in this video we're going to focus on calculating the area of a regular polygon so let's start with a regular hexagon and let's say the apothem is let's call that a let's say the apothem is 8 square root 3 and the side length is 16. with this information go ahead and calculate the area of this regular hexagon the area for any regular polygon it's one half times the apothem times the perimeter now what is the perimeter of this particular hexagon because it's a regular hexagon all six sides are congruent so they're all equal to sixteen so the perimeter is going to be six times sixteen it's adding sixteen six times six times 16 is 96 so the area of this regular hexagon is one-half times the apothem times the perimeter half of eight is four and 4 times 96 that's 384 so the area is 384 times the square root of 3. and so that's a simple way in which you can calculate the area for a regular polygon let's work on another example so let's say if we have a regular pentagon this time and let's say the apothem is nine units long and the side length of the pentagon is 10.6 go ahead and calculate the area so first let's calculate the perimeter the perimeter for this example is going to be 5 times 10.6 since we have a five-sided figure and so 5 times 10.6 that's 53 so that's the perimeter and the area is one-half times the apothem multiplied by the perimeter so the apothem is nine the perimeter is 53 so 9 times 53 times 0.5 that's equal to 238.5 so that's the area of the pentagon 238.5 square units now what would you do if you're given the side length of a regular hexagon let's say the side length is 20 and that's all you're given how can we calculate the area of the regular pentagon not the pentagon but regular hexagon feel free to pause the video and try now the first thing we need to do is calculate the angle and to do that first i need to draw a bigger picture so this is the center of the hexagon let's call this point a point m and point v so a is the center and the distance between a and m is the apothem between the center and the vertex is the radius and if this is s for the side length of a regular hexagon mv has to be s over two because the apothem is the distance between the center and it meets the side length it meets this segment right at the middle let's call this b so right at the midpoint between b and v which means b m and m v are congruent to each other so if b v is s m v has to be s divided by two and let's call this theta so we gotta calculate theta first now how can we do that there's a formula that you can use theta is 360 divided by 2n where n is the number of sides in the regular polygon so we have a hexagon and a 6 2 times 6 is 12 and 360 divided by 12 is 30 degrees so that's the angle that's step one you want to calculate that first now if you want to see it visually break up the hexagon into six congruent parts now the angle of a full circle we know it's 360 degrees and because there's six equal parts we need to divide 360 by six which will give us 60. so each of these angles is 60 degrees now if we draw a line here which is basically what we did that's the apothem and we know that this part here is 60 60 divided by 2 will give us this angle which is 30 and that's theta so that's why that angle is going to be 360 over 2n because you're going to split this part into two so now that we have the angle what do we need to do next let's focus on this triangle so the angle is 30 degrees and what is s over 2 well s we know it's 20 so s over 2 has to be half of 20 so s over 2 is 10. now how can we find the other two sides using this triangle if you have a 30-60-90 triangle there are some things that you could use if you want to avoid using trigonometry let's say if the hypotenuse is two across the 30 it's going to be half of whatever the hypotenuse is so that's going to be one for the side across the 60 it's whatever this side is times the square root of 3. so this is going to be 1 square root of 3. so this is 10 across the hypotenuse i mean across the 90 degree angle which is the hypotenuse and it's going to be twice the value of this so this is going to be 20. and across the 60 degree angle it's whatever this is times square root 3. so this is 10 square root 3. so here's what we know so far the apothem is 10 square root 3. all we need to know or all we need to find at this point is the perimeter and to calculate the perimeter it's simply n times s we have a six sided figure so n is six and the side length is twenty so the perimeter is a hundred twenty so once you have the apothem and the perimeter you can now calculate the area so it's one half ap the apothem is 10 square root 3 and a perimeter is 120. so half of 10 is 5. and 5 times 120 we know 5 times 100 is 500 and 5 times 20 is 100. so 500 plus 100 is 600 so the area is 600 times the square root of three and so as a decimal that's about 1039 square units so that's how you can calculate the area of a regular hexagon if you're given just a side length now let's work on another example very similar to last one but with a pentagon so let's say the side length of the pentagon is 30. calculate the area so first let's calculate the angle the angle is going to be 360 divided by 2m so for a pentagon we have 5 sides so n is 5 and 2 times 5 is 10. so the angle is 360 divided by 10 which is 36 degrees so that's the first thing i'm going to do now let's draw the triangle so this is r the radius this is the apothem and this is s divided by 2. so we know that s is 30 because that was given to us so s divided by 2 is 30 divided by 2 so that's 15. so i'm going to erase this and let's replace that with 15. now keep in mind the angle theta is not this angle it's this one based on the way we defined it so this is 36 degrees so the other angle is 90 minus 36 so this angle if for some reason you want to find it is 54. now we can use any one of those two angles to calculate the apothem so let's focus on 36 because that's the first one we had we really don't need this one but you could use it if you want to so which trig function sine cosine or tangent relates a and 15. so relative to 36 15 is opposite to it and a the apothem is adjacent to it and tangent is opposite over adjacent so tan 36 is equal to the opposite side which is 15 divided by the apothem so in this case we can't use the special 30 60 90 triangle we have to use trig to get the answer now before we get the answer i want to show you how to get this angle visually for those of you who prefer it that way so from the center draw a line to each vertex to every vertices of the pentagon and so 360 divided by five let's see what that is that's 72. so each of these angles is 72 degrees now if we split this triangle into two parts that's going to be 72 divided by 2 and so that's how you can get 36 if you want to do it that way i think it's just easier just to divide it by 2n so if you have a 5 sided figure just divide it by 10 and you can get that angle so here's the apothem and here's the radius of the figure so let's calculate the apothem let's cross multiply 1 times 15 is 15 and this is equal to a times tangent of 36 so this is the 36 degree angle and this side is 15. so now to get a we got to divide both sides by tan 36. 15 divided by 10 36 is 20.65 if you rounded to the nearest hundredth place and so that's the apothem in this example so now we can calculate the perimeter and then that's going to help us to calculate the area so just like before the perimeter is n times s so we have a five-sided figure and s is 30. so the perimeter is 5 times 30 or 150. so we have the perimeter and we have the apothem so now let's calculate the area the area is going to be one-half ap so the apothem is 20.65 and the perimeter is 150 so the apothem i'm gonna round it is approximately 1550 so i rounded to three sig figs i got 15 48.75 but keep in mind that this is a rounded answer so i'm gonna round this one as well let's say if we have a three-sided polygon which is basically an equilateral triangle and let's say the side length is 20. what is the area of this triangle now we're going to calculate it using two techniques the first a simple formula the area of an equilateral triangle is the square root of 3 divided by 4 times s square and so in this example s is 20. now 20 squared which is 20 times 20 that's 400 and 400 divided by four is a hundred so the area of this equilateral triangle is a hundred square root three and so that's the answer but now let's use the other formula to confirm this answer so first let's calculate theta so we know it's 360 divided by two n and for triangle there's three sides and two times three is six so 360 divided by 6 is 60 degrees so now that we have that angle let's calculate the apothem now if s is 20 this side s divided by 2 that's 20 divided by 2 so that's 10. now this other angle is 30. so we have a 30 60 30-60-90 triangle so if you want to use geometry to find a here's we can do so this is 2 across the 30 it's half of the hypotenuse across the 60 it's this number times the square root of 3. so notice that we have this value how can we find a well if we have this value and we need to get that value we need to divide by the square root of 3. so it's going to be 10 divided by the square root of 3. so the apothem is 10 over the square root of 3. now if you want to use trig to get that same answer we could use the tangent ratio tangent of 60 is equal to the opposite side that's opposite to 60 divided by the adjacent side which is the apothem now tan 60 if you type that in your calculator you should get like 1.73 which is the square root of 3. so if we cross multiply 1 times 10 is 10 and this will be a times the square root of 3. so notice that the apothem is 10 divided by the square root of 3. so both methods can work so remember sohcahtoa sine is opposite over hypotenuse cosine is adjacent over hypotenuse and tangent is equal to the opposite side divided by the adjacent side now let's calculate the perimeter so it's n times s for a triangle we have three sides and s is 20. so three times 20 is 60. so the area is going to be one half ap the apothem is 10 divided by the square root of 3 and the perimeter is 16. now one half of 10 is 5. so we have 5 times 1 over square root 3 times 60. now 5 times 60 is 300 so we have 300 divided by the square root of 3. now let's rationalize it let's multiply the top and the bottom by the square root of 3. the square root of 3 times the square root of 3 is the square root of nine which is three and three hundred divided by three is a hundred so this will give us the same answer a hundred square root three but just a lot longer but it does work for any regular polygon this formula let's say if we have a regular hexagon and this time we're not given a side length but we're given the radius of the hexagon and let's say the radius is 10. calculate the area of the regular hexagon so let's draw a picture so we're given r that's 10 and we don't know the apothem and we don't know s over 2. however we can find the angle so the angle is going to be 360 divided by 2n so n in this example is 6 and 2 times 6 is 12 so 360 over 12 that will give us an angle of 30 degrees which means this is 60. so now that we have the angle we could use the 30-60-90 triangle to get everything else so across the 30 it's going to be half of the hypotenuse so half of 10 is 5. across the 60 it's this number times the square root of 3. so it's 5 square root 3. so that tells us the apothem is 5 square root 3 and if s divided by 2 is 5 s has to be 5 times 2 which is 10. so let's draw a picture so this is 5 but this is 10 and this is 5 square root 3. now let's calculate the perimeter it's n times s so n that's six we have six sides and s is ten so it's sixty so the area is one half ap so the apothem is 5 square root 3 and a perimeter 60. so half of 60 is 30 and 30 times 5 is 150 so the area is 150 square root 3. and so that's how you can calculate the area of a regular hexagon if you're given the radius try this problem the radius of the circle is 24 centimeters what is the area of the equilateral triangle now the radius of the circle you need to realize is the apothem of the triangle and this r is the radius of let me do that better that's the radius of the triangle and this is s over 2 and here we have theta so theta is going to be 360 over 2n and n is 3. so 360 divided by 6 is 60. the apothem we see that it's 24 centimeters and so let's draw the triangle so this is 60 this is 30 and this is 24. so that's 24 the hypotenuse has to be twice the value it's 48 and across the 60 it's whatever the value across the 30 is times the square root of 3. so we know that s over 2 is 24 square root 3. so s is twice that value so s is 48 square root 3. so now that we have s and a we can calculate the area so the perimeter is going to be ns so that's 3 because we have a triangle so 3 times 48 is 144 so the perimeter is 144 square root 3. so now let's calculate the area so the area is going to be one-half ap the apothem is 24. and this is the perimeter half of 24 is 12 and 12 times 144 is 1728 times the square root of three so this is the answer now let's confirm this answer using that other formula so the area of an equilateral triangle is the square root of 3 over 4 times s squared and we have s so s is 48 times the square root of 3. now 48 squared that's 2304 and the square root of three squared that's like the square root of three times the square root of three which is the square root of nine and that's three so we gotta multiply that by this so 2304 times 3 that's 69 12 divided by 4 that will give you 17 28 square root 3 and so you can get the same answer so now let's talk about what we've learned in this video so you now know how to calculate the area of any regular polygon if you're given just a side length if you're given the apothem which is what we did in the last problem the apothem was the radius of the circle or if you're given the radius which we did like two problems ago so now at this point you should be able to calculate the area of any regular polygon in many different ways so if your teacher gives you the side length the apothem the radius you now know how to calculate the area from each perspective so that's all i have for this video thanks for watching and i wish you well on any tests you may have coming up you