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Honors Geometry Lateral Area, Surface Area and Volume

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Prisms - definitions A PRISM is any object with two parallel congruent bases with lateral sides that are parallelograms. In this case, the two blue sides are the bases. The bases are not always on the top and bottom! In the next figure, the bases are the front and the back, because they are the parallel congruent sides. In this case, the two blue sides are the bases. In this case, the two pink sides are the bases. yellow In this case, the two yellow sides are the bases.

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Naming Prisms Prisms are named by the shape of their bases. Square Prism Rectangular Prism Triangular Prism Hexagonal Prism

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Right and Oblique Prisms A right PRISM has the sides perpendicular to the bases. An oblique PRISM has sides that are NOT perpendicular to the bases. Right Prism Oblique Prism

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Cylinders CYLINDER A CYLINDER is a special case of a prism where the two parallel sides (bases) are circles. A right cylinder has its side perpendicular to its bases. A cylinder can be thought of as a circular prism. NOT An oblique cylinder has its side NOT perpendicular to its bases.

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Lateral Area of a Prism x y zz xxyy Lateral Area (LA) is the area of the side(s) of the object not including the bases. In this case, the LA = (x + y + x + y) z, but x + y + x + y = 2x + 2y. This happens to be the perimeter of the base (P). So, the LA of a right prism is given by LA = Ph.

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Lateral Area of a Cylinder I like to think of the LA of a cylinder as measuring the area of the label of a soup can. If you cut down the dotted line and peel the label off and lay it out flat, it is a rectangle. The base of the rectangle is the circumference of the base and the height of the rectangle is the height of the cylinder. h r Circumference Height Therefore, LA of a cylinder is given by LA = Ch. Since circumference is to a circle as perimeter is to a polygon, this just a variation of LA = Ph.

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Surface (Total) Area of a Prism SA = LA + 2B The Surface area of a Prism is the sum of the area of ALL of the sides. It is also the LA plus the area of the two bases. Since, however the two bases are congruent, the surface area of a right prism (if B is the area of a base) is given by SA = LA + 2B

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Example 1 Find the Surface Area LA = Ph; SA = LA + 2B LA = Ph; P = 2(12cm) + 2(10cm) = 44cm; h = 8cm LA = (44cm)(8cm) = 352cm 2 B = (12cm)(10cm) = 120cm 2 ; so 2B = 2(120cm 2 ) = 240cm 2 SA = LA + 2B = 352cm cm 2 = 592cm 2

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Example 2 Find the Surface Area LA = Ph; SA = LA + 2B LA = Ph; P = 15 in in. + 9 in. = 36 in.; h = 13 in. LA = (36 in.)(13 in.) = 468 in. 2 B = 1 / 2 (12 in.)(9 in.) = 54 in. 2 ; so 2B = 2(54 in. 2 ) = 108 in. 2 SA = LA + 2B = 468 in in. 2 = 576 in. 2

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Surface Area of a Cylinder The Surface Area of a Cylinder is much like that of a Prism. It the LA + the area of the 2 bases or (since the bases are congruent), LA + twice the area of a base (B). SA = Ch + 2B

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Example 3 Find the Surface Area LA = Ch; SA = LA + 2B LA = Ch; C = 2π(8 ft.) = 16π ft. = ft.; h = 12 ft. LA = (50.27 ft.)(12 ft.) = ft. 2 B = π(8ft.) 2 = 64π ft. 2 = ft. 2 ; so 2B = 2( ft. 2 ) = ft. 2 SA = LA + 2B = ft ft. 2 = ft. 2

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Assignment Lateral Area and Surface Area all

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Dog - PRISM

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Volume of a Prism Unit Cube Base Volume of a Prism is merely the area of a base (B) times the height. V = Bh

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Example 4 Find the Volume B = (12cm)(10cm) = 120cm 2 V = Bh V = (120cm 2 )(8cm) = 960cm 3 h = 8cm

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Example 5 Find the Volume V = Bh B = 1 / 2 (12in)(9in) = 54in 2 V = (54in 2 )(13in) = 702in 3 h = 13in.

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Volume of a Cylinder Like a prism, the volume of a Cylinder is the area of a base times its height. V = Bh

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Example 6 Find the Volume V = Bh V = (64π ft 2 )(12 ft) = 768π ft 2 = ft 2 B = π(8ft) 2 = 64π ft 2 h = 12 ft

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Assignment Volume - all

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Pyramids and Cones Square or Rectangular Pyramid Triangular Pyramid Cone For pyramids and cones V = 1 / 3 Bh

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Example 7 6 mm Find the Volume V = 1 / 3 Bh B = (6mm) 2 = 36mm 2 V = 1 / 3 (36mm 2 ) (6mm) = 72mm 3

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Example 8 5 cm 8 cm 6 cm Find the Volume V = 1 / 3 Bh B = 1 / 2 (5cm)(8cm) = 20cm 2 V = 1 / 3 (20cm 2 )(6cm) = 40cm 3 h = 6cm

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Example 9 15 cm 9 cm Find the Volume V = 1/3(81π)15 V = 405π cm 3

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Assignment Pyramids and Cones - all

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Spheres r SA = 4πr 2 V = 4 / 3 πr 3

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Example 10 7 cm Find the Surface Area SA = 4πr 2 SA = 4π(7cm) 2 SA = 196πcm 2 SA = cm 2

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Example cm V = 4 / 3 πr 3 Find the Volume V = 4 / 3 π(12cm) 3 V = 2304πcm 3 V = cm 3

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Assignment Spheres - all

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