7.6 SAT Math - Quadrilaterals

A quadrilateral is any four-sided figure.  The interior angles of a quadrilateral, regardless of shape, always add up to 360°.  To find the perimeter of any shape, add up all the lengths of the sides.

The quadrilaterals you'll be dealing with most are parallelograms, rectangles, and squares.

Parallelogram

A parallelogram has two pairs of parallel sides and opposite angles equal each other:

Consecutive angles in a parallelogram, that is, angles that share a side, are supplementary, which means they add up to 180°. (Complementary angles, on the other hand, add up to 90°.)

Rectangle

A rectangle is a parallelogram in which all the angles are right angles:

As a result, opposite sides of a rectangle are equal, so the short sides both equal the same thing, and the long sides both equal the same thing.

Square

A square is a parallelogram, and also a rectangle, in which all the angles are right angles and all the sides equal each other:

The diagonal of a parallelogram extends from one corner to the opposite corner:

            -SAT Math Hint: The diagonal of a square forms a special situation:

Because all sides of a square are equal, if you split a square on its diagonal, you have formed two isosceles right triangles, also known as two 45-45-90 triangles.  This means you should be able to quickly and easily find the length of a side or the length of the diagonal depending on the information given. 
-SAT Math Tip: Not Enough Information.  One of the favorite ways the testmakers to try to trick you on the SAT is to give you a picture with virtually no information on it, like this:

The problem will then say something along the lines of "If the circle above has radius 4, what is the area of the square?"  Your first instinct will be to panic, or assume you're missing something, or assume there is a typo and they forgot to tell you something important…but you're wrong.  There is plenty enough information given here to solve this problem, you just have to remember Rule #3: Fill in Everything You Know.  If you do not remember the rule about squares split on their diagonal, or your 45-45-90 triangle relationships, you cannot solve this problem.  If you do remember that rule, you can simply cut the square on its diagonal, which also happens to be a diameter of the circle, solve for a side of the square, and from there find the area.  This is one of their favorite problems to put on the test.  It looks impossible if you don't know your rules, and is extremely easy if you do.  Fill in what you know, and the answer will often present itself.
(Did you find the answer to the above question?  Give it a try.

            The answer is 32.)

Though the official way to find perimeter is to add up all the sides, because of the relationships between their sides, you can shortcut the perimeter of a rectangle with the formula 2b + 2h, and you can shortcut the perimeter of a square with the formula 4s, where s is a side.

Area of a Parallelogram

The area of any parallelogram (including rectangles and squares) is bh, or base times height.  Keep in mind, on a parallelogram that is not also a rectangle, the height could be exterior:

Because the base and height of a square equal each other, you can also find the area of a square by the simple expression s².