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

hello stranger, let's start in this video, we are going to talk about the

volumes

of certain spatial figures called

prisms

have two faces in parallel planes and these faces are congruent polygons, for example, in the prism on the left we have a triangular face at the top and a congruent triangular face at the bottom, similarly in the prism on the right, we have a five sided regular polygon, so it is called a pentagon, that pentagon shape is in a plane at the top top and a parallel plane at the bottom of the prism those faces that are congruent polygons can be any shape

prisms

are named according to that shape on the left here I have a rectangular prism a hexagonal prism and a pentagonal prism actually we've already learned find the volume of a box and a box is an example of a rectangular prism.

We learned that the volume of a box can be thought of as the length times the width times the height, but we also learned that it can be thought of as the

area

of the base times the height. because this face on top has dimensions w times l i could refer to this

area

as capital b b is the

area

of the base well it turns out no matter what shape the base is 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 are going to find the volume of the triangular prism given we know that our height in this diagram is equal to 7 meters we are going to need to find the

area

of that triangle the

area

of the triangle is going to be the half of b times h the height of the triangle and the base of the triangle are two different quantities and the base and height I'm talking about here you have to be careful not to confuse those two anyway the

area

of the triangle is the half of six by four so that's half of 24 which is 12 and

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're going to take that bathroom

area

se and multiply it by the height of the prism to make it 12 square meters by 7 meters which will be 84 cubic meters remember that volume is always measured in cubic units so we can apply the same idea to find

volumes

of boxes to

volumes

of any prism and , actually right circular

cylinders

work the same way, we can also find the volume of a right circular cylinder by finding the

area

of the base times 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 it's a circle of radius r the

area

of the base is pi r squared i'll just give you the name capital b for the

area

of the base so we know the volume of the cylinder is the

area

of the base times the height, which is exactly why the formula for the right circular cylinder looks like the volume equals pi r squared h the

area

of the base times the height we can also find the

area

of the

surface

of a right circular cylinder pretty easily i'm going to take out the netting which is where the object separates so it looks like a squashed soda can we've taken a

part

the top and bottom and folded the sides of the can and it forms a rectangle one dimension of that rectangle is h the other dimension we'll need if we're going to get the

surface

area

notice that that's 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 will be 2 pi r per h ok but that doesn't include the

area

of the top and bottom let's see this formula for

surface

area

here 2 pi r h are the sides of the can we just found where do you think they are getting the 2 pi r squared?

Well those are the two circles the

area

of this circle is pi r 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 mostly i just need you to recognize that it's the correct formula and use it so let's try here we have a right circular cylinder that has a height of 10 and a radius of its base is 2 meters so let's find the volume and the

surface

area

the volume will be pi times r is 2 so we'll have 2 squared times h is 10 so the volume will be 4 times 10 which is 40 and then multiplied by pi and the units are given in meters cubic look when you're working with any object that rotates in it pi comes out a lot remember pi is the number it's about 3.14 but it's a decimal that goes on forever without repeating endlessly there's no way we can write everything so we write the letter greek pi this is the only way to give an exact solution very often in your homework they will ask you to leave pi in your answer or they will have you say give an exact solution and then give your answer like this however if they ask you to approximate 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 because that won't round to 3.14 and it will give you a slightly different answer and might confuse you so just type 3.14 so I'm going to find the value approximate this volume using 3.14 as an estimate of pi.

I'm going to have to put 40 times 3.14 in my calculator and let's see that I get it right that turns out to be 125.6 and I still need to put the cubic meters so if they ask you for an exact answer of the volume you would say 40 pi cubic meters if they ask you for a approximation using 3.14 is pi you will say 125.6 cubic meters now let's find the

surface

area

for this right circular cylinder so the formula for

surface

area

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

area

of the

surface

is going to be 2 times pi times r which is 2 times h which is 10 plus 2 times pi times 2 squares uared remember the order in which you multiply doesn't matter but you have to multiply before you add so I'm going to having 2 times 2 is 4 times 10 is 40 pi plus 2 squared is 4 times 2 is 8 so 8 pi you can add these terms together they are like terms like you have an x you only have one pi and you state that 40 pi plus 8 pi make a total of 48 pies in all, what would be the units?

Surface

area

is an

area

so it is measured in square units so it will be square meters this would be your exact solution a lot of times people get annoyed that they think an exact solution must be a decimal but no, when I say exact, leave pi in your answer, so how do we do it? get the approximate solution so if i ask you to approximate using 3.14 for pi you will type it into 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 got 150.72 and that's still going to be in square meters.

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