Geometry Polygons

Welcome back to Educator.com.0000

For the next lesson, we are going to go over polygons.0002

We are going to talk about the different types of polygons and the interior and exterior angles of polygons.0004

First, let's talk about what is a polygon and what is not a polygon.0012

A polygon is a closed figure, formed by coplanar segments, such that the sides are non-collinear,0018

and each side intersects exactly two other sides at their endpoints.0027

Basically, a polygon ("poly" meaning many) is a closed shape (meaning it has to close), each side being a straight lines, and where no sides overlap.0033

As long as we have a shape that is closed (nothing open--nothing can get through), with no overlapping, and...0055

like this one...see how it is not straight...these are examples of polygons, and these are not.0069

Now, it is OK if polygons look funny; if they look like this, that is OK.0078

As long as it is a closed figure, each side is a straight line segment, and none of the sides overlap, then it is a polygon.0082

So, here, because of that it is not a polygon; because of the overlap, it is not a polygon;0094

and because this side right here is not straight, that is not a polygon.0101

The two types of polygons are convex and concave; a convex polygon is when all of the sides are on the outside of the shape.0109

What that means...maybe if I explain "concave," it will be easier to understand.0125

A concave polygon is when two sides go in towards the center of the polygon.0132

See how, right here, these two sides are angled towards the center; that would make this concave.0140

The same thing happens here: we have this angle going towards the center.0148

Think of a cave, like a mountain, or in the rocks; see how it goes inwards, and it creates a little cave?0153

So, any time it does that, it is a concave polygon.0163

If it doesn't, then it is a convex; so all of the angles are pointing away from the center.0166

All of these angles are pointing away from the center, and that is convex.0175

And this explanation here: No line containing a side of the polygon contains a point in the interior of the polygon.0184

It just means that, if you were to draw each of these sides or extend them into lines, it is not going to cut through the polygon.0193

If we make this into a line, it is not going to cut through the inside of the polygon.0211

The same thing happens here, and the same thing here.0216

With this one, however, if I draw a line, see how it cuts in the polygon; that is what it means--that is what this explanation is saying.0221

The easiest way to remember: just think of the cave--it is creating a little space right there, like a cave, so they are concave polygons.0235

Now, a regular polygon is a polygon with all sides congruent and all angles congruent; it is equilateral, and it is equiangular.0246

That shows that it is equilateral, and this shows that it is equiangular.0262

Any time that it is equilateral and equiangular, it is a regular polygon.0269

And in order for it to be equilateral and equiangular, it has to be convex; you can't have a concave polygon that is equilateral and equiangular.0281

Maybe it could be equilateral, but not equiangular.0290

The interior angle sum theorem is to figure out the sum of all of the angles inside the polygon.0294

If I have a triangle (which is actually going to be the first polygon that we are going to use), I have three angles in the triangle.0304

The interior angle sum theorem is going to give me the total, or the sum, or the angle measure, of all three angles combined.0318

So, if I have a quadrilateral (four angles), what do all four angles of the polygon add up to?0329

First, let's start with triangles; a triangle has three sides...number of triangles: a triangle only has one triangle.0337

We are going to talk about this in a little bit, but the number of triangles would just be one.0351

The sum of angle measures: we know that all three angles of a triangle add up to 180.0356

Next will be a quadrilateral, a four-sided polygon: number of sides: 4.0368

Now, if I have a quadrilateral, I have two triangles; so the sum of the angle measures is going to be 360.0377

I know that all of the angles added up together in a quadrilateral are going to add up to 360.0394

And then, a pentagon is the next one: it has 5 sides; the number of triangles is going to be 3 (let's see if I can draw this: that would be 1, 2, 3).0400

The sum of the angle measures: here, every time we add a triangle...every time we have one more side,0427

it is like we are adding another triangle in the polygon, and then we add another 180,0436

because, for every triangle that exists in the polygon, there is an additional 180.0442

We always start with the triangle, because that is the first polygon.0451

Then, when you get to quadrilateral, the next one, it is going to be plus 180.0454

Then, to get to the next one, we are going to do + 180, which is going to be 540;0462

so the angle sum of a pentagon is going to be 540 degrees.0471

How about the next one, which is a hexagon?--6 sides: 1, 2, 3, 4...0480

Again, we just add 180; it is going to be 720; and so on.0498

For each triangle that exists, again, it is going to be 180 degrees.0511

But what if I ask you for a 20-sided polygon--what is the interior angle sum of a 20-sided polygon?0516

There is a formula to go with this, and that is right here.0527

Because a triangle has 3 sides, but only one triangle exists; that is 180.0533

For every additional triangle, it is going to be an additional 180; so here, isn't this 2 times 180?--because it is 180 + 180, which is 360.0542

Here, from a 5-sided polygon, there are three triangles that exist, so isn't that 3 times 180 (180 + 180 + 180, which is 540)?0555

So, it is 180 times the 3; here, there are four triangles, so I have to do 4 times 180.0568

So, if I want to find a 20-sided polygon, how many triangles exist?0577

Well, look at the pattern: 3:1, 4:2, 5:3, 6:4, 20...it is 2 less, so it is going to be 18.0588

Now, again, this is to get 360 here; so we just do 2 times 180, which is going to equal 360; 3 times 180...0601

the number of triangles times 180...4 times 180 was 720.0615

So here, all I have to do is multiply 18 times 180.0621

So first, I have to figure out how many sides I have; this is going to be n.0629

And then, subtract the 2 to figure out how many triangles exist in that polygon; and then just multiply it by 180.0640

So then, 180 times 18...0...this is 64...8 + 6 is 14...then put the 0 here; 0, 8, 1; 0, 4, 12...it is going to be 3240 degrees.0649

All 20 angles of a 20-gon are going to add up to 3240 degrees.0684

Looking at the formula, it is going to be the number of sides; subtract 2 (you are going to solve this out first)0695

to figure out how many triangles you have in that polygon; and then just multiply it by 180; that is it.0705

It is just the number of sides, minus 2: take that number and multiply it by 180, and that is going to give you the interior angle sum.0712

Now, the exterior angle sum theorem ("exterior" meaning outside): whatever you have...it can only be one exterior angle0726

from each side or vertex...then if this right here is 1, this is 2, this is 3, this is 4, and this is 50742

(there have to be 5 of them, because there are only 5 sides here; it is a pentagon)--all 5 angles here are going to add up to 360.0757

And that is the exterior angle sum theorem; the interior angle sum theorem is different,0767

because depending on the number of angles, depending on what the polygon is, the interior angle sum is going to be different.0771

The more angles the polygon has, the greater the sum is going to be.0779

But the exterior angle sum is always going to be 360--always, always, no matter what type of polygon you have,0785

whether you have a triangle (if you have a triangle, it doesn't matter if you measure the exterior angle this way,0795

as long as you do the same for each vertex: let's say 1, 2, 3 here; the measure of angle 1, plus...they are all going to add up to 360)...0802

Here, the measure of angle 1 + 2 + 3 + 4 + 5 are all going to add up to 360; and that is the exterior angle sum theorem.0821

The first example: Draw two figures that are polygons and two that are not.0832

You can just draw any type of figures that you want.0838

The first two that are polygons...you can just draw...it doesn't matter...any type of polygon.0844

You can draw something like that, as long as it is closed, each side is a straight line segment, and no sides are overlapping.0854

Then, two examples, two figures, that are not polygons would be exactly those things.0874

Maybe like that...that would not be a polygon; if I have two sides crossing like that, that is a non-polygon.0882

If I have, I don't know, something like this, that wouldn't be a polygon...use any examples that are something like this.0895

These are polygons; these are not polygons.0911

Moving on to the next example: Find the sum of the measures of the interior angles of each convex polygon.0916

The first one is a heptagon: now, a heptagon is a 7-sided polygon; this has 7 sides.0924

Remember: if the number of sides is 7 (n is 7), we have to figure out how many triangles.0933

Remember: you subtract 2, so the number of triangles is going to be 5; and then, you are going to multiply that by 180.0940

Now, the formula itself is going to be that the sum is equal to 180 times (n - 2).0957

This is 7 - 2; that is 5; so it is the same thing as 180 times 5.0971

Then, you can just do it on your calculator; I have a calculator here; 180 times 5 is going to be 900, so the sum is 900 for a heptagon.0978

The next one is a 28-gon; now, once you pass 12-sided polygons, there is no name for it, so you would just write 28 with "-gon."0995

This is a polygon with 28 sides, so n is 28; the number of triangles is 26; you are going to multiply that by 180.1006

To use the formula, you are going to do 180 times number of sides; that is 28 - 2, so 180 times 26...use your calculator...is going to be 4680 degrees.1021

So, all the interior angles of a 28-gon are going to add up to 4680 degrees.1051

And the next one, x-gon: now, for this one, we don't know how many sides there are in this polygon.1061

So, you are just going to use the formula; and so, we know that n is going to be x; the number of sides is x.1067

In the formula, you are just going to replace the n with the x.1079

It is supposed to be the number of sides, minus 2; instead, we are going to say x - 2.1083

And that would be it; you are just replacing the n with whatever they give you as the number of sides, and that would be x.1088

Given the measure of an exterior angle, find the number of sides of the polygon.1104

Before we start with these numbers, with these examples here, I want to first use a triangle.1112

Now, we know, from the exterior angle theorem, that the sum of the exterior angles is always 360.1120

The sum of the measures of all of the exterior angles is going to be 360.1127

If I have a triangle, here, here, and here: those are my three exterior angles: 1, 2, 3.1134

How would I be able to find the number of sides?1146

Well, in this case, how would I find each of these angle measures?1153

Wouldn't I have to do 360 degrees, divided by the number of exterior angles?1161

This is going to be what?--each of these angles has to be 120.1172

Now, again, this is going to be for a regular polygon; for a regular polygon, this is going to be 120;1181

if all of these exterior angles have the same measure, then it will be 120 each.1188

So, that way, it will total to 360.1194

Well, it is like they are giving you the measure of each exterior angle; so how would we figure it out...1197

if I said that each exterior angle has an angle measure of 120--each exterior angle of a polygon is 120--find the number of sides.1205

Well, you would have to do the same thing: 360 (because that is the total) divided by 120, and that is going to give you 3.1218

So, you know that there are three sides here.1227

The same thing works for this: 36 is the angle measure of each exterior angle.1233

So, if 360 degrees is the sum of all of the exterior angles, divide it by 36 to find the number of sides; you get 10.1241

That means that the polygon has 10 sides.1256

The same thing works here: 360, divided by 15 degrees (you can use your calculator), is going to give you 24 sides.1266

If there are 24 sides, each angle of the sides (because if there are 24 sides, that means that there are 24 angles)--1293

each exterior angle--has a measure of 15 degrees, which will then1303

(since there are 24 of them) add up to 360 when you multiply these two together.1308

The same thing works for this one, x: each exterior angle measure is going to be x degrees--we just have to divide it.1314

And since we can't solve that out, this will just be the answer; you are just simplifying it out as much as possible, and that will be it.1328

There is nothing else that you can do with that.1337

The fourth example: Find the sum of the interior angles of each polygon.1342

Now, notice how both of these polygons are not regular polygons; it doesn't look like it is equilateral; it doesn't look like it is equiangular.1348

But it is OK, because we are just looking for the sum of all of the interior angles.1358

Since it is not regular, we would not be able to find the measure of each angle.1367

But instead, we can find the sum--what all of them add up to--because it depends on the polygon, not the type of angles inside the polygon.1374

Here, we have 1, 2, 3, 4, 5, 6; so we have a hexagon, a 6-sided polygon.1383

And that means, in a 6-sided polygon, that we have 4 triangles.1391

Remember: we subtract 2; we get 4 triangles; and then we have to multiply this by 180.1399

4 times 180...this is 32...720; that means that the sum of all of the angles inside here is going to be 720.1409

Now, if, let's say (I am just going to add to this problem here), this was a regular polygon--1431

say that all of the sides are the same, and all of the angles are the same, so it is equilateral, and it is equiangular;1443

and I want to find what the measure of each angle is, then.1450

Since each of these angles are the same, and I know that all 6 angles together1453

are going to add up to 720, how can I find the measure of just one of them?1460

Since they are all the same, how can I find the measure of just this one, the measure of angle A, or the measure of 1?1466

Since they all have the 720, and they are all the same--they all have the same measure, and there are 6 of them,1478

I can just take 720 and divide it by 6; 720/6 is going to give me the measure of each of these angles.1484

So then, here, you do 720 divided by 6; each of these angles is going to be 120; 120 here, 120 here, here, here, here, and here.1499

And that is only if you have a regular polygon, meaning that all of the angles are the same.1521

All of the angles have to be the same for you to be able to divide your angle sum to figure out each of these angle measures.1527

The next one: here, this is to find the sum of all of the angles side.1539

This is a quadrilateral; we only have four angles; so this is just going to be 180 times 2: let's just use the formula...1545

180...n - 2 is the sum; 180...we have four sides, minus the 2, so that means we have two triangles;1554

180 times 2, we know, is 360 (I said 360, and I wrote 320).1569

Now, again, if all of these angles were the same, were congruent, this is equilateral and equiangular, so it is a regular polygon.1579

Then, you would take 360; you can divide it by 4; and that would just be 90 degrees; that is if each of these angles were the same.1593

Then, each of them would have a measure of 90; and we know that that would just make this a square, if it was an equilateral, equiangular quadrilateral.1606

That would make it a square; then you would know that each of these angles would have to be a right angle.1614

But for the sake of just knowing what to do if you have a polygon that is regular--1618

not just a quadrilateral, but any other type of regular polygon--you would just take the sum,1624

and divide it by the number of angles you have.1631

And that is it for this lesson; thank you for watching Educator.com.1641