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 area of a circle.
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First to review over area, remember it is how much space it is covering up.
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The area of a circle is when you have a circle and you see how much space it is using.
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For example, let's say you have a hole in your jeans and you want to cover it up.
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You cut out a circle from another pair of jeans let's say.
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Then you stitch it on to your jeans to cover up your hole.
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That, however much that circle, that patch is covering up, that is area.
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It is just how much you are covering; how much space you are using.
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Remember if you are measuring the distance around the circle, that is called circumference.
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We have circumference which is the distance around the circle
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and then area which is all of this, how much space it is using up.
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The formula to find the area of a circle is π times the radius times the radius again.
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In other words, the area is πr²; r².
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Be careful; this is not r times 2.
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It is an exponent; that means it is r times itself that many times.
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It is r²; radius times the radius.
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Circumference is 2 times π times r.
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In this case, remember how we multiplied the 2 and the r together first.
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In this case, this is 2 times r or r times 2.
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This is not r times 2; this is r times r.
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Remember keep in mind the difference between the formula for the circumference and the area.
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First let's find the area of this circle.
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The formula of area is π times r² or π times r times r.
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Remember π is 3.14; π, I am going to put in 3.14.
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The radius is 4; 4²; again be careful; this is not 4 times 2.
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This is 4 times itself; 4 times 4.
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Also for order of operations, because we have two different operations
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meaning we have two different things we can do.
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We can multiply; or we can do the exponent.
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The order of operations, remember please excuse my dear aunt sally.
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Parentheses, exponent, multiplication, division, addition and subtraction.
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It is always parentheses first; exponents next; multiplication and division; addition and subtraction.
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See how the exponent comes before multiplying.
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Be careful; you do not multiply these two numbers first.
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You always have to take care of the exponent first; then you can multiply.
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3.14 times... 4² is 4 times 4 which is 16.
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Again remember do not multiply 3.14 times 4 and then square it.
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If you do that, you are going to get the wrong answer.
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Here I want to multiply 3.14 times 16.
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4 times 6 is 24; 6 times 1 is 6; plus 2 is 8.
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6 times 3 is 18; I put a 0 right here.
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1 times 4, 4; 1 times 1, 1; 1 times 3, 3; then add.
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4 plus 0 is 4; this is 12; 8, 9, 10; 3, 4, 5.
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Since I am multiplying, I look at my problem.
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I see how many numbers are behind the decimal point.
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I only have two numbers behind decimal points.
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In my answer, I am going to place two numbers behind the decimal point which is right there.
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My answer becomes 50.24; I cannot forget my units; here it is inches.
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Area is always squared; units squared; not numbers squared; units squared.
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50.24 inches squared is my answer; that is the area of this circle.
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Next example, here I am given that the diameter...
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Remember diameter is a segment whose endpoints are on the circle; on the circle; on the circle.
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And passes through the middle, the center of the circle.
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This is a diameter; the diameter is 20 meters.
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To find the area of a circle, area equals πr², radius squared.
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I need to find the radius; I have the diameter; but I want the radius.
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How do I find the radius if I am given the diameter?
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The whole thing is 20; that is the diameter.
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I know the radius is from the center to this point right there.
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The radius is half the diameter.
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If the whole thing is 20, then the radius has to be half of that which is 10.
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Now I know my radius is 10.
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I can go ahead and plug in my numbers and solve for my area.
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π is 3.14; the radius is 10².
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Again order of operations says we have to take care of the exponents before multiplying.
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Area equals... I am going to leave this for the next step.
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10² is not 10 times 2; it is not 20; be careful.
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It is 10 times 10 which is 100; remember the shortcut.
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If we want to multiply by 10 or 100 or 1000 or 10000,
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then you just count the number of 0s in that number.
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Here I have two 0s; 100 has two 0s.
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You are going to take this decimal point then.
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Whenever you multiply a number to 100 or 10 or 1000, count how many 0s there are.
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There is two; I am going to place this decimal point.
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I am going to move it two spaces then.
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Two 0s so I am going to move it two spaces.
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Do I move it to the left or to the right two spaces?
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Since I am multiplying by 100, this number has to get bigger.
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The way to make the number bigger is to move the decimal point over to the right
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because you want the whole number to be a bigger whole number.
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I have to move it to the right two spaces; go one, two.
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My answer then becomes... that is the new spot for my decimal point.
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It is 314 is my answer; 314.
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Again two 0s here; move it two spaces to the right.
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It was here; it moved over to here, the end.
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Since it is at the end, I don't have to write it.
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It is just 314 point... same thing as if not being there.
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314, you can leave it like that.
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We are done solving; but I have to add my units now.
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It is meters; area is always squared; units squared; 314 meters squared.
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My third example, we are going to find the area of the shaded region.
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I have this rectangle and a circle here that is cut out.
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All this is missing; that is area.
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If I cut it out, then don't I have to take it away?
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I have to subtract it; it is as if I have this whole rectangle.
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It was whole before the circle was cut out.
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Find the area of the whole thing.
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Then you are going to subtract the area of the circle.
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That is going to become what you have left, the area that is shaded.
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Imagine if this rectangle was like a piece of paper and you cut out a circle.
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You have to figure out what is that area of the circle you cut out to see what you are taking away.
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Find the area of rectangle; find the area of the circle; subtract it.
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You will get the area of the shaded region.
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The area of the rectangle; this is the rectangle.
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Area is base times height or length times width; length times the width.
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That is 8 times 7 which is 56 centimeters squared.
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Centimeters squared is the area of this rectangle; that is that.
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The area of the circle, πr²; π is 3.14; the radius is 2; 2².
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I am going to take care of this first.
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Area equals 3.14... I am going to leave that; solve that out; that is 4.
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3.14 times the 4; let's do that over here; 3.14 times 4.
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4 times 4 is 16; 4 times 1 is 4; plus 1 is 5.
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This is 12; I have two numbers behind the decimal point; one, two.
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I need to place two numbers behind the decimal point in my answer.
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Area equals 12.56 centimeters squared.
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Now I have the area of the whole thing and then the area of the circle.
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I need to take away the circle from the rectangle.
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It is going to be 56 minus 12.56; I need to do that.
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56 minus... remember when you subtract decimals, you have to line them up.
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Where is the decimal in this number?
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If you don't see it, it is always at the end right there.
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Minus 12 point... make sure only when you add or subtract, the decimals have to line up... 56.
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I am missing numbers here.
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If I am missing numbers here, it is at the end of a number behind the decimal point, I can add 0s like that.
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When I subtract, this is going to borrow; this becomes the 10; this becomes 9.
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Borrow; 5; is that big enough?--yes.
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10 minus this 6 is 4; 9 minus 5 is 4; point.
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5 minus 2 is 3; 5 minus 1 is 4; it is 43.44.
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This is 43.44 centimeters squared is my answer.
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Again just find the area of the rectangle; then find the area of the circle.
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I subtract it; I have to take the circle away; I have to subtract it.
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Make sure your decimals line up when you subtract; you get this as your answer.
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That is it for this lesson; thank you for watching Educator.com.