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The given figure is made by placing a cube of side length 13 centimeters on top of another cube of side length 18 centimeters.
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Find its surface area.
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The surface area is the sum of all the areas that cover the surface of an object.
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But what does that mean?
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You could think about it like the space that you would paint if you would paint this object.
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Also, if you imagine picking it up, it’s all the places that you could touch the object.
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Our small cube has a side length of 13 centimeters.
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Because we know that it’s a cube, we can also say that each face is a square.
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And we know to find the area of a square, we take the side length and we square that value.
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The area of every face of the small cube is 13 squared.
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The question is, “How many faces of this small cube are showing?”
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Here’s the right side.
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And opposite the right side is a left face that we can’t see but is part of this cube.
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We have a front face.
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And opposite the front side is a back side that we can’t see that will also be 13 squared.
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The surface area of the part that’s made up of this small cube would be equal to five times 13 squared.
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We say five times 13 squared because we need to add 13 squared to itself five times.
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So we simplify it and say five times 13 squared.
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We won’t consider the bottom of this cube.
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We wouldn’t consider that sixth face.
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And that’s because these cubes together make up one figure.
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So we can’t separate the top from the bottom.
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We couldn’t paint underneath there because it’s attached.
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We’ll only be interested in this part, but we’ll come back to that a little bit later.
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Let’s now consider the large cube surface area.
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It has a side length of 18, which means the area of each face will be 18 squared.
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The area of the front is 18 squared.
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And that means the part opposite the front, the back side, would also be 18 squared.
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The right face is 18 squared.
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And that means we’ll have a left side with area 18 squared.
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At this point, it’s really tempting to think that we’ve covered every surface.
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However, we can’t forget the bottom of this figure, the square base, that would also measure 18 squared.
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So the large cube surface area would be five times 18 squared.
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This is because we have five faces that measure 18 squared each and we’re adding them all together.
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We can simplify that process by multiplying 18 squared by five.
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Now we’re ready to consider the area of the part shaded in pink.
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How would we calculate this surface?
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We have the square that’s the top of our large cube, a square that has a side length of 18, with the smaller square stacked on top.
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We know that that smaller square has a side length of 13.
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To find the area of this extra piece, we’ll find the area of the top of the large cube, 18 squared.
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Then, we’ll find the area of the smaller square, 13 squared.
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And we’ll subtract.
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We’ll take away that small square from the large square.
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When we subtract 13 squared from 18 squared, we get 155.
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This additional piece has an area of 155.
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To find the total surface area, we’ll need to add all these figures together.
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The surface area will equal five times 13 squared plus five times 18 squared plus 155.
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Five times 13 squared equals 845.
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Five times 18 squared equals 1620.
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And we bring down the 155.
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The surface area is equal to 2620.
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We’ll need to find the units.
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Our initial measurements were given in centimeters.
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And that means our surface area is a measure of centimeters squared.
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The surface area of this figure is 2620 centimeters squared.