# SAT Mathematics : Working with Polygons with More than 4 Sides

## Example Questions

### Example Question #1 : Advanced Polygons & 3 D Shapes

What is the sum of the angles of an octagon?

Explanation:

The sum of the interior angles of an n-sided polygon can be calculated as:

Which you can test for yourself: a triangle (3 sides) has 180 degrees: .

A rectangle has 360 degrees.

So in this case, where , you'd calculate as:

.

### Example Question #2 : Advanced Polygons & 3 D Shapes

If  represents the measure of an interior angle of a regular octagon and  represents the measure of an interior angle of a regular pentagon, what is the ratio ?

Explanation:

The total interior angle of a polygon with  sides equals . Thus, the total interior angle of an octagon is  and the total interior angle of a pentagon is . A regular polygon has the property that all interior angles are congruent, so an interior angle of a regular octagon is . Similarly, the interior angle of a pentagon is  is equivalent to .

For quicker math-by-hand, recognize that you're being asked about a ratio. If you set up the measure of one angle of a regular octagon as  (the total measure divided by the number of angles) and the measure of one angle of a regular pentagon as , then notice that with a 180 term multiplied in the numerator of each portion of the ratio, the 180s can factor out. Then you're just taking the ratio of  to , which nets quickly to .

### Example Question #3 : Advanced Polygons & 3 D Shapes

In a particular n-sided polygon, all sides and all angles are congruent. If each angle measures 144 degrees, what is the value of ?

10

6

8

12

10

Explanation:

In order to effectively tackle this question, you’ll want to be comfortable with the fact that the sum of the interior angles of any given polygon will always equal , where n is the number of sides in that polygon. This should be fairly straightforward to prove, as we can always break up any given polygon into two fewer triangles than the number of sides of the polygon, and the sum of the interior angles of a triangle is 180 degrees.

In this example, we’re being asked to think about that formula a little differently. We’ve been told that each angle measures 144 degrees, so the sum of the interior angles should be equal to , where n is the number of sides. We can also express the sum of the interior angles as , as we just mentioned. So, all we need to do from here is set those two expressions equal to one another, since all we’ve done is said the same thing two different ways!

So, if

, we can distribute to arrive at

and get all our knowns to one side and all our unknowns to the other side to find that

If we divide both sides of our equation by 36, we can see that

.

### Example Question #4 : Advanced Polygons & 3 D Shapes

The following is a regular pentagon. What is the measure of angle ?

Explanation:

This question tasks us to identify and apply knowledge of the sum of interior angles of a polygon, as well as the regular nature of our shape to solve for individual angles. The sum of the interior angles of a triangle is , where n is the number of sides in the polygon. So, the sum of the interior angles of a pentagon is . Thus, each individual angle is .

If we know that angle EDC is . Since triangle EDC is an isosceles triangle made up of the two identical sides ED and CD, angles DEC and ECD must be the same and are each . Since AED is 108°, and CED is , AEC, labeled  in our diagram, must be .

### Example Question #5 : Advanced Polygons & 3 D Shapes

A regular polygon with interior angles of  each has how many more sides than a regular polygon with interior angles of  each?

8

5

10

3

3

Explanation:

To answer this question, we’ll need to use our understanding of the interior angles of a regular polygon. Since the sum of the interior angles of a polygon is as follows:

Where  is the number of sides and, thus, the number of vertices in the polygon, if all angles of the polygon are the same (as is the case in a regular polygon), each angle will be , since the  each angle.

So, if , the number of sides must be 5, since:

(*notice - at this step, we’ve just expressed the sum of the angles two different ways, as (each angle)*(number of angles), and using the formula where the sum of the angles is We could have begun our work at this step, if this setup feels more intuitive to you!)

Similarly, if each angle in a regular polygon is , that polygon must be 8-sided, since:

So, an 8-sided polygon has 3 more sides than a 5 sided polygon. (Once you’ve completed the math - be sure to answer the question you’ve been asked!)

### Example Question #6 : Advanced Polygons & 3 D Shapes

What is the measure, in degrees, of each interior angle of a regular convex polygon that has twelve sides?

120

175

150

135

150

Explanation:

The sum of the interior angles, in degrees, of a regular polygon, is given by the formula , where  is the number of sides. The problem concerns a polygon with twelve sides, so we will let . The sum of the interior angles in this polygon would be .

Because the polygon is regular (meaning its sides are all congruent), all of the angles have the same measure. Thus, if we divide the sum of the measures of the angles by the number of sides, we will have the measure of each interior angle. In short, we need to divide  by , which gives us .

### Example Question #7 : Advanced Polygons & 3 D Shapes

In the figure above, polygon ABDFHGEC is a regular octagon. What is the measure, in degrees, of angle FHI?

50

45

30

40

45

Explanation:

Angle FHI is the supplement of angle FHG, which is an interior angle in the octagon. When two angles are supplementary, their sum is equal to 180-degrees. If we can find the measure of each interior angle in the octagon, then we can find the supplement of angle FHG, which will give us the measure of angle FHI.

The sum of the interior angles in a regular polygon is given by the formula , where  is the number of sides in the polygon. An octagon has eight sides, so the sum of the angles of the octagon is . Because the octagon is regular, all of its sides and angles are congruent. Thus, the measure of each angle is equal to the sum of its angles divided by 8. Therefore, each angle in the polygon has a measure of . This means that angle FHG has a measure of 135-degrees.

Now that we know the measure of angle FHG, we can find the measure of FHI. The sum of the measures of FHG and FHI must be 180-degrees because the two angles form a line and are supplementary. We can write the following equation:

Measure of FHG + measure of FHI = 180

135 + measure of FHI = 180

Subtract 135 from both sides.

Measure of FHI = 45 degrees.

### Example Question #8 : Advanced Polygons & 3 D Shapes

What is the average (arithmetic mean) of all 15 interior angles of a quadrilateral, pentagon, and hexagon?

Explanation:

The 5 angles of a pentagon add to 540

The 6 angles of a hexagon add to 720

### Example Question #9 : Advanced Polygons & 3 D Shapes

Each interior angle of a regular polygon has measure . How many sides does the polygon have?

Explanation:

The easiest way to work this is arguably to examine the exterior angles, each of which forms a linear pair with an interior angle. If an interior angle measures , then each exterior angle, which is supplementary to an interior angle, measures

The measures of the exterior angles of a polygon, one per vertex, total ; in a regular polygon, they are congruent, so if there are  such angles, each measures . Since the number of vertices is equal to the number of sides, if we set this equal to  and solve for , we will find the number of sides.

Multiply both sides by :

The polygon has 72 vertices and, thus, 72 sides.

### Example Question #10 : Advanced Polygons & 3 D Shapes

A regular polygon has a measure of  for each of its internal angles.  How many sides does it have?