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**Volume of a Triangular Prism**. Following along are instructions in the video below:

in this video were gonna focus on calculating the volume of a triangular prism so lets start with an example problem so lets say that this triangular prism has a height of 20 centimeters and lets say this part is 12 centimeters and here this is 10 how can we calculate the volume of this triangular prism given the following information the volume of a triangular prism is the volume is the area of the base multiplied by the height and that is amuse capital H instead of low case H so 20 centimeters is the height of the prism 12 is the base of the triangle and 10 centimeters is the height of the triangle so capital B represents the area of the base or basically the area of this triangle to calculate the area of the triangle its 1/2 base times height so in this case is gonna be 1/2 the base is 12 and the height is 10 half of 12 is 6 and 6 times 10 is 60 so the area is 60 square centimeters so thats the area of the base so the volume of the prism is going to be the area of the base which is 60 square centimeters times the height of the prism which is 20 centimeters so 60 times 26 times 2 is 12 and then we need to add the 2 zeros so the volume is going to be 1,200 cubic

centimeters and so thats a simple way in which you can calculate the volume of a triangular prism now lets work on another example so lets say the height of the prism and this example is 10 inches and lets say the base of the triangle is 4 inches and the hypotenuse is 5 inches go ahead and calculate the volume of the triangular prism so once again we need to calculate the area of the base first so weve got to calculate the area of the right triangle so we have the base of the right triangle misson is the height and so we need to use the Pythagorean theorem to calculate the height first so we know that C squared is equal to a squared plus B squared C is the hypotenuse which is 5 we could say that a represents H in this example and B is 4 so 5 squared is 25 4 squared is 16 and 25 minus 16 is 9 now we need to take the square root of both sides so the square root of 9 is 3 so the height of the triangle is 3 so now we can calculate the area of the base which is the area of the right triangle so thats 1/2 base times height so 1/2 of 4 times 3 half of 4 is 2 and 2 times 3 is 6 so the area of the base is 6

square inches so now we can calculate the volume and its supposed to be capital B instead of lowercase B so the area of the base is 6 and the height of the prism is 10 so 6 times 10 is 60 so the volume of this triangular prism is 60 cubic inches and so thats the answer now lets try a different example so lets say if you have a prism that looks like this and lets say the height of the prism is 12 inches and the side of the triangle is 8 inches and youre told that its an equilateral triangle so with that information what is the area of the prison I mean nothing area but what is the volume of the prism rather so if we have an equilateral triangle that means that all three sides have the same length so each side is 8 inches and we know the volume of the prism is base times height so how can we calculate the area of the base what is the area of an equilateral triangle the area of an equilateral triangle is the square root of 3/4 times s squared where s is the side length in this case s is 8 so 8 squared is 64 and if we take 64 and divided by 4 thats 16 so the area of the base is 16 square root 3 square inches so now we can calculate

the volume of the prism so capital B thats the area the base thats 16 square root 3 and then the height of the prism is 12 inches so 16 times 12 thats 192 so the volume is gonna be 192 times the square root of 3 and its gonna be cubic inches so this is the final answer its actually the exact answer and if you want the decimal equivalent of that answer its about 300 32.55 cubic inches lets look at another example so lets say this time were given the values of all three sides of the triangle lets say the height of the triangle is 18 centimeters so everything is in centimeters calculate the volume of the prison so we know first we need to calculate the area of the base so how can we calculate the area of a triangle if were only given all three sides and its not a right triangle we need to use Herons formula so first you need to calculate s which is basically half of the perimeter of the triangle its a plus B plus C divided by two we can call this a B and C so its gonna be 9 + 10 + 11 divided by 2 9 + 10 is 19 and 19 + 11 thats 30 30 divided by 2 is 15 now once you have s you can calculate the area using this formula so its

times s minus a times s minus B times s minus C all inside of a square root symbol so keep in mind that s is 15 so the area is gonna be 15 times 15 minus a so thats 15 minus 9 times 15 minus B and B is 10 and then 15 minus 11 so 15 minus 9 thats 6 15 minus 10 is 5 15 minus 11 is 4 so now how can we simplify what we now have well we can take the square root of 4 so thats 2 15 Im gonna write that as 5 times 3 + 6 is 3 times 2 now we can take out 25 the square root of 25 is 5 and the square root of nine is Street so were left with 2 times 5 times 3 times the square root of 2 2 times 5 is 10 10 times 3 is 30 so the area is 30 square root 2 and Im just gonna check that with my calculator make sure I didnt miss anything and I got the same answer so now we can calculate the volume of the prism so its base times height the area of the base is 30 square root 2 and the height of the prism is 18 so 30 times 18 thats 540 so the answer is 540 square root 2 cubic centimeters or we could say thats approximately 760 3.7 cubic centimeters

tags:

volume of a triangular prism, triangular prism, prism, volume, geometry, practice problems, examples, problems, right triangle, equilateral triangle, area, i…

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