Polygon Part II: Interior/Exterior angles, Central Angles and Diagonals and Practice Problems

Learn the Basics here: Polygon Part I

Some important notes:

#1: Sum of all the interior angle of an n-sided polygon is \((n-2) * 180\)

#2: To find one interior angle of an n-sided regular polygon, you use :

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

or \({180 -\frac{360}{n}}\)", the latter will always give you one exterior angle of a regular n-gon.

#3:  Interior and its exterior angles are supplementary to each other. 

Interior angle A + exterior angle A = 180 degrees. 

#4: In every convex polygon, the exterior angles always add up to 360 degree.

Proof without words

#5: The central angle of a regular n-sided polygon is : \({\frac{360}{n}}\), same method as finding the exterior angle of a regular n-gon. Get more details on central angle here.

#6: Since \({\frac{360}{n}}\) will give you one exterior angle of a regular n-gon , 360 divided by one exterior angle of a regular n-gon will give you how many sides of that polygon. 

#7: To find how many diagonals an n-sided polygon has, you use: 

\(nC2 - n\) (Any two vertices except the sides will render one diagonal; however, order doesn't matter, thus choose 2.)

b. \( \frac{n(n-3)}{2}\) Any vertex, except its neighboring vertices and itself, can connect with other vertex to form a diagonal and there are n vertices; however, since order doesn't matter, AC is the same as CA so you divide the number by 2.

Questions to ponder: (Answer and solutions below.)

#1: The sum of the diagonals of two regular polygons is 44 and the sum of each of their interior angles is 264, what is the sum of their sides?

#2: If an exterior angle of a regular n-gon is 72, what is the measure of its interior angle? How many diagonals does that n-gon have? 

#3: What is one interior, exterior angle as well as how many diagonals are there for a 20 sided regular polygon?

#4: If each of the exterior angle of a regular polygon is 30, how many sides does that polygon have? 

#5: If the sum of all the interior angles of a polygon is 1440, how many sides does the polygon have? 

#6: How many degrees are there in the sum of a pentagon + a heptagon + a nonagon?

#7: The sum of the interior angles of a regular polygon is 720, what is the measure of one interior angle of that polygon? 

#8: 2000 Mathcounts State Target #8: (Check out Mathcounts here) : The total number of degrees in the sum of the interior angles of two regular polygons is 1980. The sum of the number of diagonals in the two polygons is 34. What is the positive difference between the numbers of sides of the two polygons?

                                                                               

# 9: Both pentagon and hexagon are regular. What is angle ABF?

#10:   B is the center of this regular pentagon. 

What is angle A, B and C? 

Answer key and solutions:

#1:  16: Learn the common polygon property by heart and check what the question is asked for.

In this case, let's see a few common polygons:

pentagon   5 sides     diagonals    5     interior angle  108 degrees

hexagon    6 sides     diagonals    9     interior angle  120  degrees

octagon     8 sides     diagonals   29    interior angle  135  degrees

nonagon    9 sides     diagonals   27    interior angle  140 degrees

decagon    10 sides   diagonals   35    interior angle  144 degrees

so the two polygons asked are hexagon and decagon, the sum of their sides are 6 + 10 = 16

#2: Interior and exterior angles are supplementary so 180 - 72 = 108 degrees  for its interior angle.

It's a pentagon and there are 5 diagonals in a pentagon.

#3: \(\frac{360}{20}\) = 18 degrees for the exterior angle

180 - 18 = 172 degrees for the interior angle of a 20-sided polygon.

#4: \(\frac{360}{30}\) = 12 sides 

#5:\(\frac{1440}{180}\) = 8 ; 8 + 2 = 10. It's a decagon (10 sides) -- when you find the sum of the interior angles you use (n - 2) * 180 so now you do the reverse.

#6: [ (5-2) + (7-2) + (9-2)] x 180 = (3 + 5 + 7) x 180 = 5 x 3 x 180 = 2700 degrees.

#7:\(\frac{720}{180}\)= 4 and 4 + 2 = 6 so this is a regular hexagon and one of its interior is \("\frac{720}{6}\) = 120 degrees. 

Or you can also do \(\frac{720 + 360}{180}\)

= 6 because all the interior angles and their exterior angles are supplementary and the sum of any exterior angle of a convex polygon is 360 degrees.

Using this method, you get how many sides instantly.

#8: This one is tricky because there are actually two different polygon pairs that the total number of degrees add up to 1980 degrees. One is a regular hexagon and a nonagon.  (720 + 1260)

Another is a heptagon and an octagon (900 + 1080). Looking at the sum of their diagonals, the latter is the right pair.  (14 + 20 = 34) so the answer is 8 - 7 = 1

#9: 360 - 108 - 120 = 132 degrees. 

#10: To find the central angle B, you do  \(\frac{360}{5}\)

= 72 degrees. BA and BC are both radius so the angle is

congruent. \(\frac{180-72}{2}\) = 54 degrees

Prime Factorization : Part II

   Learn basic facts here.











Another fun trick is to find the total number of factors any number has. To do this, take the prime factorization of the number, add 1 to each of the exponents of the prime factor, and multiply these new numbers together.

Example:

#1: 12 = 2² x 3¹
(2+1) x (1+1) = 3 x 2 = 6
Therefore, 12 has 6 factors. {1, 2, 3, 4, 6, 12}

#2: 8= 2³ , (3+1) = 4 Therefore, 8 has 4 factors. {1, 2, 4, 8}

Problems to ponder: answers and solutions below.
1. What is the smallest number with 5 factors?
2. What is the smallest number with 6 factors?
3. What is the smallest number with 12 factors?
4. How many factors does any prime number have?
5. How many even factors does 36 have? (tricky question)
6. How many perfect square factors does 144 have? ( don’t just list them, think how you get the answers)









Answers:
1. 16  Since 5 itself is a prime, the smallest number that has 5 factors would look like this: n⁴  put in the smallest prime number and the answer is  2⁴ = 16 

2. 12  The number 6 can be factor as 6 x 1 or 3 x 2 so you can have either n⁵ or x² * y¹
so either 2⁵ or 2² * 3 so the answer is 12. 

3. 60  12 = 12 x 1 = 6 x 2 = 4 x 3 = 3 x 2 x 2  The last would give you the smallest number which is  
2² * 3¹ * 5¹ = 60

4. 2 Any prime number has two factors, which are 1 and itself. The smallest prime number is 2, which is the oddest prime -- the only even number that is a prime. (Why??)

5. 6  Any even number multiples another integer will give you another even number. To get only even factors, you  need to always leave the smallest even number, which is 2, with all the other factors. 
36 = 2² * 3²
= 2 (2¹ *3²)  There are(1 + 1) ( 2 + 1 ) = 6 even factors

6. 6  The number 144 = 2⁴ * 3²
To get only square factors, you need to keep the smallest square number of all the prime numbers, leave out the others that are not square and find how many factors that new arrangement has. 
( 2²)² * (3²)¹  There are (2 + 1) ( 1 + 1)= 6 square factors.

See what they are on the left.

Polygons: Part I: Interior and Exterior angles, Sum and Diagonals

Some helpful links: Looking at Polygons by Kadie Armstrong, a mathematician.

Year 7 Interactive Maths: Polygons

Notes below. It takes a few seconds to download the pages. Please be patient.
Send me feedback or notify me of errors. Answer key is below for self-study.Polygons, Interior Angles, Exterior Angles, Sum













Answer Key to Polygons, Interior Angles, Exterior Angles and Sum

Math Cartoons from Calvin and Hobbes by Bill Waterson

 by Bill Watterson 

Solutions:
Calvin and Hobbes Cartoon 2You and Mr. Jones are both traveling towards each other. Mr. Jones is traveling at 35 mph, and you are traveling at 40 mph. In other words, every hour, the distance between you to lessens by 35 + 40 = 75 miles.

You can therefore think of this problem as one car covering 50 miles while going at 75 mph. The time this will take is distance/rate, or 50/75 = 2/3 hours.

2/3 hours times 60 minutes/hour = 40 minutes.

Add this to the starting time of 5:00, and you will pass Mr. Jones at 5:40.

Calvin and Hobbes Cartoon 

There are actually two answers ( 10 or 10/3 ) to this problem, because there are two different ways the points can be lined up.

If the points are in the order A-B-C and BC = 5 inches (given) , which means AB must also be 5 inches. This way, AC will be 10 inches, which satisfies the condition that AC = 2AB.

If the points are in the order B-A-C and BC = 5 inches, which means that AB + AC = 5 inches (make sure you see why – the parts of a whole add up to the whole) and, since AC = 2AB, AB = 5/3 and AC = 10/3 inches.

Once again, Jack and Joe are driving towards each other. Since Jack is driving at 60 mph, and Joe is driving at 30 mph, the distance between them is decreasing by 90 mph. Therefore, the combined result is one car driving at 90 mph.

Distance = rate x time, so distance = 90 mph x 10/60 hours (10 mins is 10/60 or 1/6 of an hour) The answer is 15 miles.

Here are some other problems that you can try. Have fun. (Answer key and solutions below.)

#1: Mary drove 50 miles at an average speed of 40 miles per hour to her friend's house, and drove back home on the same route at an average speed of 60 miles per hour. What was her average speed for the entire trip, in miles per hour? 

#2: John traveled 60 miles an hour and Edwards traveled 45 miles an hour. Both departed at the same spot in the same direction. After 45 minutes of driving, how far were they apart?

#3: A car drives in a circular path with a diameter of 800 feet. If the car completes exactly 2 laps each minute, what is the car's speed, in feet per second? Express your answer to the nearest whole number.

#4: If it takes you an hour total to go to work and your driving speed is 30 miles per hour to work and 
45 miles per hour back home, how far away is your home to the office where you work? 

#5: Driving to work, if your speed is 20 miles per hour, you'll be 10 minutes late; however, if your driving speed is 30 miles per hour, you'll be 10 minutes earlier. What speed will get you to work on time?

#6: If you walk from your house to the school at the rate of 4 km/hour, you'll reach the school 15 minutes earlier than the scheduled time. If you walk at the rate of 3 km/hour, you'll reach the school 15 minutes late. What is the distance of the school from your house? 















 Answer key and solutions: 
#1. Solution I:
Average speed, or rate, is equal to total distance divided by total time: R = D/T.
We know the total distance: Mary drives 50 miles to her friend’s house, and 50 miles back, for a total distance of 100 miles.

Finding the total time is a little trickier. Here, the equation has to be split up into two parts. Just as R = D/T, RT = D, and T = D/R (you can find these equations simply by rearranging the variables). Therefore, the time it takes for Mary to reach her friend’s house can be found as follows:

T = D/R
T = 50 miles / 40 mph
T = 5/4 hours

And similarly, the time it takes for Mary to return home can be found as follows:

T = D/R
T = 50 miles / 60 mph
T = 5/6 hours

Combining these two results, we find that her total time is 5/4 hours + 5/6 hours, or 25/12 hours.
Now we have the total distance and the total time, so her average speed for the entire trip is 100 miles / (25/12 hours), or 48 mph.

Solution II: 

Two segments of the same length (to work and back home).
Use harmonic mean:   mph

#2. 11.25 , very straight-forward questions, just remember that the time given is in minutes, not hours, so you need to convert minutes to hours.

#3. The first step in any rate problem is to see which of the three variables – rate, distance, and time – we have.

Since we’re looking for the car’s speed, or rate, that’s going to be our unknown. The question tells us that the car completes 2 laps each minute, which gives us distance and time. Notice that the problem wants the rate in feet per second, so we’ll need to do a few conversions at the end.

First, to find distance, we need to convert the phrase ‘2 laps’ into units of feet. The car is on a circular path with diameter 800 feet. In other words, 1 lap is equal to the circumference of a circle with diameter 800 feet, which is equal to 800 π feet.

Therefore, two laps is equal to 1600π  feet.

Now we’ve got total distance. We know that this distance is covered in 1 minute, or 60 seconds. Therefore, the car’s speed is equal to 1600 π feet / 60 seconds, or approximately 84 feet/second.

# 4: Let D be the distance. Times 90 (LCM) on both sides.
3D + 2 D = 90 ; 5D = 90; D = 18 miles

#5: Use harmonic mean [See solution II on question #1], since both time late and early is the same. You can just ignore it. 
The answer is 24 mpr. 

#6: D = RT; We can set up the equation and make the time on both sides = to you'll walk to the school on time. 
    (Remember to convert 15 minutes to or hour)
D = 6 miles

Special Right Triangles: 30-60-90 and 45-45-90 Degrees Right Triangles

Please give me feedback on the comment.

Thanks a lot!!  Mrs. Lin







These are the two most common right triangles.

For 45-45-90 degrees, the ratio is  1 - 1 - √ 2

For 30-60-90 degrees, the ratio is  1 - √ 3 - 2

Other concepts to remember are that in any triangle a. larger angle corresponds to longer side and b.same 
angles have the same side length.

Level 0 skills check practices: 

Find the missing side length (answers below)

I: 45-45-90 degrees right triangle

a. 3 - ___- ___            b. 5 - ___- ___         c.  √ 2 - ___ - ___      d. tricky : ___ - ___ - √3

e. ___ - ___ - 4           f. ___ - ___ - 6

II. 30-60-90 degrees right triangle

a. 3 - ___- ___            b.  ___- ___  - 4        c.  √ 2 - ___ - ___      d. ___ - ___ - 5

e.  ___ - 6 - ___           f. ___ - 3 - ___

Answer key and some notes:

I: 45-45-90 degrees right triangle

a. 3 - 3- 3 √ 2              b. 5 - 5 - 5 √ 2         c.  √ 2 - √ 2  - 2       d.√ 6 /2 - √ 6 /2 - √3

e. 2√ 2 - 2√ 2 - 4         f. 3√ 2 - 3√ 2  - 6

Notes:
a. Given the side length to 45 degrees, the easiest way to get the 90 degree side length is to time that number by √ 2 .

b. Given the side length to 90 degrees, the easiest way to get the 45 degree side length is to divide that number by 2 and then times √ 2 .

II. 30-60-90 degrees right triangle

a. 3 - 3√3 - 6            b.  2 - 2√3 - 4        c.  √ 2 - √ 6  - 2√ 2       d. 2.5 - 2.5√3 - 5

e.  2√3 - 6 - 4√3         f.  √3 - 3 -  2√3

Notes:
a. Given the side length to 30 degrees, the easiest way to get 90 degrees side length is to times 2 to the 30 degree side length. To get the side length to 60 degrees, times √3 to the side length to 30 degrees.

 b. Given the side length to 90 degrees, divide 90 degree side length by 2 to get the side length to 30 degrees.Times √3 to the side length of 30 degrees to get the side length of 60 degrees.

c. Given the side length to 60  degrees, divide that number by 3 and then times √3  to get the side length of 30 degrees. Times 2 to get the side length of 90 degrees.