Quadrilaterals on the SAT and ACT

On the SAT and ACT, you’ll have to be familiar with your shapes. Unfortunately, this knowledge goes significantly beyond distinguishing a circle from a triangle, though you’ll still have to know that.
Let’s talk about a popular one: quadrilaterals. “Quadrilateral” is just a fancy, polysyllabic word for any four-sided polygon. There are a few different types of quadrilaterals that you should be familiar, but first, let’s discuss some basic properties that all quadrilaterals have in common.
 

• All interior angles of a quadrilateral add up to 360 degrees.
• Find the perimeter of any quadrilateral by adding up the four sides.

There are three basic types of quadrilaterals you will find on the test: parallelograms, rectangles, and squares.

By definition, a parallelogram is any quadrilateral whose opposite sides are both parallel and equal in length. Notice in the diagram that the opposite angles are equal, and the consecutive angles–those  that share a side–are supplementary (they add up to 180 degrees).
Area of a Parallelogram: base times height, or bh. (Simpler than it looks)



Rectangles are special parallelograms in which all the angles are right angles. Note that they still retain the characteristic that opposite sides are equal.
Area of a Rectangle: base times height, or bh.


A square is a special type of rectangle in which all the sides are equal.
Area of a Square: base times height, or side^2

Granted, the SAT and ACT will not simply test you on your ability to spot these special types of quadrilaterals. You’ll have to use your knowledge of their properties to perform measurements and calculations. Here are some familiar situations for quadrilateral measurement:

1. Diagonals

You’ll often have to find the diagonal of a rectangle to yield new information about a shape. The diagonal is just the line that extends from one corner to another, and you can calculate a rectangle’s diagonal by using the Pythagorean Theroem: a² + b²= c². All rectangles are made up of two congruent right triangles. In fact, all squares are made up of two special congruent right triangles called 45-45-90 triangles. These special isosceles triangles are so named because they are made up of one 90 degree and two 45 degree angles. To save yourself time, remember this diagram:

With this information, you could find out the sides of a square (and therefore its area and perimeter) with just its diagonal measurement. For example, if the diagonal of a given square is 5, then I can use this equation to yield side n:
n*rad(2) = 5
n=5 / rad(2)
rationalize the denominator…
n=(5 *rad 2 ) / 2

1. The Square in Circle Problem

You may see a square inscribed in a circle like the one above, and the question may ask you for the area of the square provided that the radius of the circle is 2.5. To solve this problem, remember that the diameter of this circle must be equal to the diagonal of the square–draw a line from two opposite corners and see for yourself. To solve it, just recognize that a radius of 2.5 yields a diameter of 5, which also happens to be the length of our diagonal. Use the steps above to find side n¸and simply square that n.
Those are the quadrilateral basics. Whenever you’re stumped on a word problem with measurements, remember the properties of quads and it may get you out of a jam!