Geometry Part 28 Volumes of Prisms and Cylinders, and Surface Area of Cylinders

Hello, I miss, let's start in this video. We will talk about

volumes

of certain space figures called

prisms

.

Prisms

have two faces in parallel planes, and these faces are congruent polygons, so, for example, in the prism on the left we have a triangular face above and a congruent triangular face below, similarly in the prism on the right we have a five-sided regular polygon, ie Pentagon whose pentagonal shape is in a plane at the top and in a parallel plane at the top Bottom of the prism The faces, which are congruent polygons, can be of any shape.

Prisms

are named after this shape. Here on the left I have a rectangular prism, a hexagonal prism and a pentagonal prism and a box is an example of a rectangular prism. We learned that the volume of a box can be thought of as length times width times height, but we also learned that you can think of it as base times height. Since this

surface

has dimensions w by l above, I could use this

surface

denote as capital letters. b b is the

area

of ​​the base. Well it turns out that regardless of the shape of the base, we can still find the volume the same way by multiplying the

area

of ​​the base times the height, so let's try this, we'll find the volume of the given triangular Finding the prism we know our height in this diagram equals 7 meters which we need to find the

area

In this triangle the

area

of ​​the triangle is half a small b times h.

The height of the triangle and the base of the triangle are two different sizes, and the base and height I'm talking about here, you have to be careful not to confuse the two, the

area

of ​​the triangle is five-thirty by four anyway, so half of 24 so 12 and the

area

is always measured in square units so the

area

of ​​that base is 12 square meters if we want the volume of the prism we take that

area

of ​​the base and multiply it by the height of the prism so That's 12 square meters by 7 meters, that's 84 cubic meters. Remember that volume is always measured in cubic units so we can apply the same idea to find the volume of boxes to the volume of any prism and actually right circular

cylinders

work the same way.

We can still find the volume of a right circular cylinder by finding the

area

of ​​the base multiplied by the height, it just so happens that for a right circular cylinder the

area

of ​​the base is the

area

of ​​a circle well if there is a is a circle of radius r, the

area

of ​​the base is pi r squared. I'm just going to name it capital b for the

area

of ​​the base so we know that the volume of the cylinder is base times height, which is why the formula for the right circular cylinder is: volume equals pi r squared h base times height that I'm going to pull out the mesh where you're pulling the object a

part

to make it look like a crushed soda can, we've taken the top and bottom a

part

and unfolded the sides of the can and what emerges is a rectangle one dimension of that rectangle is h the other dimension we need if we want to get the

surface

note that this is exactly the distance around the circle we call the circumference of a circle and you probably remember that the circumference is two pi times the radius, so the

area

of ​​the sides of the can if you imagine a soda can being flattened the

area

here is 2pi r by h okay but that doesn't include the F Let's look at this

surface

formula here. 2 Pi r h are the sides of the can we just found squared The

area

of ​​this circle is pi r squared we have two of them so the total

surface

area

is two pi r h plus two pi r squared.

Most of the time you just have to realize that it's the right formula and apply it so let's try that here we have a right circular cylinder that has a height of 10 and a radius of its base is 2 meters so we find the volume and the

surface

. The volume will be pi times r equals 2 so we have 2 squared times h equals 10. So the volume will be 4 times 10 which is 40 and then times pi and the units are in cubic meters. You see, when you're working with an object spinning in it, pi comes up a lot. Remember that pi is the number which is approximately 3.14 but it is a decimal number that goes on forever without repeating without ending there is no way to write it all down so let's write the greek letter pi this is the only way to give exact solution very often when you do your homework they will ask you to leave pi in your answer or they will say give exact solution and then give your answer like this if they ask you however , appreciating the answer, they will often tell you to use 3.14 for pi, and when they do, don't make the mistake of using the pi button on your calculator, as this is not limited to 3, 14 and rounds it will give you a slightly different answer and it might confuse you so just put in 3.14 so I'll find the approximate value of this volume using 3.14 as an estimate for pi which I egg Calculator needs to enter 40 times 3.14 and let's see what I get, okay, that's 125.6, and I still have to enter the cubic meters.

So if they ask you for an exact answer for volume you would say 40 pi cubic meters if they ask you for an approximation with 3.14 pi they say 125.6 cubic meters Now let's do the

surface

area

for this right circular cylinder find so the formula for

surface

area

is pi r h plus 2 pi r again squared r is 2 meters h is 10. so our

surface

area

will be 2 times pi times r which is 2 times h which is 10 plus 2 times pi times 2 is sq Remember the order you multiply in doesn't matter, but you must multiply before you add, so I have 2 times 2 is 4 times 10 is 40 pi plus 2 squared is 4 times 2 is 8 so 8 pi you you can add these together they are like terms just like if you had an x ​​you would only have a pi so put 40 pis plus 8 pis total 48 pies what would be the units so the

surface

is an

area

, so it's measured in square units, so become be this square meter.

This would be your exact solution. A lot of times people get bothered by thinking that an exact solution has to be a decimal, but no, when I say exactly pi stays in your answer, so how do we do? Get the approximate solution. So if I ask you to approximate 3.14 for pi, put that in your calculator, so 48 times 3.14 would be the approximate value, not the exact value of the

surface

area

, so on my calculator I have 150.72 and that will still be in square meters. I hope you found this video helpful, if you are di d please give a thumbs up to help other students find the video