Triangular Prism Volume Surface Area Base and Lateral Area Formula Basic Geometry

in this video we're going to talk about how to calculate the volume and the surface area of triangular prism but first let's draw picture my picture is not perfect but you can see what I'm trying to draw and let's draw rectangular prism now if you recall maybe from previous video that the volume of rectangular prism is the length times the width time the height now look at the triangular prism notice that it's half of rectangular prism let me draw it inside the rectangular prism in blue as you can see it's about half of the volume of the rectangular prism so it turns out that the volume of triangular prism is 12 length time width time height but we're going to use different symbols to represent it just want you to compare the two so you can easily calculate the volume of triangular prism now throughout this video we're going to use this as the base of the triangle and this is going to be the height of the triangle now sometimes you can call this the height of the prism itself but don't want to confuse this AG with this one so I'm just going to call it the length of the prism just to distinguish it perhaps you've been taught in class that to find the volume of any prism is the area of the base times the height the base of the triangle of prism is basically triangle and the area of triangle is 12 base time height so the area of the base which is the area of the triangle is 1 12 base time height now this this is the height of the prism which we defined it to be so this and blue is it's not the height of the triangle it's the height of the prism which I'm going to call it the length of the prism so notice that this formula is very similar to this one it's 1/2 times the three dimensions so that's how you could find the volume of triangular prism now let's talk about how to find the surface area of rectangular prism so I'm going to draw this same picture and we're going to use the same dimensions so once again we're going to Define this to be the length of the prism this is the height of the triangle and the base of the Triangular base now in order to find the surface area of prism it's going to be the area of the base plus the lateral area of the prism now the area of the base is basically triangle so the area of triangle is 1/2 base time height so capital represents the area of the base lowercase is simply the length of one side of the triangle the base of the triangle so just make sure you distinguish capital from lowercase so this is just the area of one triangle but we have the triangle in front and we have the triangle in the back so there's two of them therefore we need to multiply this quantity by two to find the lateral area is basically the perimeter of the prism times the height and remember you need to distinguish the height of the triangle from the height of the prism the height of the prism we defined it in this video to be not not the height of the triangle so that's the lateral area is the perimeter of the base multiplied by the height or the length of the prism the perimeter is basically the perimeter of the base is the perimeter of the triangle so it's that length in blue let's call the hypotenuse of the Triangle so the perimeter is going to be + + so therefore here's your simplified equation you need the surface area is going to be the base time the height 2 * 12 is basically 1 so you can cancel it plus the perimeter times the length of the prism where the perimeter is just plus plus so I'm going to write the equation like this but if you want to learn it this way you could simplify your calculations if you want to replace with plus plus plus so you can also use this formula if you find it easier but I'm going to refer to this one most of the time now that we have all the formulas that we need let's work on some examples so let's draw our prism first and let's say the length of the prism is in the base of the prism is 3 in and the height is 4 in go ahead and calculate the volume and the surface area of the prism so first we need to identify what we have the length is 6 in the base is 3 in and the height is 4 in so keep in mind this is we're going to Define this as and this is so now just write the formula the volume is going to be 12 base time height time the length of the prism so it's 12 * 3 * 4 * 62 of four is two so we're left with 2 * 3 * 6 2 * 3 is 6 so we have 6 * 6 which is 36 and don't forget to write the units inches time inches time inches is inches Cube or cubic inches so the volume is 36 cubic in now let's find the surface area of this object so the first thing you want to do is you want to find the length of the hypotenuse which we'll call if you recall whenever you have right triangle you can use the pyan theorem to find hypotenuse so is equal to the square root of A2 + B2 or you can say that c^2 is 2 + b^2 in our example is three is four don't confuse this with this mean you can make it the same and you can make this but just understand that this that I've highlighted there you can use it to represent four or three it doesn't matter 3 is 9 4 squ is 16 9 + 16 is 25 and if you take the square root of both sides this will give you five so that's the length of the hypotenuse so now that we know that is let me put this in different color is equal to five we can now find the surface area but let's calculate the perimeter first so remember the perimeter is going to be the perimeter of the Triangular base highlighted in blue so the perimeter is + + so the length of the base is three the height that's four and the length of the hypotenuse of the triangle that's five 3 + 4 is 7 7 + 5 is 12 so the perimeter is 12 in now that we have the perimeter we can calculate the surface area the surface area is going to be the area the base which is it simplifies to base time height when you incorporate both triangles not just one remember it's 1/2 base time height but there's two triangles so 2 * half is whole so it's going to be base time height that's the area of the two triangles plus the lateral area which is the perimeter times the length of the prism so is three the height is four and the perimeter that's going to be we said it's 12 and the length is 6 3 * 4 is 12 12 * 6 is 72 and 12 + 72 is 84 so the surface area is going to be 84 Square in anytime you're dealing with area you're going to the units should be squared they could be square feet square in square yards whenever you're dealing with volume it's going to be cubic feet cubic Ines cubic yards so just keep that in mind now let's work on some practice problems all want you to do in this example is simply to find the volume of this triangular prism that's it so let's say this side is 10 cm this side is five and this portion is four that is the measure of the red line that's in the middle so go ahead and just find the volume only of this prism you don't have enough information to find surface area the first thing we need to do is identify the three dimensions that we have this is the length of the prism this is the base of the triangle and this is the height of the triangle so the volume is going to be 12 base time height time the length so it's 12 the base of the triangle times the height of the triangle times the length of the prism now 5 * 4 that's 20 and 1 12 of 20 is 10 so we're left with 10 * 10 which is 100 so the volume is going to be 100 Cub cm in this example so as you can see it's pretty straightforward to find the volume of triangular prism all we need to know is this formula and if you identify those three dimensions the length of the prism the base of the triangle and the height of the triangle then finding the volume will not be difficult now in this example want you to find the surface area let's say this side is 15 and this side is six this side is 10 and we have this red box here so using this information go ahead and calculate the surface area now in order to find surface area we need to find the perimeter of the Triangular base so we need all three sides we only have two sides so we got to find the missing side but notice that it's right triangle so let's redraw the right triangle we have the base of the triangle but not the height or you can call this the height and call this the base the order really doesn't matter your answer will be the same but let's say 6 is the base and this part is the height 10 which is across the 90° angle you have to see that 10 is basically the hypotenuse this picture in reality is the same as this picture which we've considered already if you notice it's just flipped on another side so the base here is six that's we have the hypotenuse 10 which is this is the hypotenuse highlighted in green and we're looking for the height but the length is still 15 so as you can see this portion highlighted in green on the right side is basically the same portion but just flipped on the ground so imagine taking this triangle and just flipping it on one side so basically the side that's highlighting green on the right is flipped on the bottom on the left but it's the same type of triangle just in case you're wondering now let's go ahead and find the surface area so we got to find first c^2 is equal to A2 + B2 so we have is 10 is six and we're going to say is the same as so let's replace squ with and b^2 is 6 2 10 2 is 100 6 that's 36 100 - 36 is 64 so 64 is equal to H2 and if we take the square root of both sides the square root of 64 is mean not but it's 8 which is in this example so the height is eight units and length so now we could find the perimeter now that we have all three sides of the triangle so the perimeter it's going to be the length of the base plus the height of the triangle plus the length of the prism so it's going to be 6 + 8 + which is 15 actually no wait take that back it's not the length of the prism it's the hypotenuse of the triangle we have to add this value to it so is 6 is 8 and the hypotenuse of the triangle is 10 6 + 8 is 14 + 10 that's 24 so that's the perimeter now we could find the surface area so it's going to be * plus the lateral area which is the perimeter times the length by the way sometimes you might get some problems that just want you to find or that just may ask for the lateral area which is just * so anytime you need to find lateral area just take the perimeter and multiply by the length of the base so is 6 is 8 the perimeter is 24 and is 15 6 * 8 is 48 and 24 * 15 is 360 but this number 360 that is the lateral area so for those of you who need to find it that's how you could find it it's 360 square units but to find surface area you need to add the lateral area plus the area of the base the two triangles so 360 + 48 that's 48 square units so that's the surface area now what do you think can we find the volume of this triangular prism it turns out that we can feel free to pause the video and calculate the volume of the prism so the volume is2 base time height time out the base of the triangle is six which we could see it here the height of the triangle we know it to be eight and then the length of the prism is 15 12 of six is three so we have 3 * 8 * 15 and 3 * 8 is 24 and 24 * 15 if recall correctly believe that's 360 is what we said before so it's 360 cubic units whatever the units are which didn't write for this problem so that's the volume of this prison for the sake of practice let's try one more problem so let's see if you recall everything that you've learned today and if you can apply it for this problem so want you to find the volume and the surface area of this triangular prism so let me just go ahead and draw everything that you need so let's say this is nine 9 and nine and this is 18 and this portion highlighted in blue let's say that's 7.8 so with this information you should be able to find everything the volume of the triangle and also the surface area so go ahead and take minute and figure that out so let's start with the volume we have the length we also have all three sides of the triangle and this is the height of the triangle so if that's the height then this portion must be the base and the base is not so this is basically an equilateral triangle because because all three sides are equal so now let's calculate the volume it's 12 * * so the base of the triangle is 9 the height of the triangle is 7.8 and the length of the triangle is 18 12 of 18 is 9 so we have 9 * 9 * 7.8 9 * 9 is 81 81 * 7.8 let's use the calculator for that so that's about 6318 cubic units so that's the volume of this prism now let's calculate the surface area so what is the first step to find surface area the first step is to calculate the perimeter and we have all three sides so we don't need to use the Pythagorean theorem to find the third side in this example so the perimeter of that triangle is going to be 9 + 9 + 9 just add up all three sides of the triangle so if you add nine three times it's basically the same as doing 9 * 3 multiplication is just repeat addition 9 * 3 is 27 so now you have the perimeter of the triangle next let's calculate the lateral area first because sometimes you may be asked just to do that if we need to find the lateral area is simply the perimeter times the length the perimeter is 27 the length of the prism is 18 so 27 * 18 that's 486 square units because we're dealing with area so let's say if this was 9 in if everything was in inches perimeter is just going to be in inches 27 in lateral area any type of area or Surface area that's going to be square inches or square units so that's the lateral area now to find surface area it's the area of the base plus the lateral area the area of the base the two triangles it's 2 * 1/ 12 base * height which is just base time height and the lateral area is perimeter times the length of the prism so it's going to be which is 9 is 7.8 the perimeter is 27 and is 18 9 * 7.8 that's about 70.2 and we know that 27 times 18 we already have 486 for that so we just got to add these two numbers 486 + 70.2 that will give us 556.org so that's it for this video thanks for watching and have good day now want to show you one of my algebra courses that might be useful to you if you ever need it so go to udemy.com now in search box just type in algebra and it should come up so it's the one the image with black background so if you select that option and if you decide to go to course content you can see what's in this particular course so the first section basic arithmetic for those of you who want to focus on addition subtraction multiplication and division and it has video quiz at the end it's multiple choice video quiz you can pause it work on the problems and see the solutions it covers long division multiplying two large numbers and things like that the next tutorial is on fractions adding subtracting fractions multiplying dividing fractions converting fractions into decimals and so forth so you can also take look at that next solve the linear equations which we covered and just more examples if you need more help 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