Quadrilaterals

There are four main quadrilaterals you will see in the GRE: squares, rectangles, rhombuses, and trapezoids.

Definitions

Quadrilateral

Any four-sided shape

Congruent

Having the same size and shape (e.g., angles, lines, shapes, etc.)

Parallel

Straight sides/lines that will never cross if their length is extended forever

Perpendicular

Straight sides/lines that meet at a 90-degree right angle

The definitions of these different types of quadrilaterals can be a bit confusing because some categories encompass others. For instance, all squares are rectangles, but not all rectangles are squares. But fear not! We’ll cover each of them with examples.

Rectangles

You probably already know what a rectangle is, but it’s good to review the basics briefly from a technical perspective.

Definitions

Rectangle

A quadrilateral with four right angles

The opposite sides of a rectangle are parallel and congruent, and the two diagonals are also congruent.

The perimeter of a rectangle is the sum of all four sides, but since the opposite pairs are equal, it can be simplified to $P=2b+2h$.

The area of a rectangle is simply the base times the height: $A=bh$.

Squares

We’ll keep this one really short 😄

Definitions

Square

A quadrilateral with four equal sides and angles

Since all sides are the same, the perimeter of a square is $P=4s$.

Since the base and the height of a square are the same side length $s$, the area of a square is $A=s^2$.

Rhombuses

You might not have heard of a rhombus before.

Definitions

Rhombus

A quadrilateral with four equal sides, with opposite sites parallel and opposite angles of the same measure

Basically, a rhombus is a square leaning over to the side to make a diamond. The lengths of the sides are all still the same, but the angles get squished, so instead of having four right angles, you end up with two identical acute angles and two identical obtuse angles.

Since the sides are all equal in length, the perimeter of a rhombus is the same as for a square: $P=4s$.

There are two main ways to get the area of a rhombus. The first is very similar to that of a rectangle: $A=bh$. However, since our shape is tilted, the height $h$ is not the full height of a side. Instead, it’s the inside height that goes up at a right angle from the base.

The second way to get the area of a rhombus is using the inside diagonals that reach from one corner across the shape to the other opposite corner. Using these $p$ and $q$ diagonals, the area is $A=pq/2$.

Note that the square form of a rhombus maximizes the area! The more the rhombus leans over and deviates from the 90-degree interior angles, the smaller the area will become.

Parallelograms

Parallelograms are quadrilaterals that have parallel opposite sides of equal lengths. The opposite angles inside a parallelogram are also of equal measure.

Definitions

Parallelogram

A quadrilateral with parallel opposite sides of equal lengths

Since the definition of a parallelogram is broad, it encompasses many of the other types of quadrilaterals: squares, rectangles, rhombuses, and even trapezoids might also be parallelograms depending on their shape.

There aren’t any special formulas to memorize for parallelograms since it’s just a general category of shape.

The key thing to remember is that if the shape is described as a parallelogram, you only need to worry about two adjacent sides and two adjacent angles, since the opposite ones will be identical.

Trapezoids

Trapezoids can be a bit ambiguous since there are various definitions. Since this is a GRE course, we’ll use the ETS definition: a quadrilateral in which at least one pair of opposite sides is parallel is a trapezoid. Just be aware that other definitions vary, sometimes saying that a trapezoid must have only one pair of parallel opposite sides, or that every quadrilateral is a trapezoid.

Definitions

Trapezoid

A quadrilateral in which at least one pair of opposite sides is parallel

The opposite parallel sides are the bases, although they might not necessarily be the top and the bottom of the shape. The formulas will still be accurate even if the shape is rotated.

The perimeter of a trapezoid is the sum of all four sides. Since the sides could all be different lengths, there is no shortcut version of this formula.

The area of a trapezoid is equal to the average of the two parallel sides (“bases”) times the height, i.e., $A=\frac{1}{2}(b_1+b_2)(h)$. Note that the height is the inside distance from one base to another, and most likely won’t be the length of one of the sides!

Example quadrilateral question

Now that you know the fundamentals, let’s try a question!

The area of the trapezoid is greater than the area of the rectangle. Both shapes have the same height.

Quantity A: $a$
Quantity B: $7$

Give it a try, and then check your answer!

First, recall the equation for the area of a trapezoid: $A_t=\frac{1}{2}(b_1+b_2)(h)$.

And the equation for area of a rectangle: $A_r=bh$.

The question tells us that the area of the trapezoid is greater than the area of the rectangle, so we can set up an inequality. Both shapes have the same height, so we can use the same $h$ for both.

$$\begin{array}{rl}{A}_t& >A_r\\ \frac{1}{2}(b_1+b_2)(h)& >b_{r}h\\ \frac{1}{2}(14+a)(h)& >10h\\ \frac{1}{2}(14+a)& >10\\ 14+a& >20\\ a& >6\end{array}$$

We’ve figured out that $a>6$, and we’re comparing it against $7$. The question doesn’t say anything about the numbers needing to be integers, so it’s completely possible that $a$ could be something like $6.1$, $7$, $8$, $13$… any number where $a>6$ and $a<14$ (since it’s shown to be smaller than the base). We don’t have enough information to determine a more specific relationship, so the answer must be D.