1. Interior Angle Measure Lesson and Questions
Watch ME As a SLIDESHOW

2. SKILL # 1 Finding the SUM of the interior angles of any polygon.
Our 1st job will be to find the SUM of the interior angles for any shaped polygon!!
360
degrees
540
degrees
1080
degrees
180
degrees

3. We should ALL know by now that the angle degrees INSIDE a triangle add up to 180.60
=180 60 60

4. It doesn’t matter what shape your triangle is in
It doesn’t matter what shape your triangle is in!! The sum of the interior angles STILL measures 180.
So what about other polygons, like a square or rectangle??
120 30 30
=180

5. Squares and rectangles have ALL 90 degree angles
Squares and rectangles have ALL 90 degree angles. So 4 x 90 = 360 degrees.
Another way, since we KNOW that ALL triangles have 180 degrees, is to see how many triangles are in a figure, and then multiply the # of triangles by 180!
OR, # of triangles x180 gives us the SUM of the interior angle measures for ANY polygon. 90 90 90 90 90 90 90 90

6. So if you count how many triangles there are inside ANY polygon, and then multiply by 180 degrees, you will know the SUM of the interior angle measures. Let’s look at a rectangle.
So 2 triangles x 180 = 360 degrees inside of a rectangle. We already knew this, BUT this proves that this method works. 1
First we divide our figure into triangles.
180 degrees 2
Which gives us 2 triangles.
180 degrees

7. 3!! Look, there’s a pattern to the # of triangles you can draw
How many triangles are in a pentagon?
Look, there’s a pattern to the
# of triangles you can draw
inside a polygon.
5 Sides = 3 triangles
4 sides = 2 triangles
3 sides = 1 triangle
SO, # of sides of ANY polygon minus 2 equals the number of triangles we can draw no matter how many sides there are.

8. 8 18 That’s 20-2 = 18 triangles. 348 That’s 350-2 = 348 triangles
How many triangles can you draw in a decagon?
That’s 10-2 = 8
a 20-gon?
A 350-gon?
Once we know how many triangles we can draw, IT’S EASY, just multiply by 180 degrees to find the
SUM of the interior angles.
The formula for this is S = ( n-2) *180
18 That’s 20-2 = 18 triangles.
348 That’s = 348 triangles
S = Sum of angle measures
n = # of sides
of the polygon
You either have to memorize this formula, OR just remember to subtract 2 from the # of sides and multiply by 180.

9. Example: Find the measure of the interior angles of the indicated polygon
S = ( n-2) *180
octagon = 8
S = ( 8-2) *180
S= 6 * 180
The SUM of the interior angles of an octagon is 1080 degrees

10. You Try: Find the measure of the interior angles of the indicated polygon
S = ( n-2) *180
13-gon = 13 sides
S = (13-2) * 180
11 * 180
The SUM of the interior angles of a 13-gon is 1980 degrees

11. Practice. Do this problem before you click again to check answer.
Find the measure of the interior angles of the indicated polygon
A polygon has 14 sides. Find the sum of the interior angle measures of this polygon.
Since we know the number of sides, we just need to subtract 2 and multiply by 180 degrees.
14 – 2 = 12 ( this is how many triangles we could draw inside this figure. )
12 * 180 = 2160 degrees.
If we use the equation to solve, we get
S = ( 14-2)180 = 2160 degrees.

12. SKILL # 2
Find the # of sides of any regular polygon if we know the SUM of the angle measures inside.
We’ll use the same formula, S = ( n-2) *180
fill in for S, and solve for n.

13. What is a “REGULAR” polygon??
Regular polygon: a polygon that has ALL the Same side lengths
AND all the same Angle measures inside.
ALL
Same sides,same
angles
ALL
Same sides,same
angles
ALL
Same sides,same
angles
ALL
Same sides,same
angles
When we are dealing with a regular polygon , we can use the Sum of the angle measures to help us find the length of ONE side or ONE of the angle measures inside the figure. We will learn this in the next few slides.
Example: We know a square( which is a “regular” polygon) has a total of 360 degrees of angle measures inside. We also know that each of those angles is 90 degrees and each side is the same length. This is a good example to help you understand that if you have a “regular” polygon, it will have ALL of the same side lengths and ALL of the angle measures inside.

14. They gave us that S = 1440, sub it in for S. 1440 = ( n-2) *180
Example: The sum of the measures of the interior angles of a regular polygon is 1440 degrees. Classify the polygon by the number of sides.
S = ( n-2) *180
They gave us that S = 1440, sub it in for S.
1440 = ( n-2) *180
Solve for n
1440/180 = n-2
8 = n-2 , then add 2 to both sides and get n=10.
What’s a 10 sided figure, a decagon!!
Divide by 180
1440/180 = 8

15. - They gave us that S = 3600, sub it in. 3600 = ( n-2) *180
You Try: The sum of the measures of the interior angles of a regular polygon is 3600 degrees. This regular polygon has how many sides?
- S = ( n-2) *180
- They gave us that S = 3600, sub it in.
3600 = ( n-2) *180
Solve for n
3600/180 = n-2
20 = n-2 , then add 2 both sides and get n=22.
What’s a 22 sided figure, a 22-gon!!
Divide by 180
3600/180 = 20

16. Practice. Do this problem before you click again to check answer.
QUESTION: The sum of the measures of the interior angles of a regular polygon is 3060 degrees. This regular polygon has how many sides?
S = ( n-2) *180
They gave us that S = 3060, sub it in.
3060 = ( n-2) *180
Solve for n
3060/180 = n-2
17 = n-2 , then add 2 both sides and get n=19.
What’s a 22 sided figure, a 19-gon!!
Divide by 180
3060/180 = 17

17. SKILL # 3
Find the angle measure of ONE angle inside any regular polygon if we know the SUM of the angle measures inside.
Step 1: Figure out # of sides.
Step 2: Divide Sum of interior angle measures by # of sides.
Example: Let’s look at our square again. We know a square( which is a “regular” polygon) has a total of 360 degrees and 4 sides.
So 360 degrees divided by 4 = 90 degrees.
One angle measure inside a square is 90 degrees. And we know this is true.
SEE, dividing total angle measure by # of sides WORKS!!
Example: Let’s look at a triangle. We know a regular triangle has a total of 180 degrees and 3 sides.
So 180 degrees divided by 3 = 60 degrees.
One angle measure inside a regular triangle is 60 degrees. SEE!! It works.

18. Example: The sum of the measures of the interior angles of a regular polygon is 1440 degrees. Find the measure of ONE of the angles inside this figure.
S = ( n-2) *180
They gave us that S = 1440, sub it in for S.
1440 = ( n-2) *180
Solve for n
1440/180 = n-2
8 = n-2 , then add 2 to both sides and get n=10. So 10 Sides
1440 divided by 10 = 144 degrees.
So ONE interior angle of a decagon is 144 degrees.
Because it’s regular, ALL angles inside are the same,
144 degrees
Step1: Figure how many sides this polygon has
Divide by 180
1440/180 = 8
Step 2: Divide the Sum of the angle measures they gave us by the # of sides.
Remember: This ONLY works with regular polygons.

19. You Try: The sum of the measures of the interior angles of a regular polygon is 3600 degrees. Find the measure of ONE of the angles inside this figure.
S = ( n-2) *180
They gave us that S = 3600, sub it in for S.
3600 = ( n-2) *180
Solve for n
1440/180 = n-2
20 = n-2 , then add 2 to both sides and get n=22. So 22 Sides
3600 divided by 22 = degrees.
So ONE interior angle of a 22-agon is degrees.
Because it’s regular, ALL angles inside are the same,
163.6 degrees
Step1: Figure how many sides this polygon has
Divide by 180
3600/180 = 20
Step 2: Divide the Sum of the angle measures they gave us by the # of sides.
Remember: This ONLY works with regular polygons.

20. Practice. Do this problem before you click again to check answer.
QUESTION #3: The sum of the measures of the interior angles of a regular polygon is 540 degrees. Find the measure of ONE of the angles inside this figure.
Step1: Figure how many sides this polygon has
S = ( n-2) *180
They gave us that S = 540, sub it in for S.
540 = ( n-2) *180
Solve for n
1440/180 = n-2
3 = n-2 , then add 2 to both sides and get n=5. So 5 Sides
540 divided by 5 = 108 degrees.
So ONE interior angle of a pentagon is 108 degrees.
Because it’s regular, ALL angles inside are the same,
108 degrees
Divide by 180
540/180 = 3
Step 2: Divide the Sum of the angle measures they gave us by the # of sides.
Remember: This ONLY works with regular polygons.

21. Now, practice your skills at the following websites.
Practice Site #1
Practice Site #2
Still struggling. Check out the COOL MATH Lesson
on interior angle measures.
COOL Math Lesson
NOW, If you feel like you’re ready, do the questions on the next slide.

22. Practice problems
Directions. Write these problems down and SOLVE with work, put in your class IN BOX behind my desk. This is worth an offline completion grade. You may use your notes.
What is the SUM of the interior angles of a regular 15-gon?
How many sides does a regular polygon have with an interior angle sum of 5940 degrees?
Classify a polygon by it’s sides if its interior angle measure is 1260 degrees.
What is the SUM of the interior angles of a dodecagon( Google it if you have to).
A regular polygon has 720 degrees of total angle measures inside. What is the measure of ONE angle inside this polygon?
Use your #2 answer. What is the measure of one angle inside this polygon?
When you’re done, correct in PEN from the next slide before you pass them in.

23. Answers: What is the SUM of the interior angles of a regular 15-gon?
S = (15-2)*180 =>> 13*180 = 2340 degrees
2. How many sides does a regular polygon have with an interior angle sum of 5940 degrees?
5940 = ( n-2)*180 =>> 5940/180 = n-2
=>> 33 = n-2 n = 35, so a 35-gon
3. Classify a polygon by it’s sides if its interior angle measure is 1260 degrees.
1260 = ( n-2)*180 =>> 1260/180 = n-2 =>> 7 = n-2
n = 9, so a nonagon
What is the SUM of the interior angles of a dodecagon.
S = (12-2)*180 =>> 10*180 = 1800 degrees
A regular polygon has 720 degrees of total angle measures inside. What is the measure of ONE angle inside this polygon?
First, find how many sides:
720=(n-2)*180 =>> 720/180 = n-2 =>> 4 = n-2 =>> n = 6
Now divide total angle measures by sides:
720 degrees divided by 6 sides = 120 degrees for ONE angle.
Use your #2 answer. What is the measure of one angle inside this polygon? / 35 = degrees