The sum of the angles of a triangle is 180
For a square it’s 360
For a pentagon it’s 540
For a hexagon it’s 720
Why is that?
In: Mathematics
Adding a side to a shape is the equivalent of adding another triangle somewhere along its perimeter. The new triangle has angles that sum to 180.
Take the polygon in question, pick one point and draw lines out to each other point.
You now have a bunch of triangles.
Square is made of 2 triangles, pentagon is 3, hexagon is 4 etc.
When you add a side to a polygon you add a triangle.
If you think of each shape as being composed of a number of triangles glued together it makes sense. For example a square is two triangles sharing a side. Paste another triangle on to a square and you have a pentagon. Thus the total sum of angles increases by 180 as you progress
Take any corner of a shape.
Imagine travelling down one of the sides to that corner (and for ease imagine going clockwise).
Rather than thinking about the angle inside the shape, think of instead of the angle you had to turn at the corner to continue to follow the shape.
That is 180° minus the angle inside the shape.
But once you turn every corner of the shape, you’re back going in the same direction so the sum of all those turns is 360°.
But the sum of all those turns is also the number of corners (which is also the number of sides) × 180° minus the sum of all the angles inside the shape.
So 180° × N – sum of angles inside shape = 360°. And so the sum of angles inside a N-gon is 180° × (N – 2).
The angle of a polygon is (n-2)*180, where n is the number of sides. Why?
ELI5, you can form n triangles out of any polygon by using the 2 end points of each side to the center. the center angles will total 360 therefore we get n-2.
Maybe not so ELI5: Take a pentagon. Point A is going to be the center. Point B and C are going to be the end points of a line that’s part of a hexagon. Connect all the points, what does it form? A triangle. Now make as many triangles as you can while always using point A and the end points of a line. So if B,C,D,E,F are the points at the end of a line. You have triangles ABC, ACD, ADE, AEF, AFB. 5 triangles. The total sum of those triangles is 5*180 = 900. However, that Angle A on all the triangles. If you draw it what you’ll see is it’s a perfect circle, 360 degrees, which shouldn’t be part of the sum. 900-360 = 540.
This property is the case for all non-self-intersecting polygons. One way to think about it is with forming triangles, as others have said.
A different way is to consider “walking” along the perimeter of the polygon. Think about the exterior angles – how much you need to turn – as you make one loop around. (And choose an orientation as “positive”).
For polygons without intersections, you need to make one full turn (360 degrees). But you have n sides and n vertices (corners). The interior and exterior angle (as “signed” angles, to be specific) at each corner sum to 180 degrees. So the interior angles must sum to (180n – 360) degrees, which gives the sequence.
[removed]
Pick a point in the middle of one side of an n-sided polygon. Without changing anything else, declare that point to be a “corner”, so now you have an n+1-sided polygon. The “corner” you just added has 180 degrees. If you squish the new polygon around so that your corner actually bends to less than 180 degrees, the other corners open up so the total angle remains unchanged.
Because when you turn a full circle around a perimeter you do a full 360. Each turn you make deducts whatever you turn from a flat 180, so in total you will have 180x-360 which is the generalization of the cases you mention.
Latest Answers