this video is a basic geometry review for those who are taking the sat or the ht exam and if you're taking a geometry final exam you could benefit from this video too so let's go over some common shapes and the formulas they need to know so let's say if we have a circle i know my drone is not that great but let's work with it and let's say that the radius of the circle is five given the radius what is the circumference of the circle and also what is the area the first formula you need to know the circumference is two pi r so therefore it's going to be two pi times 5 which is 10 pi so that's the exact answer sometimes you may need to put this in your calculator and get a decimal value so you could use the fact that pi is about 3.1416 so this is going to be 31.416 so that's the circumference now what is the equation for the area of a circle the area of a circle is pi r squared so for this particular example it's pi times 5 squared 5 squared is 25 so the area is 25 pi so 25 times pi i'm going to use the exact value as a decimal that's 78.54 now what about the diameter if you know the radius of the circle what is the diameter so the radius is between the center of the circle and it touches any point on the circle that's the radius the diameter also passes through the center of the circle but is the distance from one edge of the circle all the way to the other edge and it always has to pass through the center of the circle if you have a line that touches two points in the circle but doesn't pass through the circle i mean the center of the circle that is it's a chord so this line it touches these two points on the circle but it doesn't pass the center of the circle so that makes the chord but a diameter is between two points on the edge of the circle and then diameter passes through the center of the circle so make sure you know the difference between the diameter and the chord the diameter is twice the value of the radius it's 2r so in this case 2 times 5 is 10. so for a circle these are the three main equations you need to know the circumference 2 pi r the area pi r squared and the diameter is twice the length of the radius now the next shape that we're going to talk about is the square so let's say that the side length of the square is 8. what is the area and what is the perimeter of the square the area of a square is basically side squared all sides of the square are the same so in this case s is eight so the area is just going to be eight squared which is 64 square units now to find the perimeter the perimeter is the sum of the four sides it's s plus s plus s plus s if you add s four times it's the same as four times s so the perimeter is four times eight so it's 32 units long so make sure you know those two equations four square the area is side squared the perimeter is simply the sum of all four sides or 4s now here's a question for you going back to the square let's say the area is 36 square feet what is the perimeter of the square so we know that the area is s squared by the way for each of these questions pause the video and see if you can figure it out so the area is 36 if we take the square root of both sides we can get the length of each side so each side is six units long therefore the perimeter is six plus six plus six plus six four times or four times six so it's 24 feet long that's the perimeter now going back to the circle let's say if we're given the circumference of the circle let's say the circumference is 16 pi with this information find the length of the diameter and also the area of the circle so first we need to find the radius we know that the circumference is 2 pi r and the circumference is 16 pi so what we need to do is divide both sides by 2 pi so 2 pi divided by 2 pi is 1 here the pi's cancel so the radius is 16 divided by 2 so it's 8 units long if you have the radius you can easily find the diameter the radius is twice the length of the diameter so it's 16 units so now we can find the area which is simply pi r squared so it's pi times 8 squared or simply 64 pi so that's the area now let's say if you have a rectangle and let's say the length of the rectangle is 10 and the width is 5. what is the area and what is the perimeter of the rectangle feel free to pause the video and find these two things so this is the length this is the width this side is also the width and this is the length the area is simply length times width the perimeter is 2l plus 2w so the area is going to be 10 times 5 so it's 50 square units the perimeter is going to be 2 times 10 plus 2 times 5. 2 times 10 is 20 2 times 5 is 10 20 plus 10 is 30. so the perimeter is 30 units long and the area is 50 square units so let's say if the area is 40 units long and let's say the length is 8 units what is the perimeter of the rectangle go ahead and try that problem so the left is eight we don't know the width but we could find the width by using this equation so 40 is equal to 8 times w so w is 40 divided by 8 so it's 5 units long and then once you have the width you can now find a perimeter so it's going to be 2l plus 2w so it's a 2 times 8 plus 2 times 5 and that's 16 plus 10 which is 26. so here's a practice problem you could try so go ahead and take a minute and see if you figure it out the length of a rectangle is three more than twice the width if the area is 44 square centimeters what is the perimeter of the rectangle so take a minute and work on that problem so let's draw a picture so this is the left this is the width now let's write an equation the length is three more or three plus twice the width that's 2w and we know the area is 44 square units what is the perimeter rectangle if we could find the left and the width then we could find the perimeter the area we know it's a length times width and what we can do if we want to is we can replace l with 3 plus 2w so we can get the area equation in terms of w alone so we got to solve this by substitution so 3 plus 2w times w is equal to 44. now let's distribute w so w times three is three w w times two w is two w squared now let's move the forty four from the left side to the right side so zero is equal to two w squared plus three w minus forty 44. so what we have is the trinomial or quadratic expression and we need to factor it in order to find the value of w so how can we factor this particular trinomial what we need to do first is multiply the leading coefficient which is 2 by the constant term negative 44. 2 times negative 44 is negative 88. so what two numbers multiply to negative 88 but add to the middle coefficient three this is positive eleven and negative eight eleven plus negative eight adds up to three but they multiply it to negative eighty eight so now what we're gonna do is we're going to replace the middle term 3w with 11w and negative 8w so it's going to be 0 is equal to 2w squared minus 8w plus 11w minus 44. i wanted to put the 11 next to the 44 because 44 is a multiple of 11 and 8 is a multiple of two so in the next step we're going to factor by grouping and that's why i've arranged it the way i did so in the first two terms let's take out the gcf the greatest common factor is 2w 2w squared divided by 2w that's going to be w negative 8w divided by two w is negative four in the last two terms let's take out an 11. and let's get rid of some of this stuff over here actually i'm going to need that so let's just get rid of this and i don't think i need this for now so we take out an 11 11 w divided by 11 is w and negative 44 divided by 11 and that's negative four so now let's factor w minus four when those two terms are the same that means that you're on the right track you've done everything correctly so far so if we take out w minus 4 from this term what we're going to have left over is the 2w and if we take it out from the second term we're going to have 11 left over but it's going to be plus 11. so now what we need to do is set both factors w minus 4 and 2w plus 11 equal to 0. so if we add 4 to both sides we can see that w is equal to 4. and the other equation we got to start by subtracting both sides by 11. so w 2w is equal to negative 11. and if we divide by 2 w is negative 11 over 2. now we're going to get rid of the negative answer because we're dealing with a real life object and to have a side length of negative 5.5 it doesn't make sense so we're going to choose this value w is equal to 4. so if w is four we can now find the length which is three plus two w or three plus two times four so that's three plus eight and that's eleven so the left is eleven and the width is four and we can see why the area is left times width four times eleven which is forty four so that works out now we can find the perimeter the perimeter is 2l plus 2w so that's 2 times 11 plus 2 times 4 and so that's 22 plus 8 which is 30. so that's the answer to this particular problem that's the perimeter of the rectangle by the way for those of you who are taking the act exam or the sat exam when you get a chance go to youtube and search out my act math video and sat math video you can get more examples and multiple choice practice problems if you want to practice and prepare for the math sections of those exams i'm also going to post it at the last 20 seconds in the end of this video so you could find the link there as well or you just search it to youtube it should come up but let's continue with this problem the length of a rectangle is three more than its width if the perimeter is 26 what is the area of the rectangle so go ahead and try that problem now we know the perimeter is 2l plus 2w so 26 is equal to 2l plus 2w now notice that we could simplify this equation let's divide everything by 2. so 13 is equal to l plus w our goal is to find the area of the rectangle if we could find the dimensions if we could find the length and the width then we could easily find the area now we're told that the length is 3 more than its width so l is 3 plus w so we're going to do at this point is replace l with 3 plus w so 13 is equal to 3 plus w plus w so that's 3 plus 2w now we could find the value of w let's subtract both sides by 3. so 10 is equal to 2w and if we divide both sides by 2 ten divided by two is five so w is five l is three more than w so three plus five is eight so l is eight so we have the width and we have the length now we know that the area is length times width or eight times five so it's 40 square units and that's the answer now let's talk about triangles let's say if we have a right triangle and let's say one side one of the legs is three and the other leg is four what is the hypotenuse of the triangle let's call the side a b and c according to the pythagorean theorem a squared plus b squared is equal to c squared so we can say a is 3 b is 4 and let's find c 3 squared is 9 4 squared is 16 and 9 plus 16 is 25 so if you take the square root of 25 this will give you five so the length of the hypotenuse which is the side across the 90 degree angle represented by this box that is five units long let's try another example let's say the hypotenuse is 13 units long and one of the legs is five find a missing side now you need to know some special numbers for a right triangle there's the 3 4 5 triangle there's the 5 12 13 triangle which is the one that we need so notice that the missing side is 12 and some other ones they need to know is the 7 24 25 triangle the 8 15 17 triangle there's the 9 40 41 triangle and also the 11 60 61 triangle so if you know these triangles you don't have to use the pythagorean formula you could just simply find a missing number but let's confirm it using pythagorean theorem let's prove that this is 12. so we know that a squared plus b squared is equal to c squared and our goal let's say we're looking for b so a is 5 and the hypotenuse c is 13. 5 squared is 25 13 squared 13 times 13 is 169 and if we subtract both sides by 25 169 minus 25 is 144 so now we got to take the square root of 144 which is 12. so it pays to know the special y triangles here's another example so let's say that the hypotenuse is 10 and one of the size is six what is the miss inside go ahead and pause the video and figure it out so let's write the special triangles 3 4 5 5 12 13 8 15 17 and so forth notice that if we take the 3 4 5 triangle and if we multiply everything by two notice what happens we're going to get the six eight ten triangle so multiples of the three four five triangle also apply to a right triangle here this is six this is ten the missing number is eight so x is eight so let's prove that so a squared plus b squared is equal to c squared a is six we're looking for b and the hypotenuse is ten six squared is thirty six ten squared is one hundred a hundred minus thirty six is sixty four and the square root of sixty four will give us the missing side which is eight so let me give you a few examples for all of these find the missing side find the value of x you can get started with the first example okay so let's start with the first one the first one is associated with the 7 24 25 triangle therefore x is 25. now for the second one it's associated with the 8 15 17 triangle we have 15 and 17. so the missing side has to be 8. now for the third one notice that it's proportional to the three four five triangle it's not proportional to any other right triangle so if we multiply these numbers by three we're going to get two of the numbers that we have in the triangle three times three is nine four times three is twelve five times three is fifteen so if you see two sides present then you know the third side has to be twelve the next one is simply the nine it's the 40 actually i messed up on this one i meant this to be nine it's supposed to be the 9 40 41 but i got mixed up with the 11 60 61 so that's a bad problem ignore it now the next one is associated with the 5 12 13 triangle if we multiply those numbers by 2 it's going to be 5 times 2 is 10. 12 times 2 is 24 13 times 2 is 26 so we have two of these numbers 10 and 26 so the missing side has to be 24. now for the last one let me draw because of all the clutter this is 30 40 and we're looking for x notice that this is similar to the 3 4 5 triangle so if we multiply 3 by ten we get thirty four by ten forty so five times ten is fifty so the hypotenuse is fifty here's another problem that you could try at rectangle a b c d a b is 12 units long and ac is 13 units long what is the area of the rectangle now rectangles they form 90 degree angles and look at the triangle that forms we have a right triangle if this is 12 and that's 13 what is the missing side so we know this is the 5 12 13 triangle so length bc is 5. so now we can find the area the area of the rectangle is length times width the length is 12 the width is 5. so it's just 12 times 5 which is 16. so as you can see you can find the answer quickly if you know you specify triangles you don't have to use a pythagorean theorem formula if you commit this mermaid it will save you a lot of time and on the sat test and on the act test time is a factor so you want to find the answer quickly you