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Exam 9: Surfaces and Solids

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Exam 1: Line and Angle Relationships13 Questionsadd course
Exam 2: Parallel Lines13 Questionsadd course
Exam 3: Triangles16 Questionsadd course
Exam 4: Quadrilaterals14 Questionsadd course
Exam 5: Similar Triangles12 Questionsadd course
Exam 6: Circles10 Questionsadd course
Exam 7: Locus and Concurrence4 Questionsadd course
Exam 8: Areas of Polygons and Circles5 Questionsadd course
Exam 9: Surfaces and Solids4 Questionsadd course
Exam 10: Analytical Geometry8 Questionsadd course
Exam 11: Introduction to Trigonometry4 Questionsadd course
Exam 12: Geometry Problems: Complementary Angles, Collinear Points, and Similar Triangles916 Questionsadd course
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-Use the drawing to explain the following theorem. The lateral area L of a regular pyramid with slant height of length l and perimeter P for the base is given by . [Note: Except for the number of sides for the base, the proof (argument) will not change for another regular polygon.] Given: Regular polygon with slant height of length l and length s for each side of the base Prove: -Use the drawing to explain the following theorem. "The lateral area L of a regular pyramid with slant height of length l and perimeter P for the base is given by -Use the drawing to explain the following theorem. The lateral area L of a regular pyramid with slant height of length l and perimeter P for the base is given by . [Note: Except for the number of sides for the base, the proof (argument) will not change for another regular polygon.] Given: Regular polygon with slant height of length l and length s for each side of the base Prove: ." [Note: Except for the number of sides for the base, the proof (argument) will not change for another regular polygon.] Given: Regular polygon with slant height of length l and length s for each side of the base Prove: -Use the drawing to explain the following theorem. The lateral area L of a regular pyramid with slant height of length l and perimeter P for the base is given by . [Note: Except for the number of sides for the base, the proof (argument) will not change for another regular polygon.] Given: Regular polygon with slant height of length l and length s for each side of the base Prove:

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The faces of the regular polygon are congruent triangles. For each face, the altitude has
length l while the corresponding base length is s. Thus, the area of each of the n faces is The faces of the regular polygon are congruent triangles. For each face, the altitude has length l while the corresponding base length is s. Thus, the area of each of the n faces is . The lateral area L is the sum of areas of all lateral faces. That is, , Because , the formula for the lateral area becomes . . The lateral area L is the sum of areas of all lateral faces. That is, The faces of the regular polygon are congruent triangles. For each face, the altitude has length l while the corresponding base length is s. Thus, the area of each of the n faces is . The lateral area L is the sum of areas of all lateral faces. That is, , Because , the formula for the lateral area becomes . , The faces of the regular polygon are congruent triangles. For each face, the altitude has length l while the corresponding base length is s. Thus, the area of each of the n faces is . The lateral area L is the sum of areas of all lateral faces. That is, , Because , the formula for the lateral area becomes . Because The faces of the regular polygon are congruent triangles. For each face, the altitude has length l while the corresponding base length is s. Thus, the area of each of the n faces is . The lateral area L is the sum of areas of all lateral faces. That is, , Because , the formula for the lateral area becomes . , the formula for the lateral area becomes The faces of the regular polygon are congruent triangles. For each face, the altitude has length l while the corresponding base length is s. Thus, the area of each of the n faces is . The lateral area L is the sum of areas of all lateral faces. That is, , Because , the formula for the lateral area becomes . .

-Knowing that the surface area of a sphere is given by , complete the proof of the volume formula for the sphere. Having separated the sphere into n pyramids ( each with the center of the sphere for its vertex), show that the volume of the sphere is . -Knowing that the surface area of a sphere is given by -Knowing that the surface area of a sphere is given by , complete the proof of the volume formula for the sphere. Having separated the sphere into n pyramids ( each with the center of the sphere for its vertex), show that the volume of the sphere is . , complete the proof of the volume formula for the sphere. Having separated the sphere into n pyramids ( each with the center of the sphere for its vertex), show that the volume of the sphere is -Knowing that the surface area of a sphere is given by , complete the proof of the volume formula for the sphere. Having separated the sphere into n pyramids ( each with the center of the sphere for its vertex), show that the volume of the sphere is . .

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The volume of the sphere is the approximation of the sum of the volumes of the n pyramids.
Where the areas of the bases are The volume of the sphere is the approximation of the sum of the volumes of the n pyramids. Where the areas of the bases are , , , and so on through , the corresponding altitudes all have a length approximated by h. The the volume of the sphere is appproximated by . In turn, . As n increases, the length h of the altitudes of the pyramids approaches r, the length of the radius of the sphere; that is, . As n increases, the combined base areas of the pyramids also approaches the surface area S of the sphere; that is, . As n increases, we find that or , so that . , The volume of the sphere is the approximation of the sum of the volumes of the n pyramids. Where the areas of the bases are , , , and so on through , the corresponding altitudes all have a length approximated by h. The the volume of the sphere is appproximated by . In turn, . As n increases, the length h of the altitudes of the pyramids approaches r, the length of the radius of the sphere; that is, . As n increases, the combined base areas of the pyramids also approaches the surface area S of the sphere; that is, . As n increases, we find that or , so that . , The volume of the sphere is the approximation of the sum of the volumes of the n pyramids. Where the areas of the bases are , , , and so on through , the corresponding altitudes all have a length approximated by h. The the volume of the sphere is appproximated by . In turn, . As n increases, the length h of the altitudes of the pyramids approaches r, the length of the radius of the sphere; that is, . As n increases, the combined base areas of the pyramids also approaches the surface area S of the sphere; that is, . As n increases, we find that or , so that . , and so on through The volume of the sphere is the approximation of the sum of the volumes of the n pyramids. Where the areas of the bases are , , , and so on through , the corresponding altitudes all have a length approximated by h. The the volume of the sphere is appproximated by . In turn, . As n increases, the length h of the altitudes of the pyramids approaches r, the length of the radius of the sphere; that is, . As n increases, the combined base areas of the pyramids also approaches the surface area S of the sphere; that is, . As n increases, we find that or , so that . , the corresponding altitudes all have a length approximated by h. The the volume of the sphere is appproximated by The volume of the sphere is the approximation of the sum of the volumes of the n pyramids. Where the areas of the bases are , , , and so on through , the corresponding altitudes all have a length approximated by h. The the volume of the sphere is appproximated by . In turn, . As n increases, the length h of the altitudes of the pyramids approaches r, the length of the radius of the sphere; that is, . As n increases, the combined base areas of the pyramids also approaches the surface area S of the sphere; that is, . As n increases, we find that or , so that . . In turn, The volume of the sphere is the approximation of the sum of the volumes of the n pyramids. Where the areas of the bases are , , , and so on through , the corresponding altitudes all have a length approximated by h. The the volume of the sphere is appproximated by . In turn, . As n increases, the length h of the altitudes of the pyramids approaches r, the length of the radius of the sphere; that is, . As n increases, the combined base areas of the pyramids also approaches the surface area S of the sphere; that is, . As n increases, we find that or , so that . .
As n increases, the length h of the altitudes of the pyramids approaches r, the length of the radius of the sphere; that is, The volume of the sphere is the approximation of the sum of the volumes of the n pyramids. Where the areas of the bases are , , , and so on through , the corresponding altitudes all have a length approximated by h. The the volume of the sphere is appproximated by . In turn, . As n increases, the length h of the altitudes of the pyramids approaches r, the length of the radius of the sphere; that is, . As n increases, the combined base areas of the pyramids also approaches the surface area S of the sphere; that is, . As n increases, we find that or , so that . . As n increases, the combined base areas of the pyramids also approaches the surface area S of the sphere; that is, The volume of the sphere is the approximation of the sum of the volumes of the n pyramids. Where the areas of the bases are , , , and so on through , the corresponding altitudes all have a length approximated by h. The the volume of the sphere is appproximated by . In turn, . As n increases, the length h of the altitudes of the pyramids approaches r, the length of the radius of the sphere; that is, . As n increases, the combined base areas of the pyramids also approaches the surface area S of the sphere; that is, . As n increases, we find that or , so that . .
As n increases, we find that The volume of the sphere is the approximation of the sum of the volumes of the n pyramids. Where the areas of the bases are , , , and so on through , the corresponding altitudes all have a length approximated by h. The the volume of the sphere is appproximated by . In turn, . As n increases, the length h of the altitudes of the pyramids approaches r, the length of the radius of the sphere; that is, . As n increases, the combined base areas of the pyramids also approaches the surface area S of the sphere; that is, . As n increases, we find that or , so that . or The volume of the sphere is the approximation of the sum of the volumes of the n pyramids. Where the areas of the bases are , , , and so on through , the corresponding altitudes all have a length approximated by h. The the volume of the sphere is appproximated by . In turn, . As n increases, the length h of the altitudes of the pyramids approaches r, the length of the radius of the sphere; that is, . As n increases, the combined base areas of the pyramids also approaches the surface area S of the sphere; that is, . As n increases, we find that or , so that . , so that The volume of the sphere is the approximation of the sum of the volumes of the n pyramids. Where the areas of the bases are , , , and so on through , the corresponding altitudes all have a length approximated by h. The the volume of the sphere is appproximated by . In turn, . As n increases, the length h of the altitudes of the pyramids approaches r, the length of the radius of the sphere; that is, . As n increases, the combined base areas of the pyramids also approaches the surface area S of the sphere; that is, . As n increases, we find that or , so that . .

For a right circular cone, the length of the radius of the base is r, the length of the slant height is l, and the length of the altitude is h. Given the lengths for r and h, explain why the length of the slant height is given by For a right circular cone, the length of the radius of the base is r, the length of the slant height is l, and the length of the altitude is h. Given the lengths for r and h, explain why the length of the slant height is given by . (no drawing provided) . (no drawing provided)

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The radius, slant height from the vertex of the cone to the endpoint of the radius, and the altitude form a right triangle with l for the length of the hypotenuse. Then, The radius, slant height from the vertex of the cone to the endpoint of the radius, and the altitude form a right triangle with l for the length of the hypotenuse. Then, . By the Square Roots Property, . .
By the Square Roots Property, The radius, slant height from the vertex of the cone to the endpoint of the radius, and the altitude form a right triangle with l for the length of the hypotenuse. Then, . By the Square Roots Property, . .

-Use the drawing to explain the following theorem. The total area of a right circular cylinder with base area B and lateral area L is given by . -Use the drawing to explain the following theorem. "The total area of a right circular cylinder with base area B and lateral area L is given by -Use the drawing to explain the following theorem. The total area of a right circular cylinder with base area B and lateral area L is given by . ."

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