Question

Find the sum of the measures of the interior angles of each of the following convex polygons: a pentagon, an octagon, a dodecagon, a forty-sided polygon, a fifty-two sided polygon, a hundred sided polygon.

Answer 1

Answer:

pentagon=540°

octagon= 1080°

decagon= = 1440°

forty-sided polygon= 6840°

fifty-two-sided polygon=9000°

hundred-sided polygon=17640°

Step-by-step explanation:

To find the measure of the sum of the interior angle of a polygon, we will simple use the formula;

sum of interior angle = (n-2)180

where n is the number of sides of the polygon

We are ask to find the sum of the interior angle of a pentagon, octagon, decagon, a forty-sided polygon, a fifty-two-sided polygon and a hundred-sided polygon.

First we need to define some terminology;

A penatgon is a five(5)-sided polygon

An Octagon is an eight(8)-sided polygon

A Decagon is a ten(10)-sided polygon

We can now proceed to solve

sum of interior angle = (n-2)180

Sum of interior angle of a pentagon = (5-2)180

                                                             = 3 × 180

                                                             =540°

Sum of interior angle of a pentagon = 540°

Sum of interior angle of an octagon  = (8-2)180

                                                             = 6 × 180

                                                             =1080°

Sum of interior angle of an octagon = 1080°

Sum of interior angle of a decagon  = (10-2)180

                                                             = 8 × 180

                                                             =1440°

Sum of interior angle of a decagon = 1440°

Sum of interior angle of a forty-sided polygon  = (40-2)180

                                                                               = 38 × 180

                                                                                =6840°

Sum of interior angle of a forty-sided polygon = 6840°

Sum of interior angle of a fifty-two-sided polygon  = (52-2)180

                                                                                    = 50 × 180

                                                                                     =9000°

Sum of interior angle of a fifty-two-sided polygon = 9000°

Sum of interior angle of a hundred-sided polygon  = (100-2)180

                                                                                     = 98 × 180

                                                                                      =17640°

Sum of interior angle of a hundred-sided polygon = 17640°

Answer 2

This likely has a pattern. Lets see if we can find it.

The sum of a triangle is 180
The sum of a 4 sided figure is 360
The sum of a 5 sided figure is 3*180 = 540
The sum of a 6 sided figure is 4*180 = 720
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The pattern looks like it it (( n-2 )*180) for any polygon where n is the number of sides of the polygon.

You can convince yourself (but this is not a formal proof) by starting with a 6 sided figure. Pick one point on the figure where 2 sides meet. Draw diagonals to all the points not connected to the point you've chosen. You should be able to get 4 such diagonals. You have created 4 triangles.

4*180 = 720 just what your estimate said.d

8 sides will give 6*180 = 1080 degrees and if you keep going on the above table you will see 1080 come up for 8. I'd post a diagram but I don't know how to upload it.


This procedure is extremely important. There is a new math around where these kinds of problems are solved by reducing them to much simpler questions, and getting an answer for them. The pattern is what you are looking for.

Now you can do the rest of the questions.
Pentagon: (5 - 2) * 180 = 
Octagon: (8 - 2) * 180 =
dodecagon (12 - 2)*180=
40 sides 38*180 =  
100 sides 98 * 180 =