Tuesday, June 5, 2012

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 :

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.
#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        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