If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

# Challenge problems: perimeter & area

## Video transcript

let's do some example problems here so we have the perimeter of each of the outer triangles is 30 so for example if I took the sum of this side this side and that side I will get 30 and that's true of all of these outer triangles these five outer triangles they then tell us that the perimeter of F G H I J so F G H I J the perimeter of this Pentagon right over here is 50 so if I add up that side Plus that side Plus that side Plus that side Plus that side I get 50 and then they say what is the perimeter of the star so the perimeter of the star is really the outsides if you take the bases away of each of these triangles so it's this side let me do this in a new color actually so the perimeter of the triangle I'll do it in orange it's going to be this Plus that Plus that Plus that Plus that Plus that I think you get the idea Plus that Plus that Plus that Plus that so the perimeter of the of the star so let me call this the perimeter perimeter of the star is going to be equal to the perimeter of the five triangles is equal to perimeter of five outer triangles I'll just call them five triangles like this minus their bases right if I take the perimeter of all of these guys if I added it up the part that shouldn't be part of the perimeter of the star should would be this part that part that part that part that part in that part those aren't the part those aren't part of the perimeter of the star so it should be the perimeter of the five triangles minus the length of their bases length of their five bases so what's the perimeter of the five triangles well the perimeter of each of them is 30 so the perimeter of five of them is going to be five times 30 which is 150 now we want to subtract out the lengths of their five bases now the lengths of their five bases if we add them up is the exact perimeter of this inner Pentagon right over here so this inner pentagons has a perimeter of 50 that is the sum of the five bases so that right over here is 50 so the perimeter of the star is going to be 150 minus 50 or or 100 all we did is took the perimeter of all the triangles subtracted out these bases which was the perimeter of the inner Pentagon and and and we're done now let's do the next problem what is the area of this this quadrilateral something that has four sides of ABCD and this is a little bit we haven't seen a figure quite like this just yet it on the right-hand side it looks like a rectangle on the left-hand side looks like a triangle this is actually a trapezoid but we can actually as you could imagine the way we figured out the area of several triangles just splitting it up into into pieces that we can recognize and the most obvious thing to do here is start at a and then just drop a rock drop an altitude right over here and so this line right over here is going to hit at 90 degrees and we could call this point E and what's interesting here is now we can split this up into something that we recognize a rectangle and a right triangle but you might say wait Sal how do we figure out how do we figure out what these you know we have this side on that side so we can figure out the area of this rectangle pretty straightforwardly but how would we how do we figure out the area of this triangle well if this side is 6 then that means that this that EC is also going to be 6 if a b is 6 notice we have a rectangle right over here opposite sides of a rectangle or equal so if a b a b equals 6 implies that ec is equal to 6 ec is equal to 6 so EC is equal to 6 and if EC is equal to 6 then that tells us that de is going to be 3 de is going to be 3 this distance right over here is going to be 3 and we know that because if this is 6 this has to be something that we add to 6 to get 9 9 was the length of this entire of the entire base of this figure right over here 9 was this entire distance so 9 minus 6 gives us the 3 and now we have all the information that we need to figure out the area the area of this part right over here this rectangle is just going to be six times seven so it's going to be equal to 42 plus the area of this triangle right over here plus the area of this triangle right over here and that's one-half base times height one-half the base over here is three one-half times three and the height over here is once again it's going to be seven this is a rectangle opposite sides are equal so if this is seven then this is also going to be seven one half times three times seven so it's going to be 42 let's see three times seven is twenty-one 21 divided by two is ten point five ten point five so this is going to be equal to 52 point 5 52 point five is the area of this entire figure let's do one more so here I have a bizarre-looking a bizarre-looking shape and we need to figure out its perimeter and at first it seems very daunting because they've only given us this side and this side and they've also only given us this side right over here and one thing that we are allowed to assume in this and you don't always have to make you can't always make that assumption and I just didn't draw it here ahead of time just because it would have really crowded out this this diagram is that all of the angles in this diagram are right angles so I could have drawn a right angle here right angle here right angle there right angle there but as you can see it kind of makes things a little bit it makes things a little bit messy but how do we figure out the perimeter if we don't know these little distances if we don't know these did little distances here and the secret here is to kind of shift the size because all we want to care about is the sum of the side so what I'm going to do is a little exercise and shifting the sides so this side right over here I'm going to shift it and put it right up there then this side right over here this length right over here I'm going to shift it and put it right over there then let me keep using different colors then this side right over here I'm going to shift it and put it right up here then finally I'm going to have this side right over here I can shift it and put it right over there and I think you see what's going on now now all of these sides combined are going to be the same as this side that where I'm kind of building even though this thing wasn't a rectangle it's it's it's perimeter is going to be a little bit interesting although we have to think about this too right over here now let's think about all of these sides that are going up and down so this side I can shift it all the way to the right and go right over here let me make it clear this orange side goes all the way to the end right that's just the exact same or inside now this white side I can shift all the way to the right over there then this this green side I can shift right over there and then I have and then I can shift and then I can shift this actually let me not shift to that green side yet let me just leave that green side so I haven't I haven't done anything yet let me be clear I haven't done anything yet with that and that I haven't shifted them over and let me take this side right over here and shift it over so let me take this entire thing and shift it over there and shift it over there so before I count these two pieces right over here and we know that they each have length two these are all 90-degree angles so this has linked to this is linked to before I count those two pieces I've shifted everything else so I was able to form a rectangle so at least just counting everything else I have 7 plus 6 let me say 7 plus 6 all of these combined are also going to be 7 plus 7 and then all of these characters combined are also going to be 6 plus 6 and then finally I have this 2 right here that I haven't counted before this 2 plus this 2 plus this 2 and then we have our perimeter so what does this give us 7 plus 6 is 13 plus 7 is 20 plus 6 is 26 plus 4 more is equal to 30 and we're done