MEASUREMENTS AND
TRANSFORMATIONS
What is the perimeterof a two-dimensional geometric figure?
The perimeter of a two-dimensional figure is what we perceive as the “outside” of the
object. Mathematicians describe a perimeter more precisely as the sum of the lengths
of a polygon’s sides; the perimeter of a simple closed curve is measured as its length.What is the areaof a two-dimensional geometric figure?
The area of two-dimensional figures varies depending on the object. For the simpler
two-dimensional figures (square, rectangle, parallelogram), the area can actually be
found by calculating the number of square units in the interior of the object. Such a
task can be difficult and time consuming, so mathematicians simply multiply the
height (h) times the base (b), sometimes said as length times height or length times
width (with rectangles). (For more information about the area of geometric figures,
see the Appendix 3: Common Formulas for Calculating Areas and Volumes of Shapes.)How is the area determinedfor some common two-dimensional polygons?
Finding the areas of polygons is not as simple as determining the areas of a rectangle
and square. In order to find the area of certain polygons, one essentially “breaks the
shape down” into smaller shapes with simpler formulas; these types of shapes are
called composites. The formulas for many of these polygons are all adaptations from
the rectangle area formula—or height (h) times base (b). For example, a triangle is
actually exactly half of a rectangle. Thus, the formula to find the area of a triangle is
half the base times the height (1/2bh). In the case of the trapezoid, the figure can be
divided into triangular sections, with the area equal to one half times the two bases
times the height, or 1/2(Bb)h.
The triangle formula also is used to find other regular polygon areas, but they are
less obvious. For a regular polygon, a feature called the apothemis necessary for find-
ing the area. This is the height of one of the congruent triangles inside the regular
polygon. In general, to find the polygon’s area, you need to find the area of one trian-
gle and multiply it times the number of sides. For example, to find the area of a hexa-
gon, divide the figure into six triangles, each with lequal to the height of each interior
triangle or apothem; lis also half of the smallest interior dimension of the hexagon,
called w. Thus, the area of a hexagon is the square root of 3 divided by 2 times w
squared (or 3 /2 w^2 ).
There is still another way to determine the area of a polygon by using the
Pythagorean Theorem, a method that uses length and height, with the resulting for-
188 mula for the area of a polygon looking much different. Both methods give the same