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## Practice Questions

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### Related Articles:

### Area of Regular Polygons & Circles

- Regular polygon: An equilateral, equiangular polygon
- Area of a regular polygon: If a regular polygon has a perimeter of P unit and an apothem of a units, then
*A*= ½*Pa* - Area of a circle = πr
^{2}

### Area of Regular Polygons & Circles

A regular polygon is an equilateral and equiangular polygon.

A rhombus is a regular polygon.

A square is a regular polygon.

Square ABCD, write the apothem of the square.

A regular hexagon ABCDEF, AB = 5m, find the area of this hexagon.

- m∠COD = [360/6] = 60
^{o} - ∆OCD is an equilateral triangle
- OC = CD = 5
- m∠COM = [1/2]m∠COD = 30
^{o} - OM = OC*cos30 = 5*cos30 = 4.33
- A = [1/2]Pa
- P = 6AB = 30m
- A = [1/2]*30*4.33

^{2}

The area of a cirlce = [1/2]πr

^{2}

Circle A, AB = 12 in, find the area of the circle.

- A = π(AB)
^{2} - A = 3.14*12
^{2}

^{2}

Circle A, regular pentagon BCDEF, AB = 5, find the area of the shaded region.

- m∠BAG = [1/2]*[360/5] = 36
- AG = AB*cos36
- AG = 5cos36 = 4.0
- BG = AB 8sin36
- BG = 5sin36 = 2.9
- For pentagon: P = 5*(2BG) = 5*2*2.9 = 29
- A
_{1}= [1/2]P*AG = [1/2]*29*4 = 58 - For the circle: A
_{2}= p*AB^{2}= 3.14*5*5 = 78.5 - Area of the shaded region: A = A
_{2}− A_{1}= 78.5 − 58 = 20.5

Square ABCD, circle E, EF = 4m, find the area of the shaded region.

- Area of the square: A
_{1}= AB^{2}= (2EF)^{2}= 64 m^{2} - Area of the circle: A
_{2}= πEF^{2}= 3.14*4*4 = 50.2 m^{2} - Area of the shaded region: A = A
_{1}− A_{2}= 64 − 50.2 = 13.8 m^{2}.

^{2}

If the apothem of a regular polyton is equal to the radius of a circle, then the area of the regular polygon is ______ larger than the area of the circle.

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Area of Regular Polygons & Circles

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Regular Polygon 0:08
- SOHCAHTOA
- 30-60-90 Triangle
- 45-45-90 Triangle
- Area of a Regular Polygon 3:39
- Area of a Regular Polygon
- Are of a Circle 7:55
- Are of a Circle
- Extra Example 1: Find the Area of the Regular Polygon 8:22
- Extra Example 2: Find the Area of the Regular Polygon 16:48
- Extra Example 3: Find the Area of the Shaded Region 24:11
- Extra Example 4: Find the Area of the Shaded Region 32:24

### Geometry Online Course

### Transcription: Area of Regular Polygons & Circles

*Welcome back to Educator.com.*0000

*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 polygon*0008

*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 create*0255

*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 apothem*0413

*(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 πr ^{2}.*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 triangle*0695

*(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 squared*0946

*(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 πr ^{2}--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 πr ^{2}; 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 diagram*2124

*(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

0 answers

Post by Edmund Mercado on April 4, 2012

Miss Pyo:

At 13:30, why did you choose OA instead of OH?

0 answers

Post by saloni bhurke on March 9, 2012

the definition of polygon was easy to memorize than the textual definition, thank you.