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Unit 4 Reference Sheet Circle: The set of points in a plane that are fixed distance from a given point called the center of the circle. Chord: A segment whose endpoints both lie on the same circle. Name: Radius: A segment whose endpoints are the center of a circle and a point on the circle. chord Diameter: A segment that has endpoints on a circle and that passes through the center of the circle. Secant: A line that intersects a circle at two points. Tangent: A line that is in the same plane as a circle and intersects the circle at exactly one point. The radius is perpendicular to the tangent at the point of tangency. Internal Tangent: A tangent that is common to two circles and intersects the segment joining the centers of the circles. External Tangent: A tangent that is common to two circles and does not intersect the segment joining the centers of the circles. Inscribed Angle: An angle whose vertex is on a circle and whose sides contain chords of the circle. The angle measure of the inscribed angle is ½ of the intercepted arc. Central Angle: An angle whose vertex is the center of a circle. Angle Between 2 Chords: The angle between 2 chords is equal to ½ the sum of the two intercepted arcs. Angle Between 2 Secants: The angle between 2 chords is equal to ½ the difference of the two intercepted arcs. M. Winking (Unit 4 -00) p.87 Sec 4.1 – Circles & Volume The Language of Circles 1. Name: Using the Pythagorean Theorem to find the value of x in each of the diagrams below: 1. 2. x= x= Converse of the Pythagorean Theorem. Which of the following are right triangles? 3. 11 4. 5. 6. 17 17 15 35 7. 5 12 4 15 37 6 8 8 8 Right Triangle? (circle one) YES (circle one) NO YES Right Triangle? Right Triangle? Right Triangle? (circle one) (circle one) YES NO YES NO ______8. G ______9. A A. Diameter B. Radius ______10. DE C. Center ______11. GC D. Secant ______12. JB E. Chord ______13. HJ F. Point of tangency ______14. HI G. Common external tangent ______15. AB H. Common internal tangent (circle one) NO Determine if AB is tangent to the circle centered at point C. Explain your reasoning. 16. 17. M. Winking Unit 4-1 page 88 Right Triangle? YES NO AB and AD are tangent to the circle centered at point C. Find the value of x. 18. 19. x= x= Given the center of the circle is point A, find the requested measure. 20. mEF = 21. mCE = 22. mCDF = 23. mDE = 24. mBC = 25. mFB = 26. mFBE = 27. mDFC = 28. mDFB = 29. mBEC = Determine the measure of BC . 30. 31. 32. Find the requested measure for each circle. 33. FC = 34. mBG _________ M. Winking Unit 4-1 page 89 Sec 4.2 – Circles & Volume Inscribed Angles 1. Central Angle: An angle whose vertex is the center of the circle. Name: Inscribed Angle: An angle whose vertex is on a circle and whose sides contain chords of the circle Central Angle Inscribed Angle Inscribed Angle Properties: Consider the following diagram an inscribed angle of the circle center at A. D C D D A B Consider the inscribed angle ∡𝐶𝐵𝐷 which intercepts arc ̂ that measures 70˚. 𝐷𝐶 C A C B Since the central angle ̂ ∡𝐶𝐴𝐷 intercepts arc 𝐷𝐶 then 𝑚∡𝐶𝐴𝐷 = 70°. D A B Triangle ∆DAB is isosceles because the legs are radii of the circle. The measure of angle 𝑚∡𝐷𝐴𝐵 = 110° since it forms a linear pair with ∡𝐶𝐴𝐷. C A The based angles of ∆DAB must be congruent and the interior angles of triangle must sum to 180˚. So, 110 + 𝑥 + 𝑥 = 180 In a similar fashion using addition or subtraction, it can be shown this idea extends to any inscribed angle. “An inscribed angle’s measure is exactly half of the arc measure that it intercepts.” Find the most appropriate value for ‘x’ in each of the diagrams below. (Assume point ‘A’ is the center of the circle.) 1. 2. 3. A A x= A x= M. Winking x= Unit 4-2 page 90 B Find the most appropriate value for ‘x’ in each of the diagrams below. (Assume point ‘A’ is the center of the circle.) 4. 5. 6. A A A x= x= 7. x= 8. 9. A A A x= x= 10. x= 11. 12. A A A x= x= M. Winking x= Unit 4-2 page 91 Find the most appropriate value for ‘x’ in each of the diagrams below. (Assume point ‘A’ is the center of the circle.) 13. 14. 15. A A x= A x= x= 17. 16. 18. A A A x= x= 19. x= 20. 21. A A x= A x= M. Winking x= Unit 4-2 page 92 1. Sec 4.2a – Circles & Volume Tangent Circle Construction Name: [Creating a Tangent To a Circle] Construct a line tangent to circle with center A and passing through point C. Step I: First draw a segment with end points A & C. Step 3: Create a circle centered at the midpoint of segment ̅̅̅̅ 𝐴𝐶 and with a radius from the midpoint to point A. Step 2: Create a perpendicular bisector to segment ̅̅̅̅ 𝐴𝐶 Step 4: Draw a line that passes through point C and either of the intersections of the original circle and the newly created circle (point E in the diagram). Construct a tangent line to circle with center A that passes through point C. C ● ● M. Winking Unit 4-a2 page 93 A Sec 4.3 – Circles & Volume Angles of Circles 1. Name: Tangent Line Angles Consider the tangent line ⃡𝐷𝐶 and ̂ the ray 𝐶𝐵 which intercepts arc 𝐵𝐶 and has a measure of x˚. ̅̅̅̅ to Draw an auxiliary segment ̅̅̅̅ 𝐴𝐵 and 𝐴𝐶 create an isosceles triangles. We know that 𝑚∡𝐴 = 𝑥° as a central angle and the interior angles of ∆ABC sum to 180˚. So, 𝑚∡𝐵 + 𝑚∡𝐶 = 180° − 𝑥 Also, 𝑚∡𝐵 = 𝑚∡𝐶 because they are the base angles of an isosceles triangle. So, 𝑚∡𝐵 + 𝑚∡𝐵 = 180° − 𝑥 which simplifies: 2 ∙ 𝑚∡𝐵 = 180° − 𝑥 or 𝑚∡𝐵 = ̅̅̅̅ must be perpendicular to Finally, 𝐴𝐶 ⃡𝐷𝐶 since it is tangent of the circle. The angle ∡𝐷𝐶𝐵 & ∡𝐴𝐶𝐵 must sum to 90˚. So, we can find 180° − 𝑥 𝑚∡𝐷𝐶𝐵 = 90 − 2 which simplifies: 𝑥 𝑚∡𝐷𝐶𝐵 = 2 180°−𝑥 2 “The measure of an angle formed by a tangent and a chord drawn to the point of tangency is exactly ½ the measure of the intercepted arc.” Find the most appropriate value for ‘x’ in each of the diagrams below. (Assume CE is tangent to the circle.) 1. 2. 3. x= x= M. Winking x= Unit 4-3 page 94 Find the most appropriate value for ‘x’ in each of the diagrams below. (Assume CE is tangent to the circle.) 4. 5. 6. (You may assume DF is a diameter.) x= x= x= Intersecting Chords Interior Angles Consider the intersecting chords ̅̅̅̅ 𝐵𝐸 and ̅̅̅̅ 𝐹𝐶 that ̂ intercept the arcs 𝐶𝐵 ̂. and 𝐹𝐸 Draw an auxiliary segment ̅̅̅̅ 𝐵𝐹 to create the inscribed angles that we know are half of the intercepted arc. So, 𝑚∡𝐹𝐵𝐸 = 𝑧° 2 and 𝑚∡𝐵𝐹𝐶 = 𝑦° 2 Since triangles interior angles sum to 180˚. So we can subtract the 2 angles of triangle ∆DBF to find angle 𝑚∡𝐵𝐷𝐹 = 180° − 𝑦° − 𝑧°2 2 Finally, since ∡𝐹𝐷𝐸 forms a linear pair with ∡𝐵𝐷𝐹 we can subtract from 180˚ to find: 𝑚∡𝐹𝐷𝐸 = 180° − (180° − 𝑦°2 − 𝑧°2) which simplifies to 𝑦°+𝑧° 𝑚∡𝐹𝐷𝐸 = 2 “The measure of an angle formed by two intersecting chords of the same circle is exactly ½ the measure of the sum of the two intercepted arcs.” Find the most appropriate value for ‘x’ in each of the diagrams below. (Assume point ‘A’ is the center of the circle.) 7. 8. x= x= M. Winking Unit 4-3 page 95 Find the most appropriate value for ‘x’ in each of the diagrams below. (Assume point ‘A’ is the center of the circle.) 9. 10. (You may assume BE is a diameter.) x= x= 11. 12. x= x= 13. 14. x= x= M. Winking Unit 4-3 page 96 Secant Lines Exterior Angle Consider the rays 𝐵𝐷 and 𝐵𝐹 ̂ and 𝐹𝐷 ̂ which intercepts arc 𝐶𝐸 which measure a˚ and b˚ respectively. Draw an auxiliary segment ̅̅̅̅ 𝐸𝐷 . We 𝑏° know that 𝑚∡𝐷𝐸𝐹 = and 2 𝑎° 𝑚∡𝐶𝐷𝐸 = because each is an 2 inscribed angle. 𝑏° Finally, 𝑚∡𝐷𝐸𝐵 = 180° − since the two 2 angles at point E forma linear pair. 𝑏° 𝑎° Furthermore, 𝑚∡𝐵 = 180° − (180° − 2 ) − 2 since a triangle’s interior angles sum to 180˚ and that would simplify to 𝑚∡𝐵 = (𝑏−𝑎)° 2 . “The measure of an angle formed on the exterior of a circle by two intersecting secants of the same circle is exactly ½ the measure of the difference of the two intercepted arcs.” Find the most appropriate value for ‘x’ in each of the diagrams below. (Assume point ‘A’ is the center of the circle.) 15. 16. x= 17. x= 18. x= x= M. Winking Unit 4-3 page 97 Find the most appropriate value for ‘x’ in each of the diagrams below. (Assume point ‘A’ is the center of the circle.) 19. 20. You may assume DB is tangent to the circle. x= x= 21. 22. You may assume EC and DE are tangent to the circle. You may assume DB is tangent to the circle. x= x= 23. 24. x= x= M. Winking Unit 4-3 page 98 1. Sec 4.4 – Circles & Volume Circle Segments Name: Intersecting Chords Consider the intersecting chords ̅̅̅̅ 𝐷𝐶 and ̅̅̅̅ 𝐸𝐹 that intersect at point B. Draw an auxiliary segment ̅̅̅̅ 𝐷𝐸 and ̅̅̅̅ 𝐶𝐹 to create triangles ∆DBE and ∆FBC. We know that ∡𝐷𝐸𝐵 ≅ ∡𝐹𝐶𝐵 because they are both inscribed angles that intercept ̂ . Similarly, we know the same arc 𝐹𝐷 ∡𝐸𝐷𝐵 ≅ ∡𝐶𝐹𝐵. Then, by AA we know ∆𝐷𝐵𝐸~∆𝐹𝐵𝐶 Using proportions of similar triangles: 𝒙𝟏 𝒚𝟏 = 𝒚𝟐 𝒙𝟐 We can cross-multiply to give us the following statement: 𝒙𝟏 ∙ 𝒙𝟐 = 𝒚𝟏 ∙ 𝒚𝟐 Part1 Part2 Part1 Part2 “If two chords intersect then the product of the measures of the two subdivided parts of one chord are equal to the product of the parts of the other chord.” Find the most appropriate value for ‘x’ in each of the diagrams below. 1. 2. x= x= 3. 4. x= x= M. Winking Unit 4-4 page 99 Find the most appropriate value for ‘x’ in each of the diagrams below 5. 6. x= x= Segments of Secants Consider the intersecting ̅̅̅̅ and segments of secants 𝐸𝐶 ̅̅̅̅ that intersect at point C. 𝐴𝐶 ̅̅̅̅ and 𝐵𝐸 ̅̅̅̅ Draw an auxiliary segment 𝐴𝐷 to create triangles ∆ADC and ∆EBC. We know that ∡𝐶𝐴𝐷 ≅ ∡𝐶𝐸𝐵 because they are both inscribed angles that intercept ̂ . Reflexively, we also the same arc 𝐵𝐷 know ∡𝐶 ≅ ∡𝐶. Then, by AA we know ∆𝐴𝐷𝐶~∆𝐸𝐵𝐶 Using proportions of similar triangles: 𝒙𝟐 + 𝒙𝟏 𝒚𝟏 = 𝒚𝟏 + 𝒚𝟐 𝒙𝟏 We can cross-multiply to give us the following statement: (𝒙𝟐 + 𝒙𝟏 ) ∙ 𝒙𝟏 = (𝒚𝟏 + 𝒚𝟐 ) ∙ 𝒚𝟏 Whole External Whole External “If 2 secants intersect the same circle on the exterior of the circle then the product of the ‘whole’ and the ‘external’ segment measures is equal to the same product of the other secant’s portions. Find the most appropriate value for ‘x’ in each of the diagrams below. 7. 8. x= x= M. Winking Unit 4-4 page 100 Find the most appropriate value for ‘x’ in each of the diagrams below. 9. 10. x= x= 11. 12. x= x= Segments of Secants and Tangents Consider the intersecting ̅̅̅̅ and segment of a secant 𝐴𝐶 ̅̅̅̅ that segment of a tangent 𝐴𝐹 intersect at point A. Draw an auxiliary segment ̅̅̅̅ 𝐵𝐷 and ̅̅̅̅ 𝐶𝐷 to create triangles ∆ADC and ∆ABD. We know that ∡𝐵𝐷𝐴 ≅ ∡𝐴𝐶𝐷 because they are both have a measure of half of the ̂ . Reflexively, we also intercepted arc 𝐵𝐷 know ∡𝐴 ≅ ∡𝐴. Then, by AA we know ∆𝐴𝐷𝐶~∆𝐴𝐵𝐷 Using proportions of similar triangles: 𝒙𝟏 𝒚𝟏 = 𝒚𝟏 + 𝒚𝟐 𝒙𝟏 We can cross-multiply to give us the following statement: (𝒚𝟏 + 𝒚𝟐 ) ∙ 𝒚𝟏 = 𝒙𝟏 ∙ 𝒙𝟏 Whole M. Winking Unit 4-4 page 101 External Tangent Tangent Find the most appropriate value for ‘x’ in each of the diagrams below. 13. 14. x= x= 𝒂 ∙ 𝒃 = 𝒄 ∙ 𝒅 Part1 Part2 Part1 Part2 (𝒂 + 𝒃) ∙ 𝒂 = 𝒄𝟐 (𝒂 + 𝒃) ∙ 𝒂 = (𝒄 + 𝒅) ∙ 𝒄 Whole External Whole External Whole External Tangent 2 Find the most appropriate value for ‘x’ in each of the diagrams below. 13. 14. x= x= M. Winking Unit 4-4 page 102 1. 1. Sec 4.5 – Circles & Volume Circumference, Perimeter, Arc Length Name: Cut a piece of string that perfectly fits around the outside edge of the circle. How many diameters long is the piece of string (use a marker to mark each diameter on the string)? Tape the string to this page. Tape String Here: 2. A. Find the Circumference of the following circles (assuming point A is the center): C. E. 2a. C = B. 2c. C = D. 2b. C = 2e. C = 2d. C = M. Winking Unit 4-5 page 103 3. Find the Radius of each circle given the following information: A. B. 3b. r = 3a. r = ̂ . (Assuming point A is the center) 4. Find the Arc Length of arc 𝑩𝑪 B. A. 4b. 4a. D. C. 4c. 4d. M. Winking Unit 4-5 page 104 ̂. 5. Find the Arc Length of arc 𝑩𝑪 (Assuming point A is the center) A. B. (Assume EF is tangent to the circle.) 5b. 5a. 6. Find the most appropriate value for x in each diagram. A. B. 6b. 6a. M. Winking Unit 4-5 page 105 The Babylonian Degree method of measuring angles. Around 1500 B.C. the Babylonians are credited with first dividing the circle up in to 360̊. They used a base 60 (sexagesimal) system to count (i.e. they had 60 symbols to represent their numbers where as we only have 10 (a centesimal system of 0 through 9)). So, the number 360 was convenient as a multiple of 60. Additionally, according to Otto Neugebauer, an expert on ancient mathematics, there is evidence to support that the division of the circle in to 360 parts may have originated from astronomical events such as the division of the days of a year. So, that the earth moved approximately a degree a day around the sun. However, this would cause problems as years passed to keep the seasons accurately aligned in the calendar as there are 365.242 actual days in a year. Some ancient Persian calendars did actually use 360 days in their year further supporting this idea. The transition to Radian measure of angle: Around 1700 in the United Kingdom, mathematician Roger Cotes saw some advantages in some situations to measuring angles using a radian system. A radian system simply put, drops a unit circle (a circle with a radius of 1) on to an angle such that the center is at the vertex and the length of the intercepted arc is the radian measure. So, a full circle of 360̊ is equivallent to 2π∙(1) radians. In the example at the right, an angle of 50̊ is shown. Then, a circle that has a radius of 1 cm is drawn with its center at the vertex. 1 cm 50 Arc Length 2 1 cm 0.873 cm 360 Finally, the intercepted arc length is determined to be approximately 0.873 or more precisely Similarly, it can be demonstrated the basically that 180˚ is equivalent to π radians. 6. Using the ratio of 180˚: π convert the following degree measures to radians. a. 30˚ b. 80˚ c. 225˚ d. 360˚ 7. Using the ratio of 180˚: π convert the following radian measures to degrees. a. 𝜋 4 rads b. 3𝜋 10 rads M. Winking c. Unit 4-5 5𝜋 8 d. 0.763 rads page 106 rads 5𝜋 18 radians. ̂ using similar circles or a fraction of the circumference. 8. Find the Arc Length of 𝑩𝑪 A. B. 8b. 8a. 9. Solve the following. A. Determine the perimeter of the rhombus shown. B. Find an expression that would represent the perimeter of the triangle. 9a. 9b. M. Winking C. Given the perimeter of the rectangle shown below is 32 cm2 and the length of one side is 6 cm, determine the area of the rectangle. 9c. Unit 4-5 page 107 10. Find the perimeter of each compound figure below. A. Assume the compound figure includes a semicircle. B. (Assume all adjacent sides are perpendicular.) 10b. 10a. C. Assume the compound figure includes three semicircles. Assume the compound figure includes a rectangle and 2 sectors centered at point A and C respectively. D. 10d. 10c. M. Winking Unit 4-5 page 108 Mathematically, we can determine the value of pi using the Pythagorean Theorem. Sec 4.6 – Circles & Volume Circumference, Perimeter, Arc Length Problem Hint 1. Find the area of the rectangle and write down the formula for finding the area of a rectangle. 1. 2. Find the area of the right triangle and write down the formula for finding the area of a right triangle. 3. Find the area of the acute triangle and write down the formula for finding the area of an acute triangle. 4. Find the area of the parallelogram and write down the formula for finding the area of an acute triangle. M. Winking Unit 4-6 page 109 Name: Formula Problem 5. Find the area of the trapezoid and write down the formula for finding the area of a rectangle. Hint Formula Make a Copy 4 6 Rotate Copy 8 6 Creates a Parallelogram exactly twice the size of the trapezoid 6. Find the area of the circle below. 1. Find the area and perimeter of each of the following shapes. 3 cm 11.5 cm 8 cm 4 cm Area: Area: Perimeter r: Perimeter r: 7 in 6 in 1 in Perimeter r: Area: 9 in Area: Perimeter r: Area: Perimeter r: Area: Perimeter r: 2. Solve the following area problems. Determine the area of the rhombus shown. Area: Find an expression that would represent the area of the triangle. Find the length of the radius given 2 the area of the circle is 531 cm . x = Area: 3. Find the following sector areas (shaded regions) using fractional parts. Sector Area : Sector Area M. Winking Sector Area : Unit 4-6 page 111 : 4. Find the area of each of the shaded regions. Area: x = Area: 5. Solve the following problems. Find the area of the shaded region, given that AC is tangent to the circle at point B and 𝒎∡𝑨𝑩𝑬 = 𝟕𝟎° Sector Area : Find the central angle of a sector 2 that has an area of 71 cm and a radius of 7 cm. Sector Area M. Winking Find the radius of a sector that 2 has an area of 92 cm and a central angle of 130˚. Sector Area : Unit 4-6 page 112 : 6. Find the area of the following compound figures (assume all curved shapes are semicircles). Area: x = Area: 7. Find the area of the following shaded regions 4 cm Area: Area: M. Winking Unit 4-6 page 113 1. Sec 4.7 – Circles & Volume Nets & Surface Areas 1. Sketch a NET of each of the following solids: A. B. C. D. M. Winking Unit 4-7 page 114 Name: 2. Sketch a NET of each of the following solids: 3. Find the surface area of the following solids (figures may not be drawn to scale). Surface Area Surface Area : M. Winking Unit 4-7 page 115 : 4. Find the surface area of the following solids (figures may not be drawn to scale). Surface Area : : 12 cm Surface Area 4 cm Surface Area Surface Area : M. Winking Unit 4-7 page 116 : 5. Find the surface area of the following solids (figures may not be drawn to scale). Surface Area Surface Area : Surface Area : Surface Area M. Winking Unit 4-7 page 117 : : 6. Solve the following problems. (figures may not be drawn to scale). Find the Surface Area of a baseball given that its largest circumference is 23.5 cm. Determine the Surface Area of the following rectangular prism with a missing portion. Surface Area Surface Area : Determine the amount of surface area that is water on our planet in square miles. You may assume the earth is spherical, has a diameter of 7918 miles, and that water covers 71% of the Earth’s surface. Surface Area Given the circumference of the base of a cone is 31.4cm and the slant height is 13 cm, find the surface area of the cone. Surface Area : M. Winking : Unit 4-7 page 118a : 1. Sec 4.8 – Circles & Volume Volume of Pyramids & Cones Name: 1. Find the Volume of the following solids (figures may not be drawn to scale). Volume: Volume: Volume: Volume: M. Winking Unit 4-8 page 118b 12 cm 2. Find the Volume of the following solids (figures may not be drawn to scale). 4 cm Volume: Volume: Volume: Volume: M. Winking Unit 4-8 page 119 3. Find the volume of the following solids (figures may not be drawn to scale). Using a micrometer find the volume of 6 pennies stacked directly on top of each other (which is a cylinder). Show measurements to the nearest hundredth of a millimeter. Using a micrometer find the volume of 6 pennies stacked on top of each other but so that they are slanted. Show measurements to the nearest hundredth of a millimeter. Volume: Volume: Find volume of the oblique rectangular prism. Find volume of the sphere. Volume: Volume: M. Winking Unit 4-8 page 120 4. Find the volume of the following solids (figures may not be drawn to scale). The solid below shows a gas tank for a tractor trailer truck. It is in the shape of a cylinder with a radius of 9 inches and a height of 60 inches. How many gallons of fuel will it hold if there are 231 cubic inches in one gallon? A sphere is inscribed in a cube with a volume of 27 cm3. What is the volume of the sphere? Gallons: Volume: A snowman is created from two spherical snow balls. Given the circumference of each sphere determine the volume of the snowman. Find volume of the regular hexagonal prism. Volume: Volume: M. Winking Unit 4-8 page 121 1. Sec 4.9 – Circles & Volume Volume of Pyramids & Cones Name: UsingCavalieri’sPrinciplewecanshowthatthevolumeofapyramidisexactly⅓thevolumeofaprismwiththesame Baseandheight. Considerasquarebased pyramidinscribedincube. Next, translatethepeakofthe pyramid.Cavalieri’sPrinciple wouldsuggestthatthevolumeof theobliquepyramidisthesameas theoriginalpyramid. Next,wecancreate2moreoblique pyramidswiththesamevolumeof theoriginalwiththeremaining spaceinthecube. Inthisdiagram,wecanseethe3obliquepyramidsofequalvolumepulledoutfromthecube. So,thisdemonstratesapyramidinscribedinacubehasexactly⅓thevolumethecube. Thisideacanbeextendedtoanypyramidorcone. M.Winking Unit4‐9page122 1. FindtheVolumeofthefollowingsolids(figuresmaynotbedrawntoscale). Volume: Volume: Volume: Volume: M.Winking Unit4‐9page123 2. FindtheVolumeofthefollowingsolids(figuresmaynotbedrawntoscale). Volume: Volume: Findthevolumeoftheregularoctahedron. Findthevolumeoftheirregularsolid.Thebase hasanareaof80cm2andaheightof9cm. Volume: Volume: M.Winking Unit4‐9page124 3. FindtheVolumeofthefollowingsolids(figuresmaynotbedrawntoscale). Volume: Volume: ConsidertriangleABCwithverticesatA(0,0), B(4,6),andC(0,6)plottedandacoordinategrid. Determinethevolumeofthesolidcreatedby rotatingthetrianglearoundthey‐axis. Volume: M.Winking Unit4‐9page125 UsingCavalieri’sPrinciplewecanshowthatthevolumeofaspherecanbefoundby ∙ First,considera hemispherewitha radiusofR.Createa cylinderthathasabase withthesameradiusR andaheightequalto theradiusR.Then, removeaconefromthe cylinderthathasthe samebaseandheight. Next,consideracross sectionthatisparallel tothebaseandcuts throughbothsolids usingthesameplane. Cavalieri’sPrinciplesuggestsifthe2crosssectionshavethesameareathenthe2solidsmusthavethesamevolume. Theareaofthecrosssectionofthesphereis: ∙ UsingthePythagoreantheoremweknow: or So,withsimplesubstitution: ∙ Theareaofthecrosssectionofthesecondsolid is: ∙ ∙ Usingsimilartrianglesweknowthath=bandthen,using simplesubstitution ∙ ∙ VolumeofHemisphere=VolumeofCylinder–VolumeofCone= ∙ ∙ WealsoknowthatR=b=h.So,VolumeofHemisphere= ∙ ∙ ∙ ∙ Tofindthevolumeofacompletesphere,wecanjustdoublethehemisphere:VolumeofSphere= M.Winking Unit4‐9page126 ∙ ∙