Quadrilaterals and Polygons | SAT Geometry | SAT Math

Introduction

Quadrilaterals questions (questions about four-sided figures) on the SAT typically focus on rectangles (and squares, which are a type of rectangle); other quadrilaterals (e.g., parallelograms and trapezoids) are not common. On rare occasions, you will encounter polygons with more than four sides; the most common question surrounding these will involve the number of degrees in a polygon, so we’ve reproduced that formula in this lesson after introducing it in the Angles lesson. As with triangles, area comes up often; perimeter, somewhat less often.

Approach Question

An architect is sketching the plan for a family room with rectangular dimensions and a floor area of 192 square feet. Feeling bold, the architect decides to increase both the length and width of the floor plan by 50%. What is the floor area, in square feet, of the new planned family room after the increase?

A. 242
B. 256
C. 288
D. 432

Explanation

Almost without exception, the most challenging SAT problems involving rectangles or squares will involve some understanding of a change in dimensions. As an example of this, remember how the Circles lesson challenged you to recognize that, say, tripling a circle’s radius increases its area by a factor of $9$ (not $3$ ). This is because area is a two-dimensional quantity, so we square the multiplying factor to account for both dimensions.

The same principle applies here. The most common mistake would be to assume we can simply multiply the area of $192$ by $1.5$ to account for a $50%$ increase; you can be sure the SAT will include a trap answer to this effect, and indeed, here the trap answer is $288$ . This answer is wrong because it doesn’t take into account the 50% increase affecting both dimensions of the rectangle (length and width). So, how do we account for this double increase? There are a couple of good approaches.

The textbook approach would be to use an equation using $l$ for length and $w$ for width. We know that, because of the area formula, $lw = 192$ in this case. If we’re increasing both length and width, the new length can be represented by $1.5 l$ , and the new width by $1.5 w$ . This means that the new area is equal to $(1.5 l) (1.5 w) = 2.25 lw$ . We already know that $lw = 192$ , so we can substitute $192$ for $lw$ in the second equation, yielding $(2.25) (192) = 432$ . The answer is 432.

If you prefer not to use variables, you could take a trial-and-error approach in which you supply a length and width such that the area is $192$ . If you play around with numbers on your calculator (or in your head, if you’re exceptional at mental math!), you could note that the dimensions could be $12$ and $16$ , since $12 \times 16 = 192$ . (You could also use $8$ and $24$ , $6$ and $32$ or other pairs all the way down to $1$ and $192$ if you prefer.) If you start with $12$ and $16$ , you could then increase each of those values by $50%$ , yielding $18$ and $24$ . $18$ times $24$ would give you the correct answer as well.

Finally, you could take a more global, logical approach that multiplies $150%$ times $150%$ right away. You would need to convert the percents to decimals, finding that $1.5 \times 1.5 = 2.25$ . This means the overall area must increase by a factor of $2.25$ , so you multiply that factor by $192$ . This also gives us $432$ . As is typical, there is more than one way to approach this SAT problem; pick the one that suits you best!

Definitions

Polygon: A shape with three or more sides. Technically, you are likely to encounter only convex polygons, meaning that all of the sides of the polygon point away from its center. In a concave polygon, should you encounter it, one or more of the sides would point inward toward the center of the shape. This sort of polygon could still be divided into smaller shapes as described in Strategy Insights.
Similar: We covered this definition under triangles, but are repeating it here to make sure you’re aware that similarity pertains to all polygons. We know two polygons are similar if all their corresponding angles are congruent (note that this means all rectangles are similar by definition because they always have only right angles). Once we know the polygons are similar, we can conclude that their corresponding sides are proportional. So, for example, if two rectangles have corresponding lengths of $6$ and $8$ , their width must be in the same basic $3 : 4$ ratio to each other.
Parallelogram: A polygon with two parallel and congruent pairs of opposite sides. Subcategories of parallelogram include rectangle, rhombus, and square. Opposite angles in a parallelogram must be congruent; adjacent angles must be supplementary.
Rhombus: A parallelogram with all congruent sides. All of the properties of a parallelogram are true of the rhombus, with the addition that its diagonals are perpendicular.
Trapezoid: A quadrilateral with exactly one pair of parallel sides. Trapezoids on the SAT are extremely rare; if they appear, they will usually be isosceles; because they have two congruent sides and their unequal sides are parallel to each other, they must have two pairs of congruent angles.

Topics for Cross-Reference

Variations

The main alternative to a simple quadrilateral question is one that pairs a quadrilateral with another figure, usually a triangle or a circle. When a question has multiple figures, typically the first thing to identify is what those shapes have in common. For example, when a square is inscribed in a circle, the diagonal of the square is the same as the diameter of the circle.

Also, as noted in the introduction, you may occasionally encounter a polygon with more than four sides. If such a question doesn’t ask you about the total degrees in the polygon, it will likely allow you to divide the polygon into smaller shapes, like triangles and rectangles. In that case, make sure to recopy the polygon on your scratch paper so you can carve it up effectively!

Strategy Insights

Draw! Remember how the UnCLES method says, “If it’s a geometry problem, draw a figure.” In most cases, drawing the figure on your scratch paper will be crucial to understanding the problem thoroughly.
With polygons of more than four sides or with less common quadrilaterals, divide the figure into rectangles and triangles to help you understand the figure’s area. For example regular hexagon divides nicely into six equilateral triangles.
Always watch out for changes in dimension from one-dimensional line segments to two-dimensional shapes. We can’t emphasize this enough!

Flashcard Fodder

The most important formulas for quadrilaterals are for rectangles and squares:

Area of a rectangle = $l e n g t h \times w i d t h$ (this is in the SAT reference list)
Perimeter of a rectangle = $2 l + 2 w$
Area of a square = $s i d e^{2}$
Perimeter of a square = $4 s$

We’ll also repeat here the formulas regarding angles in a polygon.

Total degrees in the angles of a polygon: $(n - 2) (180)$ , where n is the number of sides
Degrees in each angle of a regular polygon $\frac{( n - 2 ) ( 180 )}{n}$

Sample Questions

Difficulty 1

The area of a square is 36 square inches. What is the length, in inches, of a side of the square? (Note: this is a free-response question.)

(spoiler)

The answer is 6. The SAT reference formulas include the fact that the area of a rectangle is equal to length times width; closely related is that a square’s area is $s i d e^{2}$ , since length and width are the same. This means the algebraic way to solve this question is $x^{2}$ =36, taking only the positive answer as the result (since an answer of -6 makes no sense when talking about the length of a square’s side). If you recognized that you need to take the square root of 36, you may have arrived at the answer with the equation.

Difficulty 2

A rectangle has a length of a units and a width of (a-6) units. If the area of the rectangle is 72 units squared, how many units is the width of the rectangle?

A. 6
B. 9
C. 12
D. 18

(spoiler)

The answer is 6. This question illustrates how a question about two-dimensional area can incorporate a quadratic equation (although there is a way to answer it without using the equation). Since the area of a rectangle is length times width, we can set up an equation that multiplies length by width, both in terms of a, and sets it equal to $72$ . This will give us a quadratic equation, and the solution will come from factoring:

$x (x - 6) = 72$

$x^{2}$ $- 6 x = 72$

$x^{2}$ $- 6 x - 72 = 0$

$(x - 12) (x + 6) = 0$

$x = 12$ or $x = - 6$

As with the previous question, we discard the negative solution; only 12 makes sense. However, alert! Did you use the UnCLES method well, noticing that the question asks for the length of the width, not the length? The width is $a - 6$ , so if $a = 12$ , the correct answer is $6$ .

We can quickly check our answer by multiplying $6$ and $12$ , our width and length, to confirm that the area is $72$ . We could have also worked backward from the answers. If you want to work backwards in the case of a real-life word problem with number answers, it’s best to start in the middle. Say you started with $9$ as the width. That would make the length $15$ , but $9$ times $15$ is too large to be our area. You know the answer must therefore be smaller, so it must be $6$ !

Difficulty 3

What is the length of one side of a square that has the same area as a circle with radius $4$ ?

A. $4$
B. $4 π$
C. $π$
D. $4 π$

(spoiler)

The answer is $4 p i$ . The most natural place to start here is to determine the second part mentioned: what is the area of a circle with radius $4$ ? If you learned well from our Circles lesson, you will quickly know that the answer is 16pi. So we need a square whose area is $16 π$ . Since, as already discussed, the area of a square is equal to $s^{2}$ , we conclude that $s^{2}$ =16pi. Taking the positive square root of both sides yields 4sqrt pi. Watch out for the answers close to this, involving just one small mistake or the other: $4 π$ and simply $4 π$ .

Difficulty 4

Rectangles MNOP and WXYZ are similar. The length of each side of MNOP is 5 times the length of the corresponding side in WXYZ. If the area of rectangle WXYZ is 42, what is the area of rectangle MNOP? (Note: this is a free-response question.)

(spoiler)

The answer is 1,050. We return here to the idea of dimensions from our approach question. If you thought the answer here was $210$ , you missed the element of moving from one dimension to two. Since the sides of the larger square are five times as long as the sides of the smaller square, the larger square is $25$ times (not $5$ times) as large in area, because we need to square that factor of $5$ . So if the smaller square’s area here is $42$ , we must multiply that value by $25$ to get the correct answer.

Difficulty 5

Square C has an area of $c$ units squared. Square D has an area $9, 801$ times as large as that of Square C. The function $f$ gives the perimeter of Square D. Which of the following defines $f$ ?

A. $99 c$
B. $198 c$
C. $396 c$
D. $9, 801 c$

(spoiler)

The answer is 396c. The question is difficult because (like the previous question) it involves dimensional analysis, but, unlike the previous question, calls on you to work that process in reverse. To deal with the very large factor of $9, 801$ times the area, remind yourself how a square’s area is affected by increasing its side lengths. We know that doubling a square’s side length quadruples its area and that tripling its side lengths increases its area by a factor of $9$ . You can see from this pattern that we square the factor of increase used for the sides in order to get the area. So, whatever factor the larger square’s sides are multiplied by in this case, it must be the square root of $9, 801$ . The calculator will tell you that: $99$ .

We now know that, if the side of the smaller square has length $c$ , a side of the larger square has length $99 c$ . All that remains is to get the perimeter (and hopefully you use the UnCLES well and don’t fall for the answer of $99 c$ ). The perimeter of a square is $4$ times the length of one of its sides, so $396 c$ does the trick.

For Reflection

How will you approach quadrilateral and polygon questions on test day? Write down at least three takeaways from this module.
Rate the difficulty of these questions for you from 1 (no problem) to 5 (problem!). This will help you decide when to answer them and when to skip them on test day.
This lesson continued the emphasis on dimensional analysis that the SAT prioritizes–for example, understanding that doubling the length of a square’s sides quadruples its area. The approach question, as well as the Difficulty 4 and 5 questions, in this lesson test this concept. If you still feel uncertain about the concept, we recommend reviewing those questions.

Introduction

Approach Question

An architect is sketching the plan for a family room with rectangular dimensions and a floor area of 192 square feet. Feeling bold, the architect decides to increase both the length and width of the floor plan by 50%. What is the floor area, in square feet, of the new planned family room after the increase?

A. 242
B. 256
C. 288
D. 432

Explanation

Definitions

Polygon: A shape with three or more sides. Technically, you are likely to encounter only convex polygons, meaning that all of the sides of the polygon point away from its center. In a concave polygon, should you encounter it, one or more of the sides would point inward toward the center of the shape. This sort of polygon could still be divided into smaller shapes as described in Strategy Insights.
Similar: We covered this definition under triangles, but are repeating it here to make sure you’re aware that similarity pertains to all polygons. We know two polygons are similar if all their corresponding angles are congruent (note that this means all rectangles are similar by definition because they always have only right angles). Once we know the polygons are similar, we can conclude that their corresponding sides are proportional. So, for example, if two rectangles have corresponding lengths of $6$ and $8$ , their width must be in the same basic $3 : 4$ ratio to each other.
Parallelogram: A polygon with two parallel and congruent pairs of opposite sides. Subcategories of parallelogram include rectangle, rhombus, and square. Opposite angles in a parallelogram must be congruent; adjacent angles must be supplementary.
Rhombus: A parallelogram with all congruent sides. All of the properties of a parallelogram are true of the rhombus, with the addition that its diagonals are perpendicular.
Trapezoid: A quadrilateral with exactly one pair of parallel sides. Trapezoids on the SAT are extremely rare; if they appear, they will usually be isosceles; because they have two congruent sides and their unequal sides are parallel to each other, they must have two pairs of congruent angles.

Topics for Cross-Reference

Variations

Strategy Insights

Draw! Remember how the UnCLES method says, “If it’s a geometry problem, draw a figure.” In most cases, drawing the figure on your scratch paper will be crucial to understanding the problem thoroughly.
With polygons of more than four sides or with less common quadrilaterals, divide the figure into rectangles and triangles to help you understand the figure’s area. For example regular hexagon divides nicely into six equilateral triangles.
Always watch out for changes in dimension from one-dimensional line segments to two-dimensional shapes. We can’t emphasize this enough!

Flashcard Fodder

The most important formulas for quadrilaterals are for rectangles and squares:

Area of a rectangle = $l e n g t h \times w i d t h$ (this is in the SAT reference list)
Perimeter of a rectangle = $2 l + 2 w$
Area of a square = $s i d e^{2}$
Perimeter of a square = $4 s$

We’ll also repeat here the formulas regarding angles in a polygon.

Total degrees in the angles of a polygon: $(n - 2) (180)$ , where n is the number of sides
Degrees in each angle of a regular polygon $\frac{( n - 2 ) ( 180 )}{n}$

Sample Questions

Difficulty 1

The area of a square is 36 square inches. What is the length, in inches, of a side of the square? (Note: this is a free-response question.)

(spoiler)

Difficulty 2

A rectangle has a length of a units and a width of (a-6) units. If the area of the rectangle is 72 units squared, how many units is the width of the rectangle?

A. 6
B. 9
C. 12
D. 18

(spoiler)

$x (x - 6) = 72$

$x^{2}$ $- 6 x = 72$

$x^{2}$ $- 6 x - 72 = 0$

$(x - 12) (x + 6) = 0$

$x = 12$ or $x = - 6$

Difficulty 3

What is the length of one side of a square that has the same area as a circle with radius $4$ ?

A. $4$
B. $4 π$
C. $π$
D. $4 π$

(spoiler)

Difficulty 4

Rectangles MNOP and WXYZ are similar. The length of each side of MNOP is 5 times the length of the corresponding side in WXYZ. If the area of rectangle WXYZ is 42, what is the area of rectangle MNOP? (Note: this is a free-response question.)

(spoiler)

Difficulty 5

Square C has an area of $c$ units squared. Square D has an area $9, 801$ times as large as that of Square C. The function $f$ gives the perimeter of Square D. Which of the following defines $f$ ?

A. $99 c$
B. $198 c$
C. $396 c$
D. $9, 801 c$

(spoiler)

For Reflection

How will you approach quadrilateral and polygon questions on test day? Write down at least three takeaways from this module.
Rate the difficulty of these questions for you from 1 (no problem) to 5 (problem!). This will help you decide when to answer them and when to skip them on test day.
This lesson continued the emphasis on dimensional analysis that the SAT prioritizes–for example, understanding that doubling the length of a square’s sides quadruples its area. The approach question, as well as the Difficulty 4 and 5 questions, in this lesson test this concept. If you still feel uncertain about the concept, we recommend reviewing those questions.