Nets and Surface Area
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Common Core Standard: Represent three-dimensional figures using nets made up of rectangles and triangles, and
use the nets to determine the surface area of these figures. Apply these techniques in the context of solving realworld and mathematical problems.
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Net – a net is a three dimensional figure that has been laid flat in a two dimensional way
o
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The length, width, and height of the original 3D figure remain the same in the net
Why nets?
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Nets can be used to determine the surface area of a three dimensional figure
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Surface area – is the area of all the surfaces or faces of a three dimensional figure
Nets will also enable you to visualize the space a three dimensional figure would take up on a two
dimensional plane
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You see nets when you take apart cardboard boxes
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For instance, this is a net of a rectangular prism
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Example: Determine the surface area of the rectangular prism above using the net
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Using this net, we can see that this particular rectangular prism is made up of 6 rectangles
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We know that surface area is the area of all the surfaces
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Thus, to determine the surface area, we would need to determine the area of each surface, which a
rectangle, and we know how to determine the area of a rectangle
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A = length multiplied by width
1st: Determine the area of each rectangle:
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The order of the rectangle area does not matter (Commutative Property of Addition)
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You can also notice that the rectangle opposite has the same dimensions
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The left and right rectangles have the same dimensions or are congruent: 4 cm by 6 cm
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This means that I only need to determine the area of one of these rectangles, then I can
multiply it by 2, since there are two with the exact same dimensions or are congruent
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Area of Left Rectangle: 4 cm • 6 cm = 24 cm2
So, the area of both the right and left rectangles is 24 cm2 • 2 = 48 cm2
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Area of the top rectangle is 4 cm •8 cm = 32 cm2
2
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This matches the third rectangle, so the area of the top and third rectangles is 32 cm • 2 =
64 cm2
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Area of the bottom rectangle is 8 cm • 6 cm = 48 cm2
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The bottom rectangle and the second rectangle are congruent, so to determine the area of
both 48 cm2 • 2 = 96 cm2
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2nd: Add the areas together
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2
2
2
32 cm + 64 cm + 96 cm = 192 cm
2
Real World Example: Icilynne bought a present that came in a box. The box has 6 equal sides which all are the
shape of a square, and one side measures 1.75 in. Icilynne needs to wrap the present, but doesn’t know how much
wrapping paper will be needed. Determine the surface area of the box to determine the amount of wrapping paper.
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1st: Determine the shape
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This is a cube, which means that all faces are squares
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2nd: Draw or imagine the net
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3rd: Determine the area of each square
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For cubes, all faces are congruent, so only need to determine the area for one face then multiply
that by 6 to determine the entire surface area, since there are 6 faces
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Area of a square is determined by multiplying the length by the width or by squaring either the
length or the width
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(1.75 in)2 = 3.0625 in2 is the area of one face of the cube
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3.0625 in2 • 6 = 18.375 in2 for the surface area
Thus, the surface area is 18.375 in2. This means that Icilynne would need to have at least 18.375 square
inches of wrapping paper.