Well, if you’ve been watching our free math videos in order, congratulations! We’ve now graduated to Geometry. The first subject we will investigate is Polygons.
Polygons. In this lesson, we will expand beyond triangles and quadrilaterals to the entire realm of polygons. A polygon is any closed figure with all line-segment sides. So here we have an example of a polygon with three sides, a triangle with four sides, a quadrilateral and then one with five sides and six sides.
And you could imagine, you could have many, many more sides. A geometric figure is not a polygon if first of all, the figure doesn’t close. So for example, something like this, not a polygon, has none of the polygon properties. If the sides cross, so we don’t have to worry about that because I’m not gonna make us think about that.
And if not all the sides are straight. Well this final one is interesting. The test could give us something that is a combination of a polygon and a circle. And of course we’d have to solve that using both polygon properties, as well as circle properties. So this is not strictly a polygon either, we’d have to use circle properties as well, to figure out something about the shape.
The only polygons that will appear on the test are convex polygons, that is polygons in which every single vertex points outward. A concave polygon has one or more vertex that points inward. Well, it turns out those technically are true polygons. But the test is not gonna ask
about them, so we don’t need to worry about them. So let’s talk about the names of the polygons.
Obviously, if a polygon has 3 sides, we call it a triangle. If it has 4 sides, we call it quadrilateral.
If it has 5 sides, we call it a pentagon, that’s a term you need to know. Of course, if it has six sides, it’s a hexagon. That’s a term you need to know. 7 sides is kind of an irregularity.
We don’t need to, for a variety of reasons and we’ll talk about this more in the next video, we don’t worry about seven-side shapes that often. They don’t come up that often. But an eight-sided shape, the octagon, that’s the term you need to know. What is true for all polygons? Well first of all, any segment that can connects two non-adjacent vertices is called a diagonal.
So triangles as we found out don’t have diagonals. Quadrilaterals have exactly two diagonals. Pentagons have five diagonals. Hexagons have 9 diagonals. For higher polygons, we could count diagonals using the techniques in the counting modules.
That actually becomes a much harder problem to figure out, say how many diagonals does a polygon with 20 sides have? This is something we are not going to worry about right now, we are going to wait until we get to the accounting problems. That is really much more of an accounting problem rather than a geometry problem. So don’t worry about that right now.
Angles in Polygons
Angles in polygons. Of course, the sum of the three angles in a triangle is 180 degrees. The sum of the four angles in a quadrilateral is 360 degrees because we can divide a quadrilateral into two triangles along a diagonal. We can extend this pattern to higher polygons. Any pentagon can be broken into three triangles, so for any single vertex, we can draw two diagonals that divides it into three triangles and therefore the sum of the angles in the pentagon must be the sum of the angles in three triangles, 3 *180= 540, that’s a good number to know.
Any hexagon can be broken into four triangles, so from any vertex, we can draw three different diagonals, divide the shape into four different triangles. The sum of the angles in the hexagon must be the sum of the angles of those four triangles, 4 * 180 is 720, that’s a good number to know also. You might see where this pattern is going. So, with a quadrilateral, there are four sides, there are four vertices.
From any vertex, we can draw only one diagonal that divides into two triangles for a sum of 2 * 180, which is 360. For a pentagon, from any one vertex, we can draw two diagonals that divides it into three triangles, so the sum of the angles is 3 * 180 which is 540. For a hexagon, from any vertex we can draw three diagonals.
It divides the shape into four triangles and the sum of the angles is 4 * 180, which is 720. Now where is this pattern going? If we have an n sided quadrilateral, an n sided polygon with n vertices, that means that we could draw from any one vertex. We could draw n- 3 diagonals.
The n- 3 diagonals would divide the shape into n- 2 triangles and then the sum of the angles would simply be (n- 2)180. Because we add two minus one triangles. So the sum of the angles in an n-sided polygon is n- 2 * 180. That’s an important formula to know. And again, I would urge you, don’t simply memorize it, make sure you understand where it comes from.
For example, consider an 18-sided polygon in which all the angles are equal. What does each angle equal? Pause the video and see if you can work this out for yourself. Okay, so we know that the sum of the angles must equal (n- 2) * 180, so (16)(180), I’m not gonna multiply that, I’m just gonna leave like that (16)(180). And if all 18 angles are equal, then we’ll divide that sum of the angles by 18 to get one of the singles angles. When we only divide by 18 we can very easily cancel, 180 divided by 18 is just 10.
So that equals 160 degrees. And that is the measure of each one of those 18 angles. And here’s the actual shape, an 18-sided polygon in which all the angles are equal. Notice that it looks awfully circle-like. And that is true of many of the higher polygons that have all equals angle and all equal sides.
In summary, a 3 side of polygon is called a triangle, a 4 side polygon is called quadrilateral, 5 side of polygon is called a pentagon, 6 sided, one is called a hexagon, 8 sided on is called an octagon. Those are words you need to know. A segment from one vertex to a non-adjacent vertex is a diagonal. And the sum of the angles in an n-sided polygon equals (n- 2) * 180.