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Area and Surface Area Prisms, Pyramids, and Cylinders

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Cross Sections A cross-section is the face that results from slicing through a solid shape. Cross sections can create various polygons or a circle. Different directions of slices can create different shapes.

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Cross Sections If I slice a cylinder horizontally, what shape will the cross section create? If I slice a cylinder vertically, what shape will the cross section create? Answer: Circle Answer: Rectangle

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Cross Sections If I slice a rectangular prism horizontally, what shape will the cross section create? If I slice a rectangular prism vertically, what shape will the cross section create? Answer: Rectangle

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Cross Sections If I slice a rectangular pyramid horizontally, what shape will the cross section create? If I slice a rectangular pyramid vertically, what shape will the cross section create? Answer: Rectangle Answer: Triangle

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Cross Sections If I slice a triangular pyramid horizontally, what shape will the cross section create? If I slice a triangular pyramid vertically, what shape will the cross section create? Answer: Triangle

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Area When working with 2-dimensional figures, we can calculate area. ▫Area- the amount of space contained in a flat surface. Area Formulas: Triangle: ½ bh Rectangle: LW Circle: π r 2 Base(b) Height (h) Length (L) Width (W) Radius (r)

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Area Examples Calculate the Area of the following shapes. Triangle: ½ bh Rectangle: LW Circle: π r 2 10 cm 12 cm 36 in 22 in 13 ft A = 60 cm 2 A = 792 cm 2 A = 530.66 cm 2

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Area Example To find the area of an irregular shape, break it up into regular polygons. Then find the area of the separate pieces and add them together. 10 cm 35 cm 10 cm

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Surface Area Surface Area is the sum of the area of all of the faces of a 3-dinemtional figure. We often use nets to help us determine surface area.

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Surface Area vs. Area State whether each of the following shapes will have area or surface area.

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Surface Area vs. Area State whether each of the following situations will involve area or surface area. 1.I want to wrap the new PS3 that I bought as a gift. (Hint: it is in the box still) 2.I just bought property and I want to measure how much land I own. 3.I am painting a shed and I need to know how much paint to buy. 4.I am painting one wall of my house and I need to know how much paint to buy. Surface Area Area Surface Area Area

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Finding Surface Area We can calculate surface area in two ways: 1.Substitute into a formula. 2.Draw a net and find the area of each section. *If you do not know the formula, the second option will always work!

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Surface Area Using a Net Use your knowledge of area to calculate the surface area of the rectangular prism. Draw a net first to help you. 3 cm 7 cm 4cm

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Surface Area 4 cm 7 cm Base = 12 cm 2 Front= 28 cm 2 Bottom= 21 cm 2 Back = 28 cm 2 Top= 21 cm 2 3 cm 4 cm 3 cm 4 cm 7 cm Length = 7 cm Width = 3 cm Height = 4 cm Add! SA = 122 cm 2 Base = 12 cm 2

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Surface Area using a Formula Surface area of a Rectangular Prism SA = 2LW + 2LH + 2HW SA = 2(7)(3) + 2(7)(4) + 2(3)(4) SA = 122 cm 2 Length = 7 cm Width = 3 cm Height = 4 cm

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Example: 8mm 9mm 6 mm 6mm Find the AREA of each SURFACE 1. Top or bottom triangle: A = ½ bh A = ½ (6)(6) A = 18 2. The two dark sides are the same. A = lw A = 6(9) A = 54 3. The back rectangle is different A = lw A = 8(9) A = 72 ADD THEM ALL UP! 18 + 18 + 54 + 54 + 72 SA = 216 mm²

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SURFACE AREA of a CYLINDER. You can see that the surface is made up of two circles and a rectangle. The length of the rectangle is the same as the circumference of the circle! Imagine that you can open up a cylinder like so:

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EXAMPLE: Round to the nearest TENTH. Top or bottom circle A = πr² A = π(3.1)² A = π(9.61) A = 30.175 Rectangle C = length C = π d C = π(6.2) C = 19.468 Now the area A = lw A = 19.468(12) A = 233.616 Now add: 30.175 + 30.175 + 233.616 = SA = 294 in²

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There is also a formula to find surface area of a cylinder. Some people find this way easier: SA = 2πrh + 2πr² SA = 2π(3.1)(12) + 2π(3.1)² SA = 2π (37.2) + 2π(9.61) SA = π(74.4) + π(19.22) SA = 233.616 + 60.35 SA = 294 in² The answers are REALLY close, but not exactly the same. That’s because we rounded in the problem.

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Common Surface Area Formulas Rectangular Prism SA = 2LW + 2LH + 2WH Cylinder SA = 2πr 2 + 2 πrh Square Pyramid SA = 4 (½ bh) + b 2 = 2bh + b 2 Cube SA = 6s 2

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Activity Using the object you brought in and a ruler, determine how much paper would be needed to wrap the outside of your object. Hint: This means we need to fins the surface area. You can do this using a formula or by drawing a net.

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