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12-7 Surface Area of Spheres

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Objective Recognize and define basic properties of spheres Find surface area of spheres

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Spheres Sphere – In space, infinitely many points that are a given distance from a central point. Great Circle – the circle made when a plane intersects a sphere where it contains the center. A great circle has the same center and radii of the sphere it is within. Hemisphere – one of two halves of a sphere. Each great circle in a sphere separates it into two hemispheres. Great Circle

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Example 1 In the figure, O is the center of the sphere and plane R intersects the sphere In circle A. If AO is 3 cm and OB is 10 cm, find AB o R A B

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The radius of circle A is segment AB, B is a point on circle A and on sphere O. Use Pythagorean Theorem for the right triangle ABO to solve for AB OB = AB + AO 10 = AB + 3 100 = AB + 9 91 = AB 9.5 AB 222 ~ ~ 2 2 2 22 Pythagorean Theorem OB = 1=, AO = 3 Simplify Subtract 9 from each side Use a calculator AB is approximately 9.5 centimeters

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Surface Area of Spheres If a sphere has a surface area of T units and a radius of r units then T = 4 or 4 times the area of the sphere’s great circle. r 2 (4)( )(r) = surface 2 area

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Example 2 Find the surface area of the sphere given the area of the great circle A 314.2 in ~ ~ 2

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From the formula, we know that the area of the sphere is 4 times the area of its great circle. T = 4 T = 4(314.2) T = 1256.8 in The surface area is about 1256.8 in r 2 2 2 Surface area of a sphere. Surface area of GS is approx. 314.2 Multiply

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Example 3 Find the surface area of a baseball with a circumference of 9 inches To determine how much leather is needed to cover the ball. 9 inches

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We know that the circumference is 9 so we can use that to find the area of the great circle 9 = D D 1.4 T = 4 r T = 4 (1.4) T 25.8 ~ ~ 2 2 ~ ~ The surface area is approx. 25.8 sq. inches Circumference of a circle Use a calculator Surface area of a sphere r = 1.4 Use a calculator

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Assignment Pg 673 10-15, 17-31

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