occupies; volumes of such objects have units of length cubed. For example,
To calculate the area of a circle, the Chinese devised
the Chinese comb method in which a circle is divid-
ed into pie slices and rearranged into a rectangle to
greatly simplify the math involved.