results in numerical terms. For example, the area of a hexagon using the smallest
interior dimension would be 0.866 times the square of the smallest width, w;the area
of a regular hexagon using the Pythagorean Theorem is 2.598 times the square of its
side length. These may be different ways of presenting the solution, but both give the
same correct area measurements.
What are the measurementsof a circle?
There are many measurements of a circle. The perimeter of a circle is actually called
the circumference;to calculate this, multiply pi (π) times the diameter, d(πd), or pi
(π) times twice the radius, r(2πr). The area of a circle is calculated by multiplying pi
(π) times the radius squared, or πr^2.
How was the area of a circlefirst determined?
When it comes to determining the area of a circle, there are many historical claims to
this solution. One of the earliest techniques may have been the Chinese “comb”
method, in which a circle is cut into nwedges, each 360/ndegrees and each piece
identical (with the same area). To see how this works, take the bottom half of a unit
circle and cut it into wedges like slices of a pie; place all the wedges next to each other,
with the points up (like the teeth of a comb or animal). Then split the top half of the
circle in the same way, putting them next to each other above the other wedges, but
this time point the tips of the wedges down. Close the “teeth”; as ngoes to infinity, the
shape of the combined wedges approaches a rectangle. Taking the formula for deter-
mining the area of a rectangle (width times height), the width is πr(or half the cir-
cumference) and the height is r. Thus, the area is πr^2. 189
GEOMETRY AND TRIGONOMETRY
How is the term “surface area” used in geometry?
L
ogically, the surface area is the area of a given surface. There are several ways
to interpret this in geometry: Area can mean the extent of the surface region
on a two-dimensional plane. Surface area (often called the lateral surface area,
although there is a difference) formulas for three-dimensional objects are more
complex—all the surface areas are added around the outside of the object, from
a cube to a sphere.Surface area is commonly denoted as Sfor a surface in three dimensions,
and Afor the surface area of a two-dimensional plane (commonly, it is simply
called “the area”). But be careful: The surface area of a three-dimensional object
should not be confused with “volume”—or the total amount of space an object
occupies. (For more about volume, see below.)