12-6 Surface Area and Volumes of Spheres. Find the surface area of each sphere or hemisphere. Round to the nearest tenth.

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1 1-6 Surface Area and Volumes of Spheres Find the surface area of each sphere or hemisphere. Round to the nearest tenth. 6/87,1 6/87,1 sphere: area of great circle = 36ʌ yd 6/87,1 We know that the area of a great circle is. Find r. Now find the surface area. hemisphere: circumference of great circle 6 cm Page 1

2 1-6 Surface Area and Volumes of Spheres sphere: area of great circle = 36ʌ yd 6/87,1 We know that the area of a great circle is. Find r. Now find the surface area. hemisphere: circumference of great circle 6 cm 6/87,1 We know that the circumference of a great circle is. The area of a hemisphere is one-half the area of the sphere plus the area of the great circle. Find the volume of each sphere or hemisphere. Round to the nearest tenth. sphere: radius = 10 ft 6/87,1 hemisphere: diameter = 16 cm 6/87,1 Page

3 1-6 Surface Area and Volumes of Spheres hemisphere: diameter = 16 cm 6/87,1 hemisphere: circumference of great circle = 4ʌ m 6/87,1 We know that the circumference of a great circle is. Find r. Now find the volume. sphere: area of great circle = 55ʌ in 6/87,1 We know that the area of a great circle is. Find r. Now find the volume. BASKETBALL Basketballs used in professional games must have a circumference of 9 surface area of a basketball used in a professional game? 6/87,1 We know that the circumference of a great circle is ʌr. Find r. inches. What is the Page 3

4 1-6 Surface Area and Volumes of Spheres BASKETBALL Basketballs used in professional games must have a circumference of 9 inches. What is the surface area of a basketball used in a professional game? 6/87,1 We know that the circumference of a great circle is ʌr. Find r. Find the surface area. Find the surface area of each sphere or hemisphere. Round to the nearest tenth. 6/87,1 6/87,1 Page 4

5 1-6Surface Area and Volumes of Spheres 6/87,1 6/87,1 6/87,1 sphere: circumference of great circle = ʌ cm 6/87,1 We know the circumference of a great circle is ʌr. Find r. esolutions Manual that - Powered by Cognero Page 5

6 1-6Surface Area and Volumes of Spheres sphere: circumference of great circle = ʌ cm 6/87,1 We know that the circumference of a great circle is ʌr. Find r. sphere: area of great circle 3 ft 6/87,1 We know that the area of a great circle is ʌr. Find r. hemisphere: area of great circle 40 in 6/87,1 We know that the area of a great circle is ʌr. Find r. Substitute for r in the surface area formula. hemisphere: circumference of great circle = 15ʌ mm 6/87,1 Page 6

7 1-6 Surface Area and Volumes of Spheres hemisphere: circumference of great circle = 15ʌ mm 6/87,1 Find the volume of each sphere or hemisphere. Round to the nearest tenth. 6/87,1 6/87,1 sphere: radius = 1.4 yd 6/87,1 esolutions Manual - Powered by Cognero Page 7

8 1-6 Surface Area and Volumes of Spheres sphere: radius = 1.4 yd 6/87,1 hemisphere: diameter = 1.8 cm 6/87,1 sphere: area of great circle = 49ʌ m 6/87,1 Find r. Find the volume. sphere: circumference of great circle in. 6/87,1 Find r. Manual - Powered by Cognero esolutions Find the volume. Page 8

9 1-6 Surface Area and Volumes of Spheres sphere: circumference of great circle in. 6/87,1 Find r. Find the volume. hemisphere: circumference of great circle 18 ft 6/87,1 Find r. Find the volume. hemisphere: area of great circle 35 m 6/87,1 Find r. Manual - Powered by Cognero esolutions Page 9

10 1-6 Surface Area and Volumes of Spheres hemisphere: area of great circle 35 m 6/87,1 Find r. Find the volume. FISH A puffer fish is able to ³puff up when threatened by gulping water and inflating its body. The puffer fish at the right is approximately a sphere with a diameter of 5 inches. Its surface area when inflated is about 1.5 times its normal surface area. What is the surface area of the fish when it is not puffed up? 5HIHUWRWKHSKRWRRQ3DJH 6/87,1 Find the surface area of the non-puffed up fish, or when it is normal. ARCHITECTURE The Reunion Tower in Dallas, Texas, is topped by a spherical dome that has a surface area of approximately 13,94ʌ square feet. What is the volume of the dome? Round to the nearest tenth. 5HIHUWRWKHSKRWRRQ3DJH 6/87,1 Find r. Page 10

11 1-6Surface Area and Volumes of Spheres ARCHITECTURE The Reunion Tower in Dallas, Texas, is topped by a spherical dome that has a surface area of approximately 13,94ʌ square feet. What is the volume of the dome? Round to the nearest tenth. 5HIHUWRWKHSKRWRRQ3DJH 6/87,1 Find r. Find the volume. TREE HOUSE The spherical tree house, or tree sphere, has a diameter of 10.5 feet. Its volume is 1.8 times the volume of the first tree sphere that was built. What was the diameter of the first tree sphere? Round to the nearest foot. 6/87,1 The volume of the spherical tree house is 1.8 times the volume of the first tree sphere. The diameter is 4.3() or about 9 ft. Find the surface area and the volume of each solid. Round to the nearest tenth. Page 11

12 1-6 Surface Area and Volumes of Spheres TREE HOUSE The spherical tree house, or tree sphere, has a diameter of 10.5 feet. Its volume is 1.8 times the volume of the first tree sphere that was built. What was the diameter of the first tree sphere? Round to the nearest foot. 6/87,1 The volume of the spherical tree house is 1.8 times the volume of the first tree sphere. The diameter is 4.3() or about 9 ft. Find the surface area and the volume of each solid. Round to the nearest tenth. 6/87,1 To find the surface area of the figure, calculate the surface area of the cylinder (without the bases), hemisphere (without the base), and the base, and add them. To find the volume of the figure, calculate the volume of the cylinder and the hemisphere and add them. Page 1

13 1-6Surface Area and Volumes of Spheres The diameter is 4.3() or about 9 ft. Find the surface area and the volume of each solid. Round to the nearest tenth. 6/87,1 To find the surface area of the figure, calculate the surface area of the cylinder (without the bases), hemisphere (without the base), and the base, and add them. To find the volume of the figure, calculate the volume of the cylinder and the hemisphere and add them. 6/87,1 To find the surface area of the figure, we will need to calculate the surface area of the prism, the area of the top base,manual the area of thebygreat circle, and one-kdoiwkhvxuidfhduhdriwkhvskhuh esolutions - Powered Cognero Page 13 )LUVW ILQGWKHVXUIDFHDUHDRIWKHSULVPDQGVXEWUDFWWKHDUHDRIWKHWRSEDVH

14 1-66/87,1 Surface Area and Volumes of Spheres To find the surface area of the figure, we will need to calculate the surface area of the prism, the area of the top base, the area of the great circle, and one-kdoiwkhvxuidfhduhdriwkhvskhuh )LUVW ILQGWKHVXUIDFHDUHDRIWKHSULVPDQGVXEWUDFWWKHDUHDRIWKHWRSEDVH This covers everything but the top of the figure. The corners of the top base can be found by subtracting the area of the great circle from the area of the base. Next, find the surface area of the hemisphere. Now, find the total surface area. The volume of the figure is the volume of the prism minus the volume of the hemisphere. Page 14

15 1-6 Surface Area and Volumes of Spheres TOYS The spinning top is a composite of a cone and a hemisphere. a. Find the surface area and the volume of the top. Round to the nearest tenth. b. If the manufacturer of the top makes another model with dimensions that are one-half of the dimensions of this top, what are its surface area and volume? 6/87,1 a. Use the Pythagorean Theorem to find the slant height. Lateral area of the cone: Surface area of the hemisphere: b. Use the Pythagorean Theorem to find the slant height. Page 15

16 1-6 Surface Area and Volumes of Spheres b. Use the Pythagorean Theorem to find the slant height. Lateral area of the cone: Surface area of the hemisphere: BALLOONS A spherical helium-filled balloon with a diameter of 30 centimeters can lift a 14-gram object. Find the size of a balloon that could lift a person who weighs 65 kilograms. Round to the nearest tenth. 6/87,1 1 kg = 1000 g. Form a proportion. Let x be the unknown. The balloon would have to have a diameter of approximately 139,86 cm. Use sphere S to name each of the following. Page 16

17 1-6 Surface Area and Volumes of Spheres The balloon would have to have a diameter of approximately 139,86 cm. Use sphere S to name each of the following. a chord 6/87,1 A chord of a sphere is a segment that connects any two points on the sphere. a radius 6/87,1 A radius of a sphere is a segment from the center to a point on the sphere. Sample answer: a diameter 6/87,1 A diameter of a sphere is a chord that contains the center. a tangent 6/87,1 A tangent to a sphere is a line that intersects the sphere in exactly one point. Line l intersects the sphere at A. a great circle 6/87,1 If the circle contains the center of the sphere, the intersection is called a great circle. S DIMENSIONAL ANALYSIS Which has greater volume: a sphere with a radius of.3 yards or a cylinder with a radius of 4 feet and height of 8 feet? 6/87,1.3 yards.3(3) or feet esolutions Manual=- Powered by6.9 Cognero Page 17

18 6/87,1 If the circle contains the center of the sphere, the intersection is called a great circle. 1-6 Surface Area and Volumes of Spheres S DIMENSIONAL ANALYSIS Which has greater volume: a sphere with a radius of.3 yards or a cylinder with a radius of 4 feet and height of 8 feet? 6/87,1.3 yards =.3(3) or 6.9 feet 3 3 Since 1376 ft > 40 ft, the sphere has the greater volume. FOOD Suppose the orange in the graphic is a sphere with a radius of 4 centimeters. Round to the nearest tenth. a. What is the volume of the portion of the sphere that represents orange production in California? b. What is the surface area of the portion of the sphere that represents orange production in Florida? 6/87,1 a. Find the volume of the sphere with radius 4 cm. b. Find the surface area of the sphere with radius 4 cm. Page 18

19 1-6 Surface Area and Volumes of Spheres 3 3 Since 1376 ft > 40 ft, the sphere has the greater volume. FOOD Suppose the orange in the graphic is a sphere with a radius of 4 centimeters. Round to the nearest tenth. a. What is the volume of the portion of the sphere that represents orange production in California? b. What is the surface area of the portion of the sphere that represents orange production in Florida? 6/87,1 a. Find the volume of the sphere with radius 4 cm. b. Find the surface area of the sphere with radius 4 cm. Describe the number and types of planes that produce reflection symmetry in each solid. Then describe the angles of rotation that produce rotation symmetry in each solid. Page 19 6/87,1 7KHUHLVDQLQILQLWHQXPEHURISODQHVWKDWSURGXFHUHIOHFWLRQV\PPHWU\IRUVSKHUHV $Q\SODQHWKDWSDVVHVWKURXJKWK

20 1-6 Surface Area and Volumes of Spheres Describe the number and types of planes that produce reflection symmetry in each solid. Then describe the angles of rotation that produce rotation symmetry in each solid. 6/87,1 7KHUHLVDQLQILQLWHQXPEHURISODQHVWKDWSURGXFHUHIOHFWLRQV\PPHWU\IRUVSKHUHV $Q\SODQHWKDWSDVVHVWKURXJKWK symmetry. Any plane that does not pass through the center will not produce symmetry. These are all symmetric These are not: The angle of rotation is the angle through which a preimage is rotated to form the image. A sphere can be rotated at will be identical to the preimage. Page 0

21 1-6 Surface Area and Volumes of Spheres 6/87,1 $Q\SODQHRIV\PPHWU\PXVWSDVVWKURXJKWKHRULJLQ 7DNHDORRNDWDIHZH[DPSOHV symmetric Symmetric Not symmetric Not symmetric :HVHHWKDWWKHRQO\SODQHVWKDWSURGXFHUHIOHFWLYHV\PPHWU\DUHYHUWLFDOSODQHVWKDWSDVVWKURXJKWKHRULJLQ The angle of rotation is the angle through which a preimage is rotated to form the image. A hemisphere can be rotated at any angle around the vertical axis, and the resulting image will be identical to the preimage. A hemisphere URWDWHGDERXWDQ\RWKHUD[LVZLOOQRWEHV\PPHWULF Page 1

22 :HVHHWKDWWKHRQO\SODQHVWKDWSURGXFHUHIOHFWLYHV\PPHWU\DUHYHUWLFDOSODQHVWKDWSDVVWKURXJKWKHRULJLQ anglearea of rotation is the angle 1-6The Surface and Volumes of through Sphereswhich a preimage is rotated to form the image. A hemisphere can be rotated at any angle around the vertical axis, and the resulting image will be identical to the preimage. A hemisphere URWDWHGDERXWDQ\RWKHUD[LVZLOOQRWEHV\PPHWULF Symmetric Not Symmetric CHANGING DIMENSIONS A sphere has a radius of 1 centimeters. Describe how each change affects the surface area and the volume of the sphere. The radius is multiplied by 4. 6/87,1 3 The surface area is multiplied by 4 or 16. The volume is multiplied by 4 or 64. The radius is divided by 3. 6/87,1 Page

23 3 1-6The Surface Area Volumesby of4spheres surface areaand is multiplied or 16. The volume is multiplied by 4 or 64. The radius is divided by 3. 6/87,1 3 The surface area is divided by 3 or 9. The volume is divided by 3 or 7. NAVIGATIONAL COORDINATES Latitude lines circle Earth in an east-west direction. a. Describe the longest latitude line. What is the geometric name for the circle created by this line? b. The circumference of Earth is about 5,000 miles. Describe how you can use this to find the surface area of Earth. Then find the surface area. c. If two planes are equidistant from the center of a sphere and intersect the sphere, then the circles of intersection are congruent. Describe how this property can be applied to latitude lines. 6/87,1 a. The longest latitude line is the line that goes around the middle, called the equator. This line represents a circle which contains the center of the sphere, so it is a great circle. b. Find the radius. Then use the formula for surface area of a sphere to find the surface area. Page 3

24 1-6Surface Area and Volumes of Spheres 3 The surface area is divided by 3 or 9. The volume is divided by 3 or 7. NAVIGATIONAL COORDINATES Latitude lines circle Earth in an east-west direction. a. Describe the longest latitude line. What is the geometric name for the circle created by this line? b. The circumference of Earth is about 5,000 miles. Describe how you can use this to find the surface area of Earth. Then find the surface area. c. If two planes are equidistant from the center of a sphere and intersect the sphere, then the circles of intersection are congruent. Describe how this property can be applied to latitude lines. 6/87,1 a. The longest latitude line is the line that goes around the middle, called the equator. This line represents a circle which contains the center of the sphere, so it is a great circle. b. Find the radius. Then use the formula for surface area of a sphere to find the surface area. c. Sample answer: Latitude lines the same distance from the center of Earth are congruent circles. So, for example, WKHODWLWXGHOLQHVUHSUHVHQWLQJ 1DQG 6DUHFRQJUXHQWFLUFOHV 7KLVLVWUXHEHFDXVHWKH\DUHHTXLGLVWDQWIURP WKHHTXDWRU CHALLENGE A cube has a volume of 16 cubic inches. Find the volume of a sphere that is circumscribed about the cube. Round to the nearest tenth. 6/87,1 Since the volume of the cube is 16 cubic inches, the length of each side is 6 inches and so, the length of a diagonal on a face is LQFKHV Use the length of the one side (BC) and this diagonal (AB ) to find the length of the diagonal joining two opposite vertices of the cube (AC ). The triangle that is formed by these sides is a right triangle. Apply the Pythagorean Theorem. Page 4

25 c. Sample answer: Latitude lines the same distance from the center of Earth are congruent circles. So, for example, 1-6WKHODWLWXGHOLQHVUHSUHVHQWLQJ 1DQG 6DUHFRQJUXHQWFLUFOHV 7KLVLVWUXHEHFDXVHWKH\DUHHTXLGLVWDQWIURP Surface Area and Volumes of Spheres WKHHTXDWRU CHALLENGE A cube has a volume of 16 cubic inches. Find the volume of a sphere that is circumscribed about the cube. Round to the nearest tenth. 6/87,1 Since the volume of the cube is 16 cubic inches, the length of each side is 6 inches and so, the length of a diagonal on a face is LQFKHV Use the length of the one side (BC) and this diagonal (AB ) to find the length of the diagonal joining two opposite vertices of the cube (AC ). The triangle that is formed by these sides is a right triangle. Apply the Pythagorean Theorem. The sphere is circumscribed about the cube, so the vertices of the cube all lie on the sphere. Therefore, line AC is also the diameter of the sphere. The radius is inches. Find the volume. REASONING Determine whether the following statement is true or false. If true, explain your reasoning. If false, SURYLGHDFRXQWHUH[DPSOH If a sphere has radius r, there exists a cone with radius r having the same volume. 6/87,1 Determine if there is a value of h for which the volume of the cone will equal the volume of the sphere. Page 5

26 1-6 Surface Area and Volumes of Spheres REASONING Determine whether the following statement is true or false. If true, explain your reasoning. If false, SURYLGHDFRXQWHUH[DPSOH If a sphere has radius r, there exists a cone with radius r having the same volume. 6/87,1 Determine if there is a value of h for which the volume of the cone will equal the volume of the sphere. The volume of a sphere with radius rlvjlyhqe\ of h LVJLYHQE\. The volume of a cone with a radius of r and a height. Thus, a cone and sphere with the same radius will have the same volume whenever the cone has a height equal to 4r. Therefore, the statement is true. OPEN ENDED Sketch a sphere showing two examples of great circles. Sketch another sphere showing two examples of circles formed by planes intersecting the sphere that are not great circles. 6/87,1 The great circles need to contain the center of the sphere. The planes that are not great circles need to not contain the center of the sphere. Sample answer: WRITING IN MATH Write a ratio comparing the volume of a sphere with radius r to the volume of a cylinder with Page 6 what the ratio means. esolutions Manual by Cognero radius r and- Powered height r. Then describe 6/87,1

27 1-6 Surface Area and Volumes of Spheres WRITING IN MATH Write a ratio comparing the volume of a sphere with radius r to the volume of a cylinder with radius r and height r. Then describe what the ratio means. 6/87,1 The ratio is :3. The volume of the sphere is two thirds the volume of the cylinder. GRIDDED RESPONSE What is the volume of the hemisphere shown below in cubic meters? 6/87,1 The volume V of a hemisphere is RU, where r is the radius. Use the formula. ALGEBRA What is the solution set of 3z + 4 < 6 + 7z? 6/87,1 The correct choice is A. If the area of the great circle of a sphere is 33 ft, what is the surface area of the sphere? F 4 f t G 117 f t - Powered by Cognero esolutions Manual H 13 f t J 64 f t Page 7

28 1-6 Surface Area and Volumes of Spheres The correct choice is A. If the area of the great circle of a sphere is 33 ft, what is the surface area of the sphere? F 4 f t G 117 f t H 13 f t J 64 f t 6/87,1 We know that the area of a great circle is The surface area S of a sphere is r is the radius. Use the formula.., where So, the correct choice is H. SAT/ACT If a line LVSHUSHQGLFXODUWRDVHJPHQWAB at E, how many points on line point A as from point B? A none B one C two D three E all points 6/87,1 Since the segment AB is perpendicular to line correct choice is E., all points on DUHWKHVDPHGLVWDQFHIURP DUHWKHVDPHGLVWDQFHIURPSRLQWA as from B. The Find the volume of each pyramid. Round to the nearest tenth if necessary. 6/87,1 7KHEDVHRIWKHS\UDPLGLVDVTXDUHZLWKDVLGHRI IHHW 7KHVODQWKHLJKWRIWKHS\UDPLGLV IHHW 8VHWKH Pythagorean Theorem to find the height h. Page 8

29 E all points 6/87,1 the segment is perpendicular to line 1-6Since Surface Area andabvolumes of Spheres correct choice is E., all points on DUHWKHVDPHGLVWDQFHIURPSRLQWA as from B. The Find the volume of each pyramid. Round to the nearest tenth if necessary. 6/87,1 7KHEDVHRIWKHS\UDPLGLVDVTXDUHZLWKDVLGHRI IHHW 7KHVODQWKHLJKWRIWKHS\UDPLGLV IHHW 8VHWKH Pythagorean Theorem to find the height h.. The volume of a pyramid is, where B is the area of the base and h is the height of the pyramid. 3 Therefore, the volume of the pyramid is about 58.9 ft. 6/87,1 The volume of a pyramid is, where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a right triangle with a leg of 8 inches and a hypotenuse of 17 inches. Use the Pythagorean Theorem to find the length of the other leg a. Page 9

30 3 1-6 Surface Area and Volumes of Spheres Therefore, the volume of the pyramid is about 58.9 ft. 6/87,1 The volume of a pyramid is, where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a right triangle with a leg of 8 inches and a hypotenuse of 17 inches. Use the Pythagorean Theorem to find the length of the other leg a. The height of the pyramid is 1 inches. 3 Therefore, the volume of the pyramid is 40 in. 6/87,1 The base of this pyramid is a rectangle of length 10 meters and width 6 meters. The slant height of the pyramid is 1 meters. Use the Pythagorean Theorem to find the height h. Page 30

31 1-6Surface Area and Volumes of Spheres 3 Therefore, the volume of the pyramid is 40 in. 6/87,1 The base of this pyramid is a rectangle of length 10 meters and width 6 meters. The slant height of the pyramid is 1 meters. Use the Pythagorean Theorem to find the height h. The volume of a pyramid is, where B is the area of the base and h is the height of the pyramid. Therefore, the volume of the pyramid is about 3.4 m3. ENGINEERING The base of an oil drilling platform is made up of 4 concrete cylindrical cells. Twenty of the cells are used for oil storage. The pillars that support the platform deck rest on the four other cells. Find the total volume of the storage cells. 6/87,1 The volume of a cylinder is, where r is the radius and h is the height of the cylinder. Page 31

32 1-6Surface Area and Volumes of Spheres Therefore, the volume of the pyramid is about 3.4 m3. ENGINEERING The base of an oil drilling platform is made up of 4 concrete cylindrical cells. Twenty of the cells are used for oil storage. The pillars that support the platform deck rest on the four other cells. Find the total volume of the storage cells. 6/87,1 The volume of a cylinder is, where r is the radius and h is the height of the cylinder. 3 The total volume of the storage cells is about 18,555,031.6 ft. Find the area of each shaded region. Round to the nearest tenth. 6/87,1 :HDUHJLYHQWKHEDVHRIWKHWULDQJOH 6LQFHWKHWULDQJOHLV - - WKHKHLJKWLVDOVR 7KHWULDQJOHLVD - - WULDQJOH VRWKHGLDPHWHURIWKHFLUFOHLV and the radius h is. Page 3

33 1-6 Surface Area and Volumes of Spheres 3 The total volume of the storage cells is about 18,555,031.6 ft. Find the area of each shaded region. Round to the nearest tenth. 6/87,1 :HDUHJLYHQWKHEDVHRIWKHWULDQJOH 6LQFHWKHWULDQJOHLV - - WKHKHLJKWLVDOVR 7KHWULDQJOHLVD - - WULDQJOH VRWKHGLDPHWHURIWKHFLUFOHLV and the radius h is. 6/87,1 The side length of the given square is (7) or 14 units. Page 33

34 1-6 Surface Area and Volumes of Spheres 6/87,1 The side length of the given square is (7) or 14 units. 6/87,1 $UHJXODURFWDJRQKDV FRQJUXHQWFHQWUDODQJOHV VRWKHPHDVXUHRIHDFKFHQWUDODQJOHLV Apothem LVWKHKHLJKWRIWKHLVRVFHOHVWULDQJOH$%& 8VHWKH7ULJRQRPHWULFUDWLRVWRILQGWKHVLGHOHQJWKDQG apothem of the polygon. Page 34

35 1-6 Surface Area and Volumes of Spheres AB = (DB) = 3 sin.5 COORDINATE GEOMETRY Find the area of each figure. WXYZ with W(0, 0), X (4, 0), Y(5, 5), and Z(1, 5) 6/87,1 Graph the diagram. 7KHOHQJWKRIWKHEDVHRIWKHSDUDOOHORJUDPJRHVIURP WR VRLWLV XQLWV 7KHKHLJKWRIWKHSDUDOOHORJUDPJRHVIURP WR VRLWLV XQLWV The area of a parallelogram is the product of a base b and its corresponding height h. Therefore, the area is 0 units. ABC with A (, ±3), B(±5, ±3), and C(±1, 3) 6/87,1 Graph the diagram. Page 35

36 7KHKHLJKWRIWKHSDUDOOHORJUDPJRHVIURP WR VRLWLV XQLWV The area of a parallelogram is the product of a base b and its corresponding height h. Therefore, the area is 0 1-6units Surface. Area and Volumes of Spheres ABC with A (, ±3), B(±5, ±3), and C(±1, 3) 6/87,1 Graph the diagram. The length of the base of the triangle goes from (±5, ±3) to (, ±3), so it is 7 units. The height of the triangle goes from (±1, ±3) to (±1, 3), so it is 6 units. Therefore, the area is 0.5(7)(6) = 1 units. Refer to the figure. How many planes appear in this figure? 6/87,1 A plane is a flat surface made up of points that extends infinitely in all directions. There is exactly one plane through any three points not on the same line. plane P plane ABD plane ACD plane ABC Name three points that are collinear. 6/87,1 Collinear points are points that lie on the same line. Noncollinear points do not lie on the same line. D, B, and G lie on line DG. Are points G, A, B, and E coplanar? Explain. 6/87,1 Coplanar points are points that lie in the same plane. Noncoplanar points do not lie in the same plane. Points A, B, and E lie in plane P, but point G does not lie in plane P. Thus, they are not coplanar. Points A, G, and B lie in plane AGB, but point E does not lie in plane AGB. At what point do 6/87,1 and LQWHUVHFW" Page 36

37 6/87,1 Collinear points are points that lie on the same line. Noncollinear points do not lie on the same line. 1-6Surface Area and Volumes of Spheres D, B, and G lie on line DG. Are points G, A, B, and E coplanar? Explain. 6/87,1 Coplanar points are points that lie in the same plane. Noncoplanar points do not lie in the same plane. Points A, B, and E lie in plane P, but point G does not lie in plane P. Thus, they are not coplanar. Points A, G, and B lie in plane AGB, but point E does not lie in plane AGB. At what point do and LQWHUVHFW" 6/87,1 and GRQRWLQWHUVHFW lies in plane P, but only E lies in P. If E were on, then they would intersect. Page 37