Polygons are enclosed geometric shapes that cannothave fewer than three sides. As this definition suggests, triangles areactually a type of polygon, but they are so important on the Math IICthat we gave them their own section. Polygons are named according tothe number of sides they have, as you can see in the chart below.
All polygons, no matter the number of sides they possess, share certain characteristics:
- The sum of the interior angles of a polygon with n sides is (n – 2)
. So, for example, the sum of the interior angles of an octagon is (8 – 2)
= 6
= 
. - The sum of the exterior angles of any polygon is
. - The perimeter of a polygon is the sum of the lengths of its sides. The perimeter of the hexagon below, for example, is 35.
Regular Polygons
Most of the polygons with more than four sidesthat you’ll deal with on the Math IIC will be regular polygons—polygonswhose sides are all of equal length and whose angles are all congruent(neither of these conditions can exist without the other). Below arediagrams, from left to right, of a regular pentagon, a regular octagon,and a square (also known as a regular quadrilateral):
Area of a Regular Polygon
There is one more characteristic of polygonswith which to become familiar. It has to do specifically with regularhexagons. A regular hexagon can be divided into six equilateraltriangles, as the figure below shows:
E,t If you know the length of just one side of aregular hexagon, you can use that information to calculate the area ofthe equilateral triangle that uses the side. To find the area of thehexagon, simply multiply the area of that triangle by 6.
Quadrilaterals
The most frequently seen polygon on the Math IC is the
quadrilateral,which is a general term for a four-sided polygon. In fact, there arefive types of quadrilaterals that pop up on the test: trapezoids,parallelograms, rectangles, rhombuses, and squares. Each of these fivequadrilaterals has special qualities, as shown in the sections below.
Trapezoids
A
trapezoid is a quadrilateral with one pair of parallel sides and one pair of nonparallel sides. Below is an example of a trapezoid:
In the trapezoid pictured above,
AB is parallel to
CD (shown by the arrow marks), whereas
AC and
BD are not parallel.
The area of a trapezoid is:
where
s1 and
s2 are the lengths of the parallel sides (also called the bases of the trapezoid), and
h is the height. In a trapezoid, the height is the perpendicular distance from one base to the other.
Try to find the area of the trapezoid pictured below:
To find the area, draw in the height of thetrapezoid so that you create a 45-45-90 triangle. You know that thelength of the leg of this triangle—and the height of the trapezoid—is4. Thus, the area of the trapezoid is 6+10⁄2
4 = 8
4 = 32. Check out the figure below, which includes all the information we know about the trapezoid:
Parallelogram
A
parallelogram is a quadrilateral whose opposite sides are parallel. The figure below shows an example:
Parallelograms have three very important properties:
- Opposite sides are equal.
- Opposite angles are congruent.
- Adjacent angles are supplementary (they add up to 180º).
To visualize this last property, simplypicture the opposite sides of the parallelogram as parallel lines andone of the other sides as a transversal:
The area of a parallelogram is given by the formula:
where
b is the length of the base, and
h is the height.
In area problems, you will likely have tofind the height using techniques similar to the one used in theprevious example problem with trapezoids.
The next three quadrilaterals that we’ll review—rectangles, rhombuses, and squares—are all special types of parallelograms.
Rectangles
A
rectangle is a quadrilateral inwhich the opposite sides are parallel and the interior angles are allright angles. A rectangle is essentially a parallelogram in which theangles are all right angles. Also similar to parallelograms, theopposite sides of a rectangle are equal.
The formula for the area of a rectangle is:
where
b is the length of the base, and
h is the height.
A diagonal through the rectangle cuts the rectangle into two equal right triangles. In the figure below, the diagonal
BD cuts rectangle
ABCD into congruent right triangles
ABD and
BCD.
Because the diagonal of the rectangle formsright triangles that include the diagonal and two sides of therectangle, if you know two of these values, you can always calculatethe third with the Pythagorean theorem. If you know the side lengths ofthe rectangle, you can calculate the diagonal; if you know the diagonaland one side length, you can calculate the other side.
Rhombuses
A
rhombus is a quadrilateral in which the opposite sides are parallel and the sides are of equal length.
The formula for the area of a rhombus is:
where
b is the length of the base, and
h is the height.
To find the area of a rhombus, use the same methods as used to find the area of a parallelogram. For example:
If
ABCD is a rhombus,
AC = 4, and
ABD is an equilateral triangle, what is the area of the rhombus?
If
ABD is an equilateral triangle, then the length of a side of the rhombus is 4, and angles
ADB and
ABD are 60º. Draw an altitude from
a to
DC to create a 30-60-90 triangle, and you can calculate the length of this altitude to be 2
. The area of a rhombus is
bh, so the area of this rhombus is 4
2
= 8
.
Squares
A
square is a quadrilateral in whichall the sides are equal and all the angles are right angles. A squareis a special type of rhombus, rectangle, and parallelogram:
1W"Y]2X(}h"V)H ?.D The formula for the area of a square is:
where
s is the length of a side ofthe square. Because all the sides of a square are equal, it is alsopossible to provide a simple formula for the perimeter: P = 4
s, where
s is, once again, the length of a side.
A diagonal drawn into the square will always form two congruent 45-45-90 triangles:
From the properties of a 45-45-90 triangle, we know that
. Inother words, if you know the length of one side of the square, you caneasily calculate the length of the diagonal. Similarly, if you know thelength of the diagonal, you can calculate the length of the sides ofthe square.
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本帖最后由 端木·宇 于 2008-6-18 19:04 编辑 ]
