SSAT Middle Level

Polygons

Properties of quadrilaterals, regular polygons, and symmetry

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Imagine building a fort out of cardboard or drawing a cool superhero logo. What do they have in common? Shapes! 🦸‍♂️ In math, we call these flat, closed shapes with straight sides polygons. No curves allowed! (Sorry, circles, you can't sit with us today). The word 'poly' means many, and 'gon' means angles. So, a polygon is just a shape with many angles!

Think about the shapes you see every day. A stop sign is an octagon (8 sides). A kite is a quadrilateral (4 sides). What about a slice of pizza? Well, real pizza has a curved crust, but a perfectly straight triangle slice is a polygon with 3 sides! 🍕

On the SSAT, you will be a shape detective. You'll need to know special 4-sided shapes (called quadrilaterals) like rectangles, rhombuses, and trapezoids. You'll also meet 'regular' polygons. Regular polygons are the ultimate rule-followers: every side is the exact same length, and every angle is the exact same size! A square is a regular quadrilateral. 🕵️‍♀️✨

Get ready to count sides, find missing angles, and draw cool invisible lines inside shapes called diagonals. If you learn a few secret formulas, polygon questions will become some of the easiest points you can score on the SSAT!

Key Formula

To find the sum of all the inside angles of a polygon, use

S = 180 (n - 2)

, where

n

is the number of sides. To find just ONE angle of a regular polygon, divide that sum by the number of angles:

\frac{180 ( n - 2 )}{n}

Practice Questions

4 practice questions for SSAT Middle Level

Q1 Medium

A rectangular garden has a length of

(3 x - 5)

units and a width of

(x + 7)

units. If the perimeter of the garden is

68

units, what is the length of its longer side?

15

units

19

units

21

units

23

units

25

units

Show Solution

The perimeter $P$ of a rectangle is given by the formula $P = 2 (l e n g t h + w i d t h)$ .

Given $l e n g t h = 3 x - 5$ , $w i d t h = x + 7$ , and $P = 68$ .
Substitute the values into the formula:
$68 = 2 ((3 x - 5) + (x + 7))$
Combine like terms inside the parentheses:
$68 = 2 (4 x + 2)$
Divide both sides by 2:
$34 = 4 x + 2$
Subtract 2 from both sides:
$32 = 4 x$
Divide by 4:
$x = 8$
Now find the lengths of the sides by substituting $x = 8$ :
Length = $3 x - 5 = 3 (8) - 5 = 24 - 5 = 19$ units
Width = $x + 7 = 8 + 7 = 15$ units
The longer side is $19$ units.

Answer: B

Q2 Medium

What is the measure of each interior angle of a regular octagon?

108

degrees

120

degrees

135

degrees

144

degrees

150

degrees

Show Solution

A regular octagon has $8$ equal sides and $8$ equal interior angles.

The sum of the interior angles of a polygon with $n$ sides is given by the formula $(n - 2) \times 180$ degrees.
For a regular octagon, $n = 8$ .
Sum of interior angles = $(8 - 2) \times 180 = 6 \times 180 = 1080$ degrees.
Since all interior angles in a regular octagon are equal, the measure of each interior angle is the sum divided by the number of angles:
Each interior angle = $\frac{1080}{8} = 135$ degrees.
Alternatively, you can find the measure of an exterior angle first. The sum of the exterior angles of any convex polygon is $360$ degrees.
For a regular octagon, each exterior angle = $\frac{360}{8} = 45$ degrees.
Since an interior angle and its adjacent exterior angle form a linear pair (sum to $180$ degrees), each interior angle = $180 - 45 = 135$ degrees.

Answer: C

Q3 Medium

Which of the following quadrilaterals has exactly two lines of symmetry?

A Square

B Rhombus (but not a square)

C Isosceles Trapezoid

D Kite (but not a rhombus)

E Parallelogram (but not a rectangle or rhombus)

Show Solution

Let's analyze the lines of symmetry for each quadrilateral:

A) A Square has $4$ lines of symmetry (two connecting midpoints of opposite sides, and two connecting opposite vertices).
B) A Rhombus (that is not a square) has $2$ lines of symmetry (along its two diagonals).
C) An Isosceles Trapezoid has $1$ line of symmetry (the line segment connecting the midpoints of the parallel sides).
D) A Kite (that is not a rhombus) has $1$ line of symmetry (along its longer diagonal).
E) A Parallelogram (that is not a rectangle or a rhombus) has $0$ lines of symmetry. It has rotational symmetry of order 2, but no reflectional symmetry.
Therefore, a rhombus (that is not a square) has exactly two lines of symmetry.

Answer: B

Q4 Medium

In parallelogram

P QR S

, the measure of angle

P

x

degrees and the measure of angle

Q

(x + 40)

degrees. What is the measure of angle

S

70

degrees

90

degrees

110

degrees

120

degrees

130

degrees

Show Solution

In a parallelogram, consecutive angles are supplementary, meaning they add up to $180$ degrees. Angles $P$ and $Q$ are consecutive angles.

So, $an g l e P + an g l e Q = 180$ degrees.
Substitute the given expressions for the angles:
$x + (x + 40) = 180$
Combine like terms:
$2 x + 40 = 180$
Subtract $40$ from both sides:
$2 x = 140$
Divide by $2$ :
$x = 70$
Now find the measure of angle $Q$ :
$an g l e Q = x + 40 = 70 + 40 = 110$ degrees.
In a parallelogram, opposite angles are equal. Angle $S$ is opposite angle $Q$ .
Therefore, $an g l e S = an g l e Q = 110$ degrees.

Answer: C

Tips & Strategies

Memorize the names of polygons up to 10 sides (triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, decagon). The SSAT won't tell you that a 'hexagon' has 6 sides—you have to know it!
When a question says 'regular polygon', it's giving you a huge hint! It means you can divide the total angle sum perfectly by the number of sides to find the size of just one angle.

Common Mistakes

Watch out for confusing the TOTAL sum of angles with ONE interior angle. Always read carefully to see if the SSAT wants the whole pie or just one slice!
Don't forget that a square is a super-shape! It is technically a special type of rectangle AND a special type of rhombus. But remember, a normal rectangle is NOT a square.

Frequently Asked Questions

What exactly does 'regular' mean in geometry?

A regular polygon is the ultimate rule-follower! It means every single side is the exact same length, and every inside angle is the exact same size. Think of a perfect stop sign or a square.

Do I really need to memorize the diagonal formula for the SSAT?

Yes, it is super helpful, especially for the Upper Level SSAT! The formula $\frac{n ( n - 3 )}{2}$ saves you from having to draw the shape and count every single line, which gets really messy for big shapes.

How do I remember the difference between a parallelogram and a trapezoid?

A parallelogram has TWO pairs of parallel tracks (top/bottom and left/right). A trapezoid only has ONE pair of parallel tracks, kind of like a triangle that had its top chopped off!

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