Volume and surface area
Solid Geometry Volume Volume of triangular prisms and cubes
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- Let's do some solid geometry and volume problems.
- so they tell us shown is a triangular prism.
- and so there's a couple of types of three dimensional figures that deal with triangles.
- And this is what a triangular prism looks like
- it has a triangle on one two faces and they are kind of separated, they kind of have rectangles in between.
- the other kind of triangular three dimensional figures
- as you might see would be pyramids
- this is a rectangular pyramid, cuz it has a rectangular, or it has a square base, just like that
- you could also have a triangular pyramid where literally every side is a triangle.
- but this over here is a triangular prism.
- I don't want to get to much into the shape classification.
- if the base of the triangle b is equal to 7
- the height of the triangle 'h' is equal to 3
- and the lenght of the prism 'l' is equal to 4
- what is the total volume of the prism?
- so they are saying that the base is equal to seven
- so this right over here is equal to base is equal to seven
- the height of the triangle is equal to 3
- so this right over here
- this distance right over here
- 'h' is equal to 3
- and the length of the prism is equal to 4
- so i'm assuming it is this dimension
- right over here is equal to 4
- so length is equal to four
- so in this situation what you really just have to do
- is figure out the area of this triangle right over here
- we could figure out the area of this triangle
- and multiply it by how much you go deep
- so multiply it by this length
- so the volume is going to be the area of this triangle
- let me do it in pink
- the area of this triangle
- we know that the area of a triangle
- is one half, times the base, times the height
- so the area
- this area right over here is going to be
- one half times the base times the height
- and we are going to multiply it by like kind of our depth of our triangular prism
- so we have a depth of four
- so we are going to multiply that
- times the four
- times this depth
- times the four
- and we get, let's see
- one half times four is two
- so these guys cancel out, you'll just have a two
- and then 2 times 3 is 6
- 6 times 7 is forty...
- is forty two
- and it would be in some kind of cubic unit
- so if these were in
- i don't know
- centimeters, it would be centimeters cubed
- but they are not making us focus on units in this problem
- let's do another one
- shown is a cube
- if each side is of equal length 'x' is equal to 3
- what is the total volume of the cube?
- so each side is equal length x
- which happens to equal to 3
- so this side is 3
- this side over here x is equal to 3
- every side x is equal to 3
- so it's actually the same exercise as the triangular prism
- it is actually a bit easier when you are doing it with a cube
- where you really just want to find the area of this surface right over here
- now this is pretty straight forward
- this is just a square
- it would be the base times the height
- or since they are the same it is just 3 times 3
- so the volume is going to be the area of this surface
- 3 times 3
- times the depth
- times the depth
- so we go 3 deep
- so times
- so times
- three
- and so we get 3 times 3 times 3
- so this is 27
- or you might recognize this from exponents
- this is the same thing as three to the third power
- and that is why sometimes if you have something to the third power
- they'll say you cubed it
- because literally to find the volume of a cube
- you take the length of one side and you multiply that number by itself three times
- one for each dimension
- one for the length, the width, and or I guess the height,
- the length and the depth.
- depending on how you want to define them.
- so it's literally just 3 times 3 times 3
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