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Surface Area & Volume

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Spheres A sphere is a shape where the distance from the center to the edge is the same in all directions. This distance is called the radius ( r ).

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**Cylinders A cylinder is a prism with a circular base of radius (r).**

The height (h) is the distance from the bottom base to the top of the cylinder.

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Cones A cone is a pyramid with a circular base of radius (r) and the side length (s) is the length of the side. The height (h) is the distance from the center of the base to the top of the cone.

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Pyramids A pyramid is a solid figure with a polygonal base and triangular faces that meet at a common point over the center of the base. The height (h) is the distance from the base to the apex or top of the pyramid. The side length (s) is the height of the face triangles. The area (A) of the base (b)is calculated according to the shape of the base.

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Rectangular Prisms A rectangular prism can be described as a stack of rectangles. The height (h) is the height of the prism, the width (w) is the width of the base, and the length (l) is the length of the base.

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Triangular Prism A prism can be described as a stack of shapes. The figure shows a prism of triangles stacked d thick, but any shape could be used. For a triangluar prism, the height (h) is the height of the base, the base (b) is the width of the base, and the length (l) is the length of the prism. A = bh + 3bl

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Application I A standard 20-gallon aquarium tank is a rectangular prism that holds approximately cubic inches of water. The bottom glass needs to be 24 inches by 12 inches to fit on the stand.

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**Application I Find the height of the aquarium to the nearest inch.**

V = l x w x h 4600 = (24) x (12) x h 4600 = 288 x h 16.0 inches = h V = 4600 cubic inches 12 inches 24 inches

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Application I Find the total amount of glass needed in square inches for the six faces. 12 inches 16 inches 12 inches 24 inches 24 inches TSA = 2(24)(12) + 2(12)(16) + 2(24)(16) TSA = TSA = 1728 in2

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Application II Does the can or the box have a greater surface area? a greater volume? Cylinder Square Prism 5 cm 9 cm 9 cm 6 cm 5 cm

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Application II Surface Area of can 9 cm 6 cm 9 cm 6 cm

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Application II Surface Area of box 5 cm 9 cm 5 cm 5 cm 9 cm 5 cm

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Application II Since 230cm2 > 288.2cm2, the box has a greater surface area. Which container holds more juice?

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Application II Volume of can 9 cm 6 cm 9 cm 6 cm

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Application II Volume of box 5 cm 9 cm 5 cm 5 cm 9 cm 5 cm

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**Application II Since 254.5cm3 > 225cm3, the can has greater volume.**

If the can and the box were the same price, which one would you buy?

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Application III A birthday gift is 55 cm long, 40 cm wide, and 5 cm high. You have one sheet of wrapping paper that is 75 cm by 100 cm. Is the paper large enough to wrap the gift? Explain.

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**Application III TSA = 2(55)(40)+2(55)(5)+2(40)(5) A = 75 x 100**

TSA = 5350 sq cm A = 75 x 100 A = 7500 sq cm 100cm 5 cm 75 cm 40 cm 55 cm

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Application IV If both of these cans of pizza sauce are cylinders, which is the better buy? $1.09 20.5 cm $3.49 11 cm 7.5 cm 10 cm

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**Application IV V = πr2h V = π(5)2(20.5) V = π * 25 * 20.5**

V = π cm3 V = cm3 = 461 cm3 per dollar 20.5 cm $3.49 10 cm

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**Application IV V = πr2h V = π(3.75)2(11) V = 154.7 π cm3 V = 486.0 cm3**

= 446 cm3 per dollar $1.09 11 cm 7.5 cm

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**Application IV Which jar would you buy?**

Since 461 cm2 per dollar > 446 cm2 per dollar, you should buy the first jar.

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