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**Surface Area, Lateral Area, and Volume of Prisms and Pyramids**

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**Lateral Area of a Right Prism (P = perimeter of the base, h = height)**

L = Ph (P = perimeter of the base, h = height) Perimeter = 2(5) + 2(4) = Perimeter of triangular base H = = = 19 L = 18 x 8 = H = 4 L = 19 x 4 = 76 6 8 5 4 5 4 8

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Surface Area of a Right Prism Surface Area = Lateral Area + 2 x area of the base Surface Area = Ph + 2B 6 8 5 4 Lateral area = 18 x 8 = 144 Area of base = 5 x 4 = 20 Total surface area = (20) = 184 Lateral area = 19 x 4 = 76 Area of base = ½ (6)(4) = 12 Total surface area = (12) = 100 5 4 8

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**Volume of a Right Prism V = area of base x height V = Bh**

6 8 5 4 Area of base =5 x 4 = 20 H = 8 V = 20 x 8 = 160 Area of base = ½(6)(4) = 12 H = 4 V = 12 x 4 = 48 5 4 8

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Regular Pyramids A regular pyramid has a base which is always a regular polygon. The lateral faces all intersect at a point called the vertex and form triangles. The altitude is a segment from the vertex perpendicular to the base. The slant height is the height of a lateral face. vertex altitude Lateral side Slant height Base

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**Lateral Area of a Regular Pyramid**

The lateral area of a regular polygon = ½ pl ( p = perimeter of the base, l = slant height) L = ½ pl 7 8 Perimeter = 4 x 7 = 28 Slant height = 8 Lateral area = ½ (28)(8) = 112

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**Surface Area of a Regular Polygon**

Surface area of a regular polygon = lateral area + area of the base. Surface area = ½ pl + B Lateral area = ½ (28)(8)= 112 Area of base = 7 x 7 = 49 7 8 Surface area = = 161

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**Volume of a Right Pyramid**

The volume of a right pyramid = 1/3 of the area of the base time the height. V = ⅓ Bh 6 7 8 5 Area of base = 7 x 6 = 42 Height = 5 Volume = ⅓ (42)(5) = 70

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**Formulae for Lateral Area, Surface Area, and Volume**

Prisms: Lateral area: L = Ph (p = perimeter, h = height) Total Surface area: T = Ph + 2B (B = area of base) Volume: V = Bh Pyramids: Lateral area: L = ½ Pl (p = perimeter, l = slant height) Total Surface area: T = ½ Pl + B (B = area of base) Volume: V = ⅓ Bh ( B = area of base, h = height)

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