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Polygons
A polygon is a two-dimensional figure with three or more
straight sides. (So triangles are actually a type of polygon.) Polygons
are named according to the number of sides they have.
All polygons, no matter how many sides they possess, share
certain characteristics:
- The sum of the interior angles of a polygon
with n sides is (n –
2)
. For instance, 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 is 5 + 4 + 3 + 8 + 6 + 9 = 35.
Regular Polygons
The polygon whose perimeter you just calculated was an
irregular polygon. But most of the polygons on the SAT are regular:
Their sides are of equal length and their angles congruent. Neither
of these conditions can exist without the other. If the sides are
all equal, the angles will all be congruent, and vice versa. In
the diagram below, you’ll see, from left to right, a regular pentagon,
a regular octagon, and a square (also known as a regular quadrilateral):
Quadrilaterals
Good news: Most polygons on the SAT have just four sides.
You won’t have to tangle with any dodecahedrons on the SAT you take.
But this silver cloud actually has a dark lining: There are five different
types of quadrilaterals that pop up on the test. These five quadrilaterals
are trapezoids, parallelograms, rectangles, rhombuses, and squares.
Trapezoids
A trapezoid may sound like a new Star Wars character.
Certainly, it would be less annoying than Jar Jar Binks. But it’s
actually the name of a quadrilateral with one pair of parallel sides
and one pair of nonparallel sides.
In this trapezoid, AB is
parallel to CD (shown by the arrow
marks), whereas AC and BD are
not parallel.
The formula for 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.
To find the area of a trapezoid on the SAT, you’ll often
have to use your knowledge of triangles. Try to find the area of
the trapezoid pictured below:
The question tells you the length of the bases of this
trapezoid, 6 and 10. But to find the area,
you first need to find the height. To do that, split the trapezoid
into a rectangle and a 45-45-90 triangle by drawing
in the height.
Once, you’ve drawn in the height, you can split the base
that’s equal to 10 into two parts: The base of the
rectangle is 6, and the leg of the triangle is 4.
Since the triangle is 45-45-90, the two legs must be
equal. This leg, though, is also the height of the trapezoid. So
the height of the trapezoid is 4. Now you can plug
the numbers into the formula:
Parallelogram
A parallelogram is a quadrilateral whose opposite
sides are parallel.
In a parallelogram,
- Opposite sides are equal in length: BC = AD and AB = DC
- Opposite angles are equal:
and 
- Adjacent angles are supplementary:
- The diagonals bisect (split) each other: BE = ED and AE = EC
- One diagonal splits a parallelogram into two congruent
triangles:
- Two diagonals split a parallelogram into two pairs of
congruent triangles:
and 
The area of a parallelogram is given by the formula
where b is the length of
the base, and h is the height.
Rectangles
A rectangle is a quadrilateral in which the opposite
sides are parallel and the interior angles are all right angles.
Another way to look at rectangles is as parallelograms in which
the angles are all right angles. As with parallelograms, the opposite
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.
The diagonals of a rectangle are always equal to each
other. And one 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.
Since the diagonal of the rectangle forms right
triangles that include the diagonal and two sides of the rectangle,
if you know two of these values, you can always calculate the third
with the Pythagorean theorem. If you know the side lengths of the
rectangle, you can calculate the diagonal. If you know the diagonal
and one side length, you can calculate the other side. Also, keep
in mind that the diagonal might cut the rectangle into a 30-60-90 triangle.
That would make your calculating job even easier.
Rhombus
A rhombus is a specialized parallelogram in which all
four sides are of equal length.
In a rhombus,
- All four sides are equal: AD = DC = CB = BA
- The diagonals bisect each other and form perpendicular lines (but note that the diagonals are not equal in length)
- The diagonals bisect the vertex angles
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 on the SAT (you guessed
it), you’ll probably have to split it into triangles:
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Since ABD is an equilateral
triangle, the length of each side of the rhombus must be 4,
and angles ADB and ABD are 60º.
All you have to do is find the height of the rhombus. Draw an altitude
from A to DC to
create a 30-60-90 triangle.
Since the hypotenuse of the 30-60-90 triangle
is 4, you can use the ratio 1:
:2 to calculate that the length of
this altitude is 2
. The area formula
for a rhombus is bh, so the area of
this rhombus is 4 
2
= 8
.
:2 to calculate that the length of
this altitude is 2. The area formula
for a rhombus is bh, so the area of
this rhombus is 4 2 = 8.Square
A square combines the special features of the rectangle
and rhombus: All its angles are 90º, and all four of
its sides are equal in length.
The square has two more crucial special qualities. In
a square,
- Diagonals bisect each other at right angles and are equal in length.
- Diagonals bisect the vertex angles to create 45º angles. (This means that one diagonal will cut the square into two 45-45-90 triangles, while two diagonals break the square into four 45-45-90 triangles.)
The formula for the area of a square is
where s is the length of
a side of the square.
Because a diagonal drawn into the square forms two congruent 45-45-90 triangles,
if you know the length of one side of the square, you can always
calculate the length of the diagonal:
Since d is the hypotenuse
of the 45-45-90 triangle that has legs of length 5,
according to the ratio 1:1:
, you know that 
.
, you know that . Similarly, if you know the length of the diagonal, you
can calculate the length of the sides of the square.
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