Geometry Area of Regular Polygons & Circles

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For the next lesson, we are going to go over area of regular polygons and circles.0002

To review for a regular polygon, we know that it is when you have a polygon0008

with all of the sides being congruent and all of the angles being congruent; it is equilateral and equiangular (it has to be both).0015

Now, a couple more things to review for this lesson: if this is the center of my polygon,0024

I know...let's say I am going to have a starting point right here...that to go from here,0033

all the way around, we know that this is 360 degrees; to go all the way around a full circle is 360 degrees.0042

Also, for right triangles: Soh-cah-toa is for right triangles when you are given angles and sides, and you have to find an unknown measure.0055

This is "the sine of an angle is equal to the opposite side over the hypotenuse."0079

"The cosine of the angle measure is equal to...a is for adjacent, over the hypotenuse."0089

And "the tangent of the angle measure is equal to the opposite over the adjacent side."0100

That is Soh-cah-toa; and also, for right triangles, there are special right triangles.0111

We have 30-60-90 (let me do it on this side) triangles; the side opposite the 30-degree angle...let's say this is 30,0118

and this is 60...is going to be n; the side opposite the 60...be careful here; it is not 2n; it is n√3;0136

and the side opposite the 90 is going to be 2n.0152

That is a 30-60-90 special right triangle; and also, a special right triangle is 45-45-90.0156

This is 45; this is 45; this is 90; the side opposite the 45...if that is n, then this is also n, because they are going to be congruent.0170

The side opposite the 90 is going to be n√2; so here is the second type of special right triangles.0184

So again, when you are going all the way around a full circle, that is 360 degrees.0193

For right triangles, you can either have a special right triangle--if you have a 30-60-90 or a 45-45-90--0201

then you can use these shortcuts; if not, then you would have to use Soh-cah-toa.0209

The formula for the area of a regular polygon is 1/2 times the perimeter, times the apothem.0222

Here is a new word, apothem, and we are going to talk about that in a second.0233

To explain this formula, imagine if I have my regular polygon, because this is the area of a regular polygon (I'll draw that a little better).0239

If I take my polygon, and let's say I break it up into triangles; from the center, I am going to create0255

a triangle here, a triangle here, here, here, here, and here; I know that the area of one triangle is 1/2 base times height.0269

So, if that is one triangle, 1/2 base times height...how many triangles do I have?...I have 1; I have 2, 3, 4, 5, 6;0295

so, it is 1/2 base times height, times 6; and this would be the area of this regular polygon.0310

Now, of course, that is only if it is a regular polygon; you can only do this if you have a regular polygon.0321

1/2 base times height is going to give you the area of one triangle, and you are going to multiply it by the 6 triangles that you have.0328

Well, look at how many sides I have for this polygon: this is 1, 2, 3, 4, 5, 6--this is a hexagon--this is 6 sides.0335

So, right here, my base, times the 6, is going to give me perimeter, because, if this is the base,0347

I have 6 of them together; base and 6 together are going to give me the perimeter.0366

The height is this right here; this height is now called the apothem.0378

The apothem has to go from the center to the midpoint of one of the sides; that is called the apothem.0386

So now, my h is going to turn into an a; the height of one of the triangles is now an apothem.0395

And then, 1/2 is going to be part of the formula, as well.0403

Just changing my base times the 6 to become perimeter, and then renaming the height to be the apothem0413

(the height of the triangle is an apothem of the regular polygon), and then just keeping the 1/2...0426

that now becomes the formula for the area of a regular polygon.0433

If you ever get confused with this formula right here, all you have to do is just break this up into triangles;0439

find the area of one of the triangles, and multiply it by however many triangles you have.0448

And that is going to be the exact same thing as this formula here.0451

But this is the main formula for a regular polygon: 1/2 times the perimeter times the apothem.0457

The area of a circle, we know, is πr².0477

From the center to a point on the circle is called the radius, and that is what this r is--the radius.0481

The radius squared, times π, is going to give you the area of this whole circle.0494

OK, let's work on some examples: Find the area of the regular polygon.0503

Now, to find the area of this polygon, my formula is 1/2 times the perimeter, times the apothem.0511

Now, I have that this side is 10; and again, this is a regular polygon, so then we know that all of the sides have to be 10.0526

I can find the perimeter: a = 1/2...the perimeter would be 10 times however many sides I have; that is 5, so it is 10 times the 5.0532

And then, for my apothem, remember: the apothem is from the center to the midpoint of the side, and it is going to be perpendicular.0547

And if it is the midpoint, well, we know that this whole thing is 10--this whole side has a measure of 10.0564

Each of these is going to be 5: 5 and 5.0571

But I need this right here, a; that is my apothem--that is what I need.0578

What I can do is draw a triangle from here to there; we know that it is a right triangle; and I can look for my side.0585

But for me to do that, I need to know this angle measure right here.0600

What I can do is just...let's say I have all of these triangles, again; I know that from here, all the way around here--that is a total of 360 degrees.0605

A full circle is 360 degrees; now, I just want to know this little angle measure right here.0628

What I can do: since it is a regular polygon, I can take 360 degrees, and I am going to divide it by however many triangles I have here.0637

I have 1, 2, 3, 4, 5--5 angles that make up my five triangles; so 360 divided by 5 (I have my calculator here) is going to give you 72 degrees.0652

That means that, for each of these, this is 72; this is 72; 72; 72; and this whole thing right here is going to be 72.0678

But since I am only using this half-triangle, because I want to make it a right triangle0695

(because if I use a right triangle, then I have a lot of tools to work with; I have a lot of different things that I can use;0704

I can use Soh-cah-toa; I can use special right triangles), I want to use this right triangle here;0712

so then, I am going to take this 72, and then divide it by 2 again, because I just want this angle measure right there.0722

So, 72 divided by 2, again, is going to be 36.0729

That means that this angle right here, where the arrow is pointing, is 76; that angle is 76.0738

Now, I am going to re-draw that triangle, so that it is a little bit easier to see; that is my apothem; this angle measure is 36 degrees;0746

this side right here is 5; now, since this angle measure is 36, I can't use my special right triangles,0762

because only when it is a 30-60-90 triangle or a 45-45-90 degree triangle could I use special right triangles.0773

Since I can't use special right triangles, I would have to use Soh-cah-toa.0780

Now, if you remember, Soh-cah-toa is made up of three formulas.0786

I just have to figure out which one I am going to use: I am not going to use all three--I am just going to use one of these, depending on what I have.0796

Look at it from this angle's point of view: I have the side opposite, so I have an o; and I have the side adjacent, so I have an a.0804

Then, which one uses o and a? This uses h; this uses ah; this one uses oa, so then I would have to use tangent.0820

That is going to be...the tangent of this angle measure, 36, is equal to the side opposite, which is 5, over the side adjacent, which is a.0831

Now, I can just solve for a; so I am going to multiply this side by a and multiply the other side by a.0851

That way, this is going to be a times the tangent of 36, this whole thing, equal to 5.0860

Remember: keep these numbers together--it has to be the tangent of an angle measure; you can't divide the 36.0870

You have to find the tangent of 36 on your calculator.0876

So, to find a, I am going to, on the calculator, do 5 divided by this whole thing, tan(36): with 5/tan(36), I get my as 6.88.0880

My apothem is 6.88; so now, I have my apothem--this is going to be 6.88.0919

Then, from here, I just have to solve this out: this is 1/2, times this whole thing (is the perimeter), times the apothem.0934

You could just do that on your calculator; and so, the area should be 172.05 units squared0946

(because it is still area, so make sure that you have your units squared); I'll box it to show that that is my answer.0970

So again, since we have a side, we can find the perimeter; the perimeter is fine.0981

But to find the apothem, you are going to have to take the 360, divided by however many sections that you need it to divide into.0989

So, that way, you can use this right triangle right here, and then you just use Soh-cah-toa to find the apothem.0998

All right, let's do a couple more: the next example: Find the area of the regular polygon.1007

Here, this is a; even though it is not going down to this side, it is still an apothem--we know that this is the apothem; it is 5.1016

And that is all they give us; remember: to find the area of a regular polygon, it is going to be 1/2 times the perimeter times the apothem.1032

We have the apothem, but no sides, so we can't find the perimeter.1043

So, what we can do is, again, form this triangle here; that way, we have a right triangle.1049

And then, since I need an angle measure (because if I find this side right here, then I can multiply it by 2,1058

and then that is going to give me the whole side), again, break this up into triangles--just sections--1069

so that you know what to divide your 360 by.1080

From here, going all the way around, we have 1, 2, 3, 4, 5, 6, 7, 8, which is the same as the number of sides that we have; this is an octagon.1087

So, it is 360, divided by 8; that is going to give us the angle measure of each triangle: 360/8...we get 45.1104

But remember: you have to be very careful, because that is for each of these triangles right here.1125

So, this whole thing right here is 45; each of these angles is 45, and so is this right here.1131

But again, we need to find only this angle measure, which is half of the 45,1145

because the 45 is the same as from here all the way to here; that is 45.1151

So, we are going to take the 45 and divide it by 2; and then, that angle measure right here is going to be 22.5.1159

I am going to re-draw this right triangle; this is my apothem that was 5; this angle measure is 22.5 degrees.1169

And I am looking for this side, so then, let's label that x.1187

I have a right triangle; we need to use Soh-cah-toa; so again, from this angle's point of view, what do I have?1193

I have opposite, and I have adjacent; this is the hypotenuse, so I have opposite and adjacent, so I am going to have to use "toa."1209

That means that the tangent of the angle measure, 22.5 degrees, equals...the side opposite is x, over...adjacent is 5.1221

To solve for x, I am going to multiply both sides by 5; and again, use your calculator: 22.5...the tangent of that...multiply it by 5.1235

Make sure you find the tangent of this number; don't multiply this times this and then find the tangent of that--that is going to be wrong.1257

It is 5 times the tangent of this number; and that is going to give me 2.07.1265

So, this right here is 2.07; but remember: I need this whole thing--that is my side.1275

If just this is 2.07, then the whole thing is times 2; so, my side, then (I can just write it here, because this side is the same thing as this side) is 4.14.1287

So then, go back to our formula: area equals 1/2...the perimeter would be 4.14 times...how many sides do we have?1314

This is an octagon, so we have 8 sides; so then, this together would make the perimeter...times the apothem.1330

The apothem is 5; that was given to us; so, my answer, then...what you can do is just punch in this times this times this,1339

and then just divide it by 2, because multiplying by 1/2 is the same thing as taking all of that and dividing it by 2.1358

The area is going to be 82.84 units squared.1367

Again, just to review what I just did: the apothem was given to me, but that was the only measure that I had.1381

So, I had to take the 360 and divide it by 8 to find the angle measures for each of the triangles.1390

But then, I cut that triangle, again, in half, to make it a right triangle; so then, this angle measure right here was 22.5.1401

I used Soh-cah-toa: the tangent of 22.5 equals x over 5; you solve for x, and that is going to give you...1412

remember this section right here, this little part right here; multiply that by 2 to get the full side.1424

And then, from there, since I have one of the sides, and this is equilateral, I just multiply that by 8, because there are 8 sides total.1433

That gives me the perimeter; the perimeter, times the apothem (which is 5), multiplied all together...you get 82.84 units squared.1441

The next one: Find the area of the shaded region.1453

Here we have a circle; and inscribed in that circle is a hexagon, because we have a 6-sided polygon.1458

OK, now, that means...to find the area of a shaded region, I am going to find the area of the circle.1469

It is the circle, minus the hexagon; and that is going to give us this shaded part.1485

It is like we are cutting out that hexagon from the circle.1503

To find the area of the circle, we need radius; the area of a circle, we know, is πr²--that is the area.1509

r is the radius, so it is from the center; now, ignore the hexagon for now--just look at the circle.1526

It is from the center, and it goes to a point on the circle, so that would make that the radius; it equals π(8)².1534

So, 8 squared is 64; you are going to do 64 times the π, and that will be 201.06 units squared.1549

And now, to find the area of the hexagon: the formula is 1/2 perimeter times apothem, and this is a regular hexagon.1574

Now, be careful here: this is not my apothem; the apothem has to go from the center, and it has to go to the midpoint of one of the sides.1596

I don't have the apothem, so I need to solve for it.1613

So again, I need to take my 360 (because that is a full circle), and I am going to divide it by...1618

if I cut this up into triangles, it is going to be 6 triangles; remember: it is the same number of sides that I have.1632

So, 360, divided by 6, is going to give me 60 degrees; so that means that this whole thing right here is going to be 60.1640

But since I only want to know this angle right here, this is going to be half of that, which is 30.1656

So again, let me just draw out that triangle: here is a; this is 30; and this is 8.1664

Now, if this is 30, then this has to be 60, because all three have to add up to 180; or these two have to add up to 90, since this is a right angle.1676

30-60-90--it is my special right triangle; so, I don't have to use Soh-cah-toa--I can just use my special right triangle shortcut.1687

If this is n, then the side opposite the 60 is going to be n√3; the side opposite the 90 is going to be 2n.1702

That is my rule for a 30-60-90 triangle.1712

What is given to me--what do I have?1716

I have the side that is opposite the 90, so I have a 2n; I have this right here.1719

I want to solve for the side opposite the 60, because that is the apothem.1727

Well, 2n is the same thing as 8, so I am going to make them equal to each other.1734

How do I solve for n? Divide the 2; n is 4.1740

That is this right here: 4. Then, the side opposite my 60 is going to be n√3, so that is 4√3: a, my apothem, is 4√3.1746

Now, do I know my side? I don't have my side measures, but since I have this triangle, I can just look for this right there, which is that right there.1767

Let me just highlight that: this is the same thing as that--that is the side opposite the 30, which is 4; that means that this is 4.1780

So, this whole thing right here, this side, is going to be 4 + 4, which is 8.1795

Each of these sides has a measure of 8; so, my perimeter is going to be 8 times 6; 8 times 6 is 48; my apothem is 4√3.1803

If you want to, you can just divide this by 2, just to make it a smaller number; that is 24; multiply it by 4√3.1830

Now, since we have a decimal here, you can just go ahead and turn this into a decimal, also, by just punching it into the calculator.1842

Here, you can do 4 times √3, which is 6.93; multiply it by 24; so the area of this hexagon is 166.28 units squared.1852

Now that I found the area of the circle and found the area of this hexagon, I am going to subtract it.1885

201.06 - 166.28 is going to give me 34.78 (if you round it to the nearest hundredth) units squared.1896

So, the area of the shaded region is 34.78 hundredths.1931

And the fourth example: we have the circle inscribed in a decagon; a decagon is a 10-sided polygon.1946

And again, this is regular decagon; we are dealing with regular polygons, so this is going to be regular; find the area of the shaded region.1959

This is like the opposite of what we just did: we are taking the area of the polygon, the decagon, and we are subtracting the area of the circle.1967

Let's see, this is a decagon; let's find the area first: the area of a regular polygon is 1/2 perimeter times apothem.1978

And in this case, it looks like we are given...1995

And before you start, just look to see what you have, what you need, and how you are going to find what you need.2000

Can I find the perimeter? Well, if this side is 4, can I find the perimeter of my decagon?2011

Yes, because, if this is 4, all of the sides will be 4; so I can just do 4 times however many sides I have, which is 10.2018

That is going to give me perimeter; so, area equals 1/2 times the 4 times a 10, which is the perimeter.2026

And the apothem, remember, is the segment from the center all the way to the midpoint of one of the sides, so that it is perpendicular.2035

And that is 6; so I have everything that I need--I don't have to solve anything out.2048

Here, this is going to be, let's see...1/2 times 4 times 10, so the perimeter is 40 times this 6.2058

You can just divide one of these numbers by 2--you could cut it in half and cross-cancel.2079

So then, if I take this and a 3, I know that 4 times 3 is 12; add the 0 to the end of that number;2085

the area is 120 units squared--that is the area of the decagon.2095

I am not done: I have to find the area of the circle now.2102

The circle's area is πr²; r, which is the radius, from the center to the point on the circle--2105

well, that is going to be the same number as the apothem, because this is also, for this diagram2124

(not always, but the apothem for this diagram) the radius, because it is going from the center of the circle to a point on the circle.2130

So, that is going to be 6 squared; so that is 3.14 times 36, which is 113.10 units squared.2138

And from here, I have to take the decagon, and it is like I am cutting up the circle; so I have to subtract.2163

So, it is going to be 120 - 113, and that is going to give me 6.9 units squared.2170

So, that is the area of the shaded region here.2192

That is it for this lesson; thank you for watching Educator.com.2199