WEBVTT mathematics/basic-math/pyo
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Welcome back to Educator.com.
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For the next lesson, we are going to go over the surface area of a cylinder.
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Remember surface area is the area of all the sides of the solid.
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For the cylinder, we are going to find the area of the top base, the bottom base,
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and then the lateral area which is the rectangle which makes up the side of the cylinder.
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Imagine if this is a can of soup.
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If you were to rip off the label, the label that wraps around the can, this is a cylinder.
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If you were to cut it and rip off the label, it becomes a rectangle.
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Here the lateral area, meaning this side that does not include the bases,
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if you cut it and open it up, then it becomes a rectangle.
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The lateral area for the cylinder is the rectangle that makes up the side of the cylinder.
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Then we have of course the two bases which are circles.
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Surface area is the lateral area plus the two bases, the two circles.
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To find the lateral area, if this is where we cut, then there is the cut right there.
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We need to find the measure of this length and width
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to be able to find the area of this lateral area, this rectangle.
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Here we know that this wraps around this base.
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All of this here becomes this here.
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When you are given this cylinder, to find this measure, to find that length,
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we have to be able to find the length of this circle right here.
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If we had a prism, that would be considered the perimeter of the base.
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But because it is a circle, that is called circumference.
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To find the circumference of a circle, it is going to be 2πr.
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This is 2πr; circumference, all of this; circumference equals 2πr.
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π is 3.14; there is the r; there is the radius.
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Once you find that, that will be the length of this right there.
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We know this would be the height.
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That would just be the height of the cylinder.
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To find the area of a circle, because we have to find the area of this whole thing.
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This whole thing is this whole thing right here.
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The area of a circle is πr²; a few things to look at here.
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We have circumference to help us find the length of this rectangle, the lateral area.
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We have the area of a circle because that is the area of the base.
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Be careful here not to get confused with volume.
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Surface area and volume are very different things.
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Surface area, we are just finding the space of the outside of the cylinder.
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Whereas volume measures everything inside, how much space it is covering on the inside.
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Surface area will be the lateral area; lateral area is this times this.
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It is circumference times the height; or I can just say 2πr times h.
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That is going to give me the lateral area, all of this here.
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Plus... now I have to find the area of my bases.
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Plus, since I have two bases, 2 times πr².
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The surface area SA is going to be all this here.
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Try not to let this confuse you; surface area is just very simple.
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You are just finding all the space on the outside.
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Just this right here, the area of this rectangle, plus the area of the circle which is the top.
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Plus the area of this bottom circle which is the bottom.
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It is just area of the rectangle plus the two bases of the cylinder.
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That is the surface area.
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As long as you are comfortable, you know the circumference of a circle,
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you know how to find the area of a circle, you are fine with surface area.
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Let's do a few examples.
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Again try not to get confused with the formula for volume.
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The volume is the area of a circle πr² times the height of the cylinder.
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That is the volume.
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For surface area, we are just going to take it one step at a time
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and find the lateral area, find the area of that label that wraps around the can.
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Then add it to our bases, the area of the bases.
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Surface area, we are going to do lateral area which is that label
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plus the area of the base which is a circle, πr² since it is a circle,
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and multiply that by 2 because we have two of those bases.
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Again remember a lateral area, we are finding the rectangle; that label, rectangle.
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If we cut it here, then it is as if we are cutting it here.
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We need to find this measure here which would be this measure there which is the circumference.
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The circumference is 2πr; that is 2 times 3.14 times... the radius is 4.
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2 times 3.14 is... 2 times 1 is 2; this is 6.
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I have two numbers behind decimal points; that will be 6.28.
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Times 4; this is 32; that is 8, 9, 10, 11; 24, 25.
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Again two numbers; it is 25.12.
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Easier way, you can just multiply this number with this number first.
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That is 8; then you can multiply that 8 to 3.14.
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Here our circumference is 25.12; this is 25.12.
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That is the length of this side right here; this height is 6.
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The lateral area... I am going to write that in here.
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Lateral area is 25.12 times 6; lateral area.
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25.12 times the 6; this is 12.
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6 times 1 is 6; plus 1 is 7.
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5 times 6 is 30; 6 times 2 is 12; plus 3 is 15.
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How many numbers do I have behind decimal points?--I have two.
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That means I have to place two numbers behind the decimal point here in my answer.
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25.12 times 6 is 150.72; that is only this part right here.
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Plus... now I have to find the area of my base, πr², all of this right here.
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πr²; the area of the base is πr².
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That is 3.14 times the radius which is 4².
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Remember that I have to square this first; 4 times 4 is 16.
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3.14 times 16; 4 times 6 is 24.
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6 times 1 is 6; plus 2 is 8; 6 times 3 is 18.
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Leave this space alone; 1 times 4 is 4; this is 1; this is 3.
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I am going to add them up; 4, 12, 10, 5.
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I have a total of two spaces behind decimal points.
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I have to put that decimal point right there; it becomes 50.24.
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The area of this base right here πr² is 50.24.
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I just place that right here, that number, 50.24.
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Since I have two of them, I have to multiply that by 2.
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If you want, you can just do 50.24 plus 50.24; times 2.
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4 times 2 is 8; this is 4.0; that is 10.
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All this lateral area plus 100.48 is surface area; add these two numbers together.
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Remember when you add decimals, you have to line up the decimal point.
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2 plus 8 is 10; 7 plus 4 is 11; plus the 1 is 12.
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1, 5, and 2; decimal point, just bring it straight down.
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My surface area is 251.20; my units, centimeters; it is area so it is squared.
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I know this was a lot of work; this was a lot.
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But all we did is find the area of that label.
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Remember that rectangle that wraps around the can.
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We are going to find the area of that; it is just a rectangle.
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You are just finding 2πr because that is this side right here; times the 6.
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If you want, instead of using this whole formula here, you can just maybe draw out the rectangle.
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Draw your two bases, your circles; find the area of each one.
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Then just add them together; that is all we did here.
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This is just rectangle plus the base times 2; that is surface area.
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Let's do a couple more examples; find the surface area of the cylinder.
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For this one, let's draw out our net.
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Meaning I am going to draw out my rectangle which is that label that wraps around the can.
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Then my two circles.
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First let's find the area of that rectangle; this I know is 20 meters.
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Then to find this measure here, that is just from here going all the way around.
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That is remember circumference; this is going to be 2πr.
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2; π is 3.14; r, my radius is not given; I have my diameter instead.
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The diameter is remember from here all the way to here is 12.
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My radius is half of that; r is 6; it is times 6.
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To find this, I can just multiply these two first.
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2 times 6 is 12; 12 times 3.14; 3.14 times 12.
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This is 8, 2, 6; leave the space alone; 1 times 4 is 4.
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That is 1; that is 3; now we add; 8, 6, 7, 3.
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I have two numbers total behind decimal points.
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Place the decimal point in front of two numbers.
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This side right here, my circumference, is 37.68.
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To find the area of this, just multiply this number times that number.
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It is 20 times 37.68; 37.68 times 20.
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If you want, you can drop the 0 and just multiply this number by the 2.
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But you have to remember that in your answer, after you multiply it by 2,
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you have to place a 0 at the end of that.
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That is just a little faster way to do it.
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Or I can just say that this is 0.
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Then instead of placing... because everything times this is all 0.
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Instead of writing all these 0s, I can just go ahead and start multiplying this next number, the 2.
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2 times 8 is 16; this is 12, 13, 14, 15; 6, 7.
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Two numbers behind decimal points; right there.
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The area of this is 753.6 meters squared.
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That is the area of this rectangle there.
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Now let's find the area of the base, this right here, the circle.
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That will be πr²; π is 3.14; radius is 6².
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Again exponent first; that is 3.14 times 36.
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4 times 6 is 24; 6 times 1 is 6; plus 2.
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6 times 3 is 18; put a 0 there; 3 times 4 is 12.
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3 times 1 is 3; plus the 1; 4; 9.
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Add them; 4; 10; 12, 13; 9, 10, 11; two numbers there.
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My area of this circle is 113.04 meters squared; that is the area of that.
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The area of this is also 113.04.
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Now all I have to do to find the surface area is add them together.
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753.6 or 60... remember if this is a 0 that is behind the decimal point
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and at the end of a number, you can just drop it.
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Plus my circles; 113.04 plus 113.04.
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Let's go ahead and just add them; line them up.
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753.60 with this one, 113.04 and 113.04.
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Make sure when you add decimals, the decimal points all line up.
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The decimal point in the answer is just going to go straight down.
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This is 8; 6 plus 0 plus 0; 6.
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3, 6, 9; 5, 6, 7; 7, 8, 9.
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Our surface area is 979.68 meters squared for area.
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Third example, we have a cylinder that has a radius of 6 inches and a height of 10 inches.
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Draw a net for the cylinder and find the surface area.
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Cylinder, there is my cylinder there.
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My radius is 6 inches; my height is 10 inches.
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The net is going to be rectangle for the lateral area and then my two circles, my bases.
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The surface area is going to be this plus this plus that.
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Remember to find the length here, that is going to be
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from here all the way wrapping around which is the circumference.
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That is 2πr; the radius is 6.
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This is going to be 2 times 3.14 times 6.
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I can go ahead and multiply this 2 and the 6 together.
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That is 12; times the 3.14; that is going to give this length right there.
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3.14 times 12; 8; 2; 6; 1 times 4.
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Leave this space alone; put a 0 there; 4, 1, and 3.
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Add them; it is 8; this is 6; 6, 7, 3.
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I have two numbers behind decimal points here.
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I am going to place the decimal point right there in front of two numbers.
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My circumference here, this length is going to be 37.68; my height is 10.
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To find the area of this rectangle, I have to multiply this times this, length times the width.
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Here if you just multiply by 10, all I have to do is
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take the decimal point and move it over to the right one space.
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The area of this rectangle is going to be 376.8.
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It went from in front of the 6 to behind the 6; one space.
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Then plus... I have to find the area of that circle.
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The area of a circle is πr².
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π is 3.14; the radius is 6; 3.14 times 6².
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Remember we have to take care of this exponent first.
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6 times 6 is 36; it is 3.14 times 36.
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This is 24; this is 6; add the 2; 8; this is 18.
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Leave the space alone; put a 0.
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3 times 4 is 12; 3 times 1 is 3; plus 1 is 4.
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3 times 3 is 9; 4; now we are adding.
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8 plus 2 is 10; this is 12, 13; 9, 10, 11.
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Two numbers behind decimal points; it is going to go right there.
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The area of this circle is 113.04.
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This is the same circle, same area; this is 113.04 also.
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To find the surface area, I take this which is called my lateral area.
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Lateral area plus area of my base; then add another base because I have two of them.
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If you want, you can just take this and multiply it by 2 since we have two of them.
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I just have to add this all up.
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Remember when we add decimals together, we have to line up the decimal point.
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Decimal point has to go right there; that is 04; again 113.04.
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I am missing a number right here; I can just place a 0.
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Whenever you have a number behind the decimal point
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and at the end of a number, you can add as many 0s as you need.
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We are going to add this.
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4 plus 4 is 8; 8 plus 0 plus 0 is 8 point...
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6 plus 3 is 9; plus 3 is 12; 7, 8, 9, 10.
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1 there; 0 there; 3, 4, 5, 6.
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My surface area when I add the lateral area... here is my base and my other base.
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When you add lateral area plus the base plus the base,
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we are going to get our surface area which is 602.88.
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Our units is inches; area is always squared.
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That is the surface area of this cylinder.
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That is it for this lesson; thank you for watching Educator.com.