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Course: Geometry (all content)>Unit 3

Lesson 5: Angles with polygons

Sum of the exterior angles of a polygon

Learn a simple and elegant way to find the sum of the exterior angles of any convex polygon. You will see how to redraw the angles adjacent to each other and form a circle. Then you will discover that the sum of the exterior angles is always 360 degrees. Watch this video to master this important skill in geometry. Created by Sal Khan.

Want to join the conversation?

• Is 360 degrees for all polygons ?
• You've been lied to.

It will actually work for any polygon, as long as you remember to use negative numbers for the concave angles. The answer is always 360°, and you can prove it by drawing a shape something like https://goo.gl/photos/zFSQs2XwDxwqKGwZ8 (sorry for the terrible picture). The -90° makes up for the two extra 45°s, and so it comes out even.
• I was confused by the definition of "exterior angles".

If the interior angle of one corner is, say, 90 degrees (like a corner in a square) then shouldn't the exterior angle be the whole outside of the angle, such as 270? Why is only 90 degrees counted for the exterior angle of a corner instead of 270?

In other words, exterior corners look like they are always greater than 180, but we subtract the 180. Why?
• It's just the way exterior angles are defined.

From the wikipedia article: "an exterior angle (or external angle) is an angle formed by one side of a simple, closed polygon and a line extended from an adjacent side."

See: http://en.wikipedia.org/wiki/Exterior_angle
• is a star considered as a convex polygon?
• No, it is concave because it has an angle greater than 180 degrees( also known as a reflex angle.
• What is the definition of a convex polygon?
• A convex polygon is a many-sided shape where all interior angles are less than 180' (they point outward).

Examples of convex polygons:
- all triangles
- all squares
An octagon with equal sides & angles (like a stop sign) is a convex polygon; the pentagons & hexagons on a soccer ball are convex polygons too.

There are also concave polygons, which have at least one internal angle that is greater than 180' (points inward).

Examples of concave polygons:
- a star
- a cross
- an arrow

To tell whether a shape is a convex polygon, there's an easy shortcut: just look at the pointy parts (or "vertices"). If every single one of the points sticks out, then the polygon is convex!

Hope this helps!
• The sum of interior angles of a regular polygon is 540°. Calculate the size of each exterior angle. *
• You need to know four things. the sum of all exterior angles equal 360, allexterior angles are the same, just like interior angles, and one exterior angle plus one interior angle combine to 180 degrees. Finally, the sum of interior angles is found with the formula 180(n-2) where n is the number of angles. since it tells us the sum we can find the number of angles.

180(n-2)=540
n-2 = 3
n = 5

So five corners, which means a pentagon. this means there are 5 exterior angles. since they all have to add to 360 you can divide 360/5 = 72. You can also check by adding one interior angle plus 72 and checking if you get 180.

total interior angle is 540, there are 5 angles so one angle is 108. 108+72 = 180 so this confirms that one exterior angle is 72 degrees.

Let me know if aything didn't make sense.
• At the very start of the video, Sal references to a video done "several videos ago". Could someone please link the video he's talking about?
• What is the meaning of anticlockwise?
• it is the same as counter-clockwise, which is the opposite of the direction the hands of a clock go.

Or if you start at the top of a circle, and go down and around to the left.
• What is concave and convex? (If you see this and you know the answer please answer. Thank you!)
(1 vote)
• A concave lens "caves in". We can extend this to geometry as well. Concave polygons will have a part or parts that are sticking inwards, instead of being outwards. A Concave polygon could be a boomerang shape, while a convex polygon would be any regular polygon, since it doesn't cave in. The formal definition for a polygon to be concave is that at least one diagonal (distance between vertices) must intersect with a point that isn't contained in the polygon.
• So, how would you find the exterior angle of a reflex angle? And does the sum of the exterior angles of a concave polygon always equal 360, or does it equal 540?