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SURFACE AREA

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**Surface area is how much area is on the outside of a solid**

Surface area is how much area is on the outside of a solid. We measure surface area with square units.

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What We Know: AREA is the amount of space inside a flat surface, which is measured with square units. Square Rectangle Triangle 3 units 3 units 3 units 4 units 4 units Area = b x h = 9 square units Area = b × h = 12 square units Area = (b × h) ÷ 2 = 6 square units

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What We Know: Surface —On a prism, surfaces refer to the flat faces that make up the solid. Rectangular prisms have 6 faces. All faces are rectangles. Triangular prisms have 5 faces. 2 are triangles, 3 are rectangles

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**How do we find the surface area of a rectangular prism?**

10 units 12 units 6 units

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**The front and back are identical. The left and right are identical.**

10 units 12 units 6 units We can “unfold” the prism to make its net. We can find the area of each rectangle. The front and back are identical. The left and right are identical. BACK The top and bottom rectangles are identical 6 units 10 units BOTTOM RIGHT LEFT Top = u2 Bottom = u2 TOP View 6 × 12 = 72 sq. units 6 × 12 = 72 sq. units 10 × 12 = 120 square units 10 × 12 = 120 square units 12 units 12 units 12 units 12 units Front = u2 Back = u2 + 6 units 10 units 6 units 10 units Left Side = u2 Right Side = 72 u2 6 × 10 = 60 square units 6 × 10 = 60 square units FRONT 6 units 10 units 504 u2

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To find the surface area of a rectangular prism, you are finding the area of each of the 6 rectangular surfaces and adding them up to get a total. Top = u2 Bottom = u2 10 units 12 units 6 units Front = u2 Back = u2 + Left Side = u2 Right Side = 72 u2 504 u2 Surface Area

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**Find the surface area of this rectangular prism.**

Quick Check! Find the surface area of this rectangular prism. Click to reveal the answer. Front = 9 cm × 12 cm = 108 cm2 Back = Front = 108 cm2 Left Side = 9 cm × 8 cm = 72 cm2 Right Side = Left Side = 72 cm2 Top = 8 cm × 12 cm = 96 cm2 Bottom = Top = 96 cm2 Surface Area = 552 cm2 8 cm 9 cm 12 cm

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**How do think we find the surface area of a triangular prism?**

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**+ We can find the area of each polygon. We can “unfold” the prism**

12 units 10 units 8 units 6 units We can find the area of each polygon. We can “unfold” the prism to make its net. We add up the areas of all the faces. 6 units 8 units (6 × 8) ÷ 2 = 24 u2 (6 × 8) ÷ 2 = 24 u2 12 units 10 units 12 units 6 units 12 units 10 units Rectangle 1 = 120 u2 Rectangle 2 = 72 u2 Rectangle 3 = 120 u2 Triangle 1 = u2 Triangle 2 = u2 10 × 12 = 120 u2 6 × 12 = 72 u2 10 × 12 = 120 u2 + 360 u2 8 units

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**Click to reveal the answer.**

Quick Check! What are the shapes and measurements for each of the faces of this triangular prism? List them. Click to reveal the answer. Rectangle 1 = 3 in × 3 in Rectangle 2 = 3 in × 5 in Rectangle 3 = 3 in × 4 in Triangle 1 = 3 in × 4 in Triangle 2 = 3 in × 4 in 5 inches 4 inches 3 inches 3 inches

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**Now find the surface area of this triangular prism.**

Quick Check! Now find the surface area of this triangular prism. Click to reveal the answer. Rectangle 1 = 3 in × 3 in = 9 in2 Rectangle 2 = 3 in × 5 in = 15 in2 Rectangle 3 = 3 in × 4 in = 12 in2 Triangle 1 = (3 in × 4 in) ÷ 2 = 6 in2 Triangle 2 = (3 in × 4 in) ÷ 2 = 6 in2 Total Surface Area = 48 in2 5 inches 4 inches 3 inches 3 inches

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**End of Surface Area Lesson.**

Continue with Volume

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OLUME

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**What We Need to Understand**

Volume is the amount of space inside a three-dimensional object. In order to measure volume, we need a three-dimensional unit, so we use cubes. The size of the cube depends on the unit that the object is measured with, so we can measure with cubic inches, cubic feet, cubic centimeters, etc. A cubic inch is a cube that measures an inch on each of its side; a cubic mile is a cube that measures a mile on each of its sides. (That’s BIG!)

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**5 units × 5 units = 25 square units Cubes in Bottom Layer × Height**

Now we can determine how many LAYERS of these cubes there are in the prism. The number of layers is the same as the prism’s HEIGHT. To determine the number of cubes that fill this rectangular prism, first we will find out how many cubes will fit in the bottom. The number of SQUARES that will fill the bottom (base) is the same as the AREA of the base. Since the bottom is a rectangle, we can use LENGTH × WIDTH to determine the number of squares on the base. If we know how many SQUARES are on the bottom then we could set a cube on each of those squares. Volume of Rectangular Prisms 5 units LENGTH × WIDTH 5 units × 5 units = 25 square units 25 squares 25 cubes! Cubes in Bottom Layer × Height 25 cubes × 5 = 125 cubes 5 units

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**V = L × W × H V = B.A. x H The formula:**

Volume of rectangular prism = Base Area × Height V = B.A. x H B.A. = AREA of the Base H = Height or distance between the bases The Base Area (B.A.) for any rectangular prism is Length × Width so we can also state the formula for a rectangular prism as: V = L × W × H 5 units 5 units × 5 units × 5 units = 125 cubic units

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**Let’s find the volume of this rectangular prism by using the formula**

B.A. × H V = B.A. × H V = (5 × 4) × 20 V = 20 × 20 V = 400 cm3 20 cm 4 cm 5 cm Remember that our units will always be in terms of “cubic” units

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**Volume of Rectangular Prism = B.A. x H Click to reveal the answer.**

Quick Check! A packing box is 20 cm high, 15 cm wide and 18 cm deep. Find the volume. Volume of Rectangular Prism = B.A. x H Volume = (15 x 18) x 20 Volume = 270 x 20 Volume = 5400 cm3 Click to reveal the answer.

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**V = B.A. x H Volume of Triangular Prisms**

The formula for finding the volume of a triangular prism is the same as our formula for a rectangular prism: V = B.A. x H B.A. = AREA of the Base H = Height or distance between the bases B.A. = 12 units2 (the number of cubes in one layer) V = B. A. x H V = 12 units2 × 5 units V = 60 units3 Then we can multiply that by the height, which is the number of layers. 4 units 6 units First find the area of the base, which is a triangle: B.A. = (B x H) ÷ 2 B.A. = (6 × 4) ÷ 2 B.A. = 12 units2 The area of the base tells us how many cubes are in one layer. 5 units

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CAUTION!! Don’t be fooled by a triangular prism that is not sitting on its base! We still need to find the area of the base (the triangle) and multiply by the height (the distance between the bases)

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**V = B.A. x H Let’s find the volume of this triangular prism**

V = Area of the Base × Height V = (16 cm × 10 cm ÷ 2) × 15 cm V = (80 cm2) × 15 cm V = 1200 cm3 10 cm 15 cm 16 cm Remember that our units will always be in terms of “cubic” units Continue

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**Find the volume of this triangular prism.**

Quick Check! Mark’s scout group has a pup tent that is the shape of a triangular prism. It is 8 feet long, 6 feet wide and has a height of 5 feet from the ground to the peak of the roof. How many cubic feet of air are inside the tent? Click to reveal the answer. Volume = B.A. × H Volume = (6 ft × 5 ft ÷ 2) × 8 ft Volume = 120 ft3 6 ft 5 ft 8 ft

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THE END

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