چند ضلعی ها

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, then one with five sides and six sides.

And you could imagine, you could have many, many more sides. 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.

The test is not gonna make us think about that. And in, not all the sides are straight. Well, this final one is interesting. We, the test could give us something that was 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 have to use circle properties as well to figure out something about this 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, 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 lets talk about the names of the polygons. Obviously if a polygon has three sides we call it a triangle. If it has four sides we call it a quadrilateral.

If it has five sides we call it a pentagon. That’s a term you need to know. If it has six sides it’s a hexagon. That’s a term you need to know. Seven 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 a term you need to know. What is true for all polygons? Well first of all, any segment that 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 nine diagonals For holly, for higher polygons, we could count diagonals using the techniques in the counting modules.

That actually becomes a much harder property, problem to figure out. Say, how many diagonals does a polygon with 20 sides have? That’s something we’re not gonna worry about right now. We’re gonna wait’ til we get to the, the counting problems. That’s really much more a counting problem, rather than a geometry problem. So don’t worry about that right now.

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 quadrilaterals in to two triangles along the diagonal. You can extend this pattern to higher polygons.

Any pentagon can be broken into three triangles, so for many 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 times 180, which is 540. That’s a good number to know.

Any hexagon can be broken into four triangles. So from any vertex we can drop 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 times 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 4 sides, there are 4 vertices, from any vertex we can draw only one diagonal that divides it to 2 triangles, for a sum of 2 times 180, which is 360. For a pentagon, from any one vertex, we can draw 2 diagonals, that divides it into 3 triangles.

So the sum of the angles is 3 times 180, which is 540. For a hexagon, from any vertex we can draw 3 diagonals which divides the shape into 4 triangles and the sum of the angles is 4 times 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 minus 3 diagonals.

The n minus 3 diagonals would divide the shape into n minus 2 triangles. And then the sum of the angles would simply be n minus 2 times 180. Because we’d have n minus 2 triangles. So the sum of the angles in an n sided polygon is n minus 2 times 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 minus 2 times 180.

So 16 times 180. I’m not gonna multiply that out. I’m just gonna leave it like that. 16 times 180. And if all 18 angles are equal then we’ll divide that sum of the angles by 18 to get one of the single angles.

When we divide by 18 we can very easily cancel. 180 divided by 18 is just ten. And 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 the angles are equal. Notice that it looks awfully circle like and that is true of many of the higher polygons that have all equal angles and all equal sides.

In summary, a 3-sided polygon is called a triangle. A 4-sided polygon is called a quadrilateral. 5-sided polygon is called a pentagon. 6-sided one is called a hexagon. 8-sided one 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 minus 2 times 180.