Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

ADVANCED METHODS

517

The area A of an arbitrary region in M is the limit approached by the sum B of the areas of rectangles, of the type
just described, which are inscribed
in the region:
A= lim B.
If A' is the area of the projected /
region and B' is the sum of the
areas of the projected rectangles,
evidently
A' = lim B'.
Since the area of each projected F. 4
rectangle is cos 0 times the area of the original rectangle,
B' = B cos 9.
Hence A' = lim B cos 0 = cos 0 lim B
or A'= A cos 0, q. e.d.
Let the areas of the projections of the given region on the
coordinate planes be denoted by Ay, Az,, A,~-an- let the
normals to M have the direction angles a, /, y. By Th. 1,7
A, = IA cosal, A,=_ IA cos /, A = IA cosyl.
Hence
(1) A2 = AZ2 + A,2 + A.2
Thus we have proved the theorem:
THEOREM 2. The sum of the squares of the areas of the projections of a region on the three coordinate planes equals the
square of the area of the region.
Area of a Triangle. It is now easy to write down a
formula for the area A of the triangle whose vertices are at
the points P: (x,, yl, zl), P2: (x2, Y2, z2), P3: (,3 Y3, z3). For,
the areas of the projections of the triangle on the coordinate
planes are, by Ex. 18 at the end of Ch. XVI,
A,,,-lylzll, A, == ~~[x21, I y+~x 1Y2 1l,
Av ~IY1I211, zz~- |^ 2 1 XZ21 )~i 1y22|,
* The absolute value signs are necessary since a, /, y are not necessarily acute angles.