SSAT Upper 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 Upper Level

Q1 Hard

The measure of an interior angle of a regular polygon is

14 0^{\circ}

greater than the measure of one of its exterior angles. How many sides does this polygon have?

A 12

B 15

C 18

D 20

E 24

Show Solution

Let $n$ be the number of sides of the regular polygon.

The measure of an interior angle $I$ of a regular $n$ -sided polygon is given by the formula: $I = (n - 2) \times 180/ n$ .
The measure of an exterior angle $E$ of a regular $n$ -sided polygon is given by the formula: $E = 360/ n$ .
According to the problem, the interior angle is $14 0^{\circ}$ greater than the exterior angle:
$I = E + 140$
Substitute the formulas for $I$ and $E$ into the equation:
$(n - 2) \times 180/ n = 360/ n + 140$
To eliminate the denominator $n$ , multiply every term by $n$ :
$(n - 2) \times 180 = 360 + 140 n$
Distribute $180$ on the left side:
$180 n - 360 = 360 + 140 n$
Subtract $140 n$ from both sides:
$40 n - 360 = 360$
Add $360$ to both sides:
$40 n = 720$
Divide by $40$ :
$n = \frac{720}{40}$
$n = 18$
Thus, the polygon has 18 sides.

Answer: C

Q2 Hard

In parallelogram

P QR S

, the measure of

∠ P

(5 a - 10)^{\circ}

and the measure of

∠ Q

(3 a + 30)^{\circ}

. What is the measure of

∠ S

8 0^{\circ}

9 0^{\circ}

10 0^{\circ}

11 0^{\circ}

12 0^{\circ}

Show Solution

In a parallelogram, consecutive angles are supplementary, meaning their sum is $18 0^{\circ}$ . Angles $P$ and $Q$ are consecutive angles.

So, $m ∠ P + m ∠ Q = 18 0^{\circ}$ .
Substitute the given expressions for $m ∠ P$ and $m ∠ Q$ :
$(5 a - 10) + (3 a + 30) = 180$
Combine like terms:
$8 a + 20 = 180$
Subtract $20$ from both sides:
$8 a = 160$
Divide by $8$ :
$a = 20$
Now, find the measure of $∠ Q$ :
$m ∠ Q = 3 a + 30 = 3 (20) + 30 = 60 + 30 = 9 0^{\circ}$
In a parallelogram, opposite angles are equal. $∠ S$ is opposite $∠ Q$ .
Therefore, $m ∠ S = m ∠ Q = 9 0^{\circ}$ .
(As a side note, since one angle is $9 0^{\circ}$ , and consecutive angles sum to $18 0^{\circ}$ , all angles must be $9 0^{\circ}$ , meaning $P QR S$ is a rectangle.)

Answer: B

Q3 Hard

A certain regular polygon has exactly 8 lines of symmetry. If it is rotated about its center by

x

degrees, it maps onto itself for the first time before completing a full rotation. What is the sum of

x

and the number of sides of the polygon?

A 45

B 53

C 60

D 75

E 90

Show Solution

1. Determine the number of sides ( $n$ ): For any regular polygon, the number of lines of symmetry is equal to the number of its sides. Since the polygon has exactly 8 lines of symmetry, it must be a regular octagon, meaning it has $n = 8$ sides.
2. Determine the smallest angle of rotational symmetry ( $x$ ): The smallest angle of rotational symmetry (the angle $x$ by which it maps onto itself for the first time) for a regular $n$ -sided polygon is given by the formula $36 0^{\circ} / n$ .

For this octagon, $x = 36 0^{\circ} /8 = 4 5^{\circ}$ .
3. Calculate the sum: The question asks for the sum of $x$ and the number of sides of the polygon.

Sum = $x + n = 45 + 8 = 53$ .

Answer: B

Q4 Hard

The diagonals of a rhombus measure

12

cm and

16

cm. What is the perimeter of the rhombus?

20

32

40

48

60

Show Solution

1. Properties of a rhombus: The diagonals of a rhombus bisect each other at right angles. This property forms four congruent right-angled triangles within the rhombus, where the legs of each triangle are half the lengths of the diagonals, and the hypotenuse is a side of the rhombus.
2. Calculate half-diagonals:

Half of the first diagonal = $\frac{12}{2} = 6$ cm.
Half of the second diagonal = $\frac{16}{2} = 8$ cm.
3. Find the side length ( $s$ ): Let $s$ be the side length of the rhombus. Using the Pythagorean theorem ( $a^{2} + b^{2} = c^{2}$ ) for one of the right-angled triangles formed by the half-diagonals and a side:

$s^{2} = 6^{2} + 8^{2}$
$s^{2} = 36 + 64$
$s^{2} = 100$
$s = 100$
$s = 10$ cm.
4. Calculate the perimeter: A rhombus has four equal sides. The perimeter $P$ is $4 \times s$ .

$P = 4 \times 10 = 40$ cm.

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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