Polygon Puzzle: How Many Degrees Are in a Polygon?

How many degrees are in the interior angles of a pentagon? A hexagon? An octagon? Or any polygon? Keep on reading The Math Dude to learn how to solve this polygon puzzle!

Jason Marshall, PhD
4-minute read
Episode #227

Hmm, very interesting. Now take a minute or two and see what you can come up with.

Any ideas yet?...

Isn’t this fun?...

OK, ready for the answer?...

PentagonHopefully, you noticed that a 5-sided regular pentagon can be broken up into three triangles. To see this, draw a line from the top corner of your pentagon down to each of its opposing lower corners. Once you draw this picture, you’ll see that a pentagon is indeed composed of three triangles. And from that you should see that a pentagon must contain 3 x 180º = 540º.

So if the interior angles of a 3-sided triangle add up to 180º, those of a 4-sided square add up to 360º, and those of a 5-sided pentagon add up to 540º. It looks like we keep adding 180º for each additional side. Does this trend hold up as we move to polygons with more and more sides?

How Many Degrees In a Hexagon, Octagon, Or Any Polygon?

To see, let’s make the next logical leap and move from thinking about a 5-sided pentagon to a 6-sided hexagon. Again, I encourage you to take a few minutes to think about what we’ve done so far, and see if you can puzzle out the total number of degrees contained in the interior angles of a hexagon.

Do you see the trick?...

Same idea as before...

Got it?...

To answer this question, draw yourself a lovely regular hexagon, and then draw three lines from one of its corners to the three opposing corners. When you do this, you should see that a hexagon can be broken up into four triangles. And, therefore, you should see that a hexagon must contain 4 x 180º = 720º.

HexagonDo you see a trend yet? If you really wanted to, you could continue this game for the 7-sided heptagon, the 8-sided octagon, and so on. In fact, you can do it for any and every polygon.

When you do that, you’ll find that an n-sided polygon (where n simply represents the number of sides in the polygon) can be broken down into n–2 triangles. Which means that the interior angles of an n-sided polygon will always add up to (n – 2) x 180º.

Which, lo and behold, is the answer to today’s great polygon puzzler!

Wrap Up

OK, that’s all the math we have time for today.

For more fun with math, please check out my book, The Math Dude’s Quick and Dirty Guide to Algebra. And remember to become a fan of The Math Dude on Facebook, where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too.

Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!

Soccer ball image courtesy of Shutterstock.


About the Author

Jason Marshall, PhD

Jason Marshall is the author of The Math Dude's Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.