The elements of non-Euclidean plane geometry and trigonometry, by H. S. Carslaw.

51, 52] THE AREA OF A POLYGON

89

by its defect, k being a constant depending on the unit triangle,
and the unit of angle is chosen so that a right angle has r for
its measure. The number k2 is introduced to bring the results
into agreement with the analytical work in other parts of this
book.
It follows from ~ 51 that
1. If two triangles have the same measure of area, they are
equivalent, and that if two triangles are equivalent, they have
the same measure of area.
2. If a triangle is broken up into a finite number of triangles,
the measure of area of the triangle is equal to the sum of the
measures of area of the triangles in the partition.
3. If a triangle is equivalent to the sum of two other triangles, the measure of area of this triangle is equal to the sum
of the measures of area of the other two triangles.
The measure of area of a polygon is defined to be the sum of the,measures of area of the triangles into which it is divided in any
given partition.
This sum is independent of the partition which has been
chosen. The sum of the defects of the triangles in any partition is equal to (n - 2) times two right angles - the sum of the
angles of the polygon. This is sometimes called the Defect of
the Polygon.
With regard to polygons we can now state the following
theorems:
1. If two polygons have the same measure of area, they are
equivalent. For they are each equivalent to the triangle whose
defect is the sum of the defects of the given partitions.
2. If two polygons are equivalent, they have the same
measure of area. For they can be broken up into a finite
number of triangles congruent in pairs.
3. If a polygon is broken up into a finite number of subpolygons, the measure of area of the polygon is the same as
the sum of the measures of area of the sub-polygons.
4. If a polygon is equivalent to the sum of two other polygons, its measure of area is equal to the sum of the measures
of area of these two polygons.
Rectilinear polygons with the same measure of area will be
said to have equal area. Thus equivalent polygons have equal