Lesson 7 Math13-1

Lesson 7 SPHERES Week 9 MATH13-1 Solid Mensuration

Definitions Relating to Sphere

A sphere is a three-dimensional solid bounded by a surface consisting of all points equidistant from an interior point called the center. The closed surface is called spherical surface, while the space covered by the surface is called the interior. A sphere is usually designated by its center. Hence, sphere O is the sphere whose center is at point O.

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Definitions Relating to Sphere

Radius - Line from the center to a point on the spherical surface. Diameter (AB = 2R) - Line passing through the center and joining two points on the spherical surface.

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Definitions Relating to Sphere

Any section of a sphere is a circle. Great Circle (circle C) - Formed when a plane passes through the center. Small Circle (circle F) - Formed when a plane does not pass through the center.

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Definitions Relating to Sphere

Axis of a circle of a sphere (AB) - A diameter of the sphere perpendicular to the plane of the circle of the sphere. Poles of the circle (points A and B) - Endpoints of the axis.

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Definitions Relating to Sphere Spherical distance - Measure of the minor arc of a great circle containing the two points. It is the shortest distance between two points on the surface of the sphere. Polar Distance - Spherical distance from the nearest pole. Quadrant - A polar distance along the arc of a great circle which is equal to one fourth of the circumference of a great circle (i.e. Spherical distance from north pole [or south pole] to the equator). Reference: Solid Mensuration: Understanding the 3-D Space by

Definitions Relating to Sphere

Tangent line - Intersects the sphere at exactly one point. The point of intersection is called the point of tangency. Internally tangent spheres - If they lie on one side of the plane. Externally tangent spheres - If they lie on opposite sides of the plane. Reference: Solid Mensuration: Understanding the 3-D Space by

Definitions Relating to Sphere

angle (θ = spherical CPD = APB) •Spherical   - Dihedral angle formed by the planes of two great circles. It has the same measure as the plane angle formed by the tangents to the arcs drawn through their point of intersection.

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Definitions Relating to Sphere

Arc length (arc CD = θ, less than 180°) - Generally stated as the angle subtended by the arc at the center of the sphere.

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Surface Area and Volume •Sphere   S = 4πR2 V = πR3

Hemisphere S = 2πR2 V = πR3

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Inscribed Solids Involving Spheres Sphere inscribed in a polyhedron The polyhedron is circumscribed about the sphere. - The sphere is tangent the all the faces of the polyhedron. -

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Inscribed Solids Involving Spheres Sphere circumscribed about

a

polyhedron - The polyhedron is inscribed in the sphere. - All The vertices of a polyhedron lie on the surface of the sphere.

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Inscribed Solids Involving Spheres The radius of the sphere circumscribed about a regular tetrahedron is equal to three-fourths of the altitude of the tetrahedron while the radius of the inscribed sphere is one-fourth of the altitude.

R = ¾h

R = ¼h

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Inscribed Solids Involving Spheres If a sphere is inscribed in a cylinder, the volume of the sphere is two-thirds that of the cylinder, and the surface area of the sphere is equal to the lateral area of the cylinder, or twothirds of the total area of the cylinder. Vsphere = ⅔Vcylinder S = LSA S = ⅔TSA Reference: Solid Mensuration: Understanding the 3-D Space by

Inscribed Solids Involving Spheres If a cone has its base and height equal respectively to the base and height of a cylinder, its volume is one-third that of the cylinder or one-half that of a sphere inscribed in the cylinder. Vcone = ⅓Vcylinder = ½Vsphere

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Inscribed Solids Involving Spheres  If a sphere is inscribed in a cube, the volume of the sphere is about one-half the volume of the cube, or Vsphere = (Vcube) to be exact. Vsphere = (Vcube) ≈ ½Vcube

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EXAMPLES #1, p168: An equilateral triangle with an area of 9√3 sq. units is inscribed in a circular section of the sphere. How far from the center of the sphere is the plane of the triangle if the radius of the sphere is 14 units? ANS: 2√46 units #3, p170: The radii of two spheres are 4 ft. and 8 ft. Find the volume of a sphere whose surface area is equal to Reference: Solidof Mensuration: Understanding the 3-D Space by the sum the surface areas of these

EXAMPLES #5, p171: Find the volume of a hemispherical shell in which the outer and inner surface areas are 128π in2 and 50π in2, respectively. ANS: 258π in3

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7.1 EXERCISES #9, p174: The plane of a small circle in a sphere of radius 18 inches is 6 inches from the center of the sphere. Find the area of the small circle and the polar distance of a point on the circumference of the small circle. ANS: 288π in2, 22.2 in or 70.5° #10, p174: A solid metal in the form of a sphere with radius of 16 in., is cut by a plane 4 in. from a nearer pole. Find the area of the section. Reference: Solid Mensuration: Understanding the 3-D Space by

7.1 EXERCISES #13, p174: The radii of two parallel sections on the same sphere are 9 cm and 12 cm, and the distance between them is 21 cm. Find the radius of the sphere. ANS: 15 cm #21, p175: The total area of a regular tetrahedron is 81√3 ft2. What is the volume of the sphere circumscribing about the tetrahedron? ANS: 701.2 ft 3 Reference: Solid Mensuration: Understanding the 3-D Space by

7.1 EXERCISES #26, p175: Find the volume of a rectangular solid whose dimensions are in the ratio 4:6:8 if the radius of the circumscribing sphere is 6√29 inches. #28, p175: Find the volume of a sphere inscribed in a regular tetrahedron whose edges measure 6 cm.

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SPHERICAL SEGMENT Portion of a sphere formed by passing two parallel planes through the sphere. Parallel circular sections are called bases of the spherical segment. If one of the parallel planes is tangent to the sphere, the segment is called a spherical segment of one base (also known as spherical cap).

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SPHERICAL SEGMENT

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VOLUME OF A SPHERICAL SEGMENT •Two   Bases

One Base V = ⅙πh(3r2 + h2) V = ⅓πh2(3R − h) For a hemisphere, h = R.

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ZONE Portion of the surface of a sphere bounded by two parallel planes. Parallel circular sections are called bases of the zone. Distance between the bases is the altitude of the zone. May be thought of as the lateral area of a spherical segment.

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ZONE

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AREA OF A ZONE Z = 2πRh This formula applies to all three types of a zone. two bases one base where h < R one base where h > R

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TSA OF A SPHERICAL SEGMENT •For   one-base segment: TSA = Z + B = 2πRh + πr2 For two-base segment: TSA = Z + B1 + B2 = 2πRh + π + π

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EXAMPLES #9, p179: A plane cuts a sphere of radius 5 ft. at a distance of 3 ft. from the center of the sphere. Find the sum of the areas of the two zones formed. ANS: 100π ft2

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LUNE Portion of the surface of a sphere bounded by two intersecting arcs of great circles. Partial revolution of a semicircle about its diameter as an axis. The angle of the lune (θ) is the spherical angel formed by the semicircles bounding the lune. θ = 180° for a hemisphere. θ = 360° for the surface of a sphere.

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LUNE •CAD   is the angle of the lune ACBD.

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AREA OF LUNE L = 2R2θ

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EXAMPLES #10, p181: The area of a lune is 32π cm2 and the area of its sphere is 256π cm2. Find the angle of the lune in degrees. ANS: θ = π/4 = 45°

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SPHERICAL WEDGE A solid bounded by a lune and the planes of two great circles. “Slice” of the watermelon cut through its center. The lune (“skin” of the watermelon) may be considered as the base of the wedge.

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SPHERICAL WEDGE

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VOLUME OF A SPHERICAL WEDGE V = ⅓LR Since L = 2R2θ, the formula becomes V = ⅓(2R2θ)R. Thus, the volume of a spherical wedge is given by V = ⅔R3θ. Reference: Solid Mensuration: Understanding the 3-D Space by

EXAMPLES #11, p182: An angle of a lune base of a spherical wedge is 30° and the radius of the sphere is 9 m. Find the volume of the wedge. ANS: 81π m3

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7.2 EXERCISES #1, p183: Given a sphere of radius 16 in. Find the area of the zone of the sphere whose altitude is 10 in. ANS: 320π in2 #2, p183: Find the altitude of the zone whose area is 240π in2 if the radius of the corresponding sphere is 12 in.

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7.2 EXERCISES #4, p183: The volume of a spherical segment of a sphere of radius 16 in if the altitude of the segment is 12 in. #8, p183: A cylindrical post, 12 feet long and 5 inches in radius, is surmounted by a part of sphere 6 inches in radius. Find the surface area of the exposed part of the sphere and the volume of the whole post including the part of the sphere. Reference: Solid Mensuration: Understanding the 3-D Space by

7.2 EXERCISES #9, p183: A cutting plane divides the surface of a sphere into two zones with areas 24π in 2 and 72π in2. Find the area of the section formed. ANS: 18π in2 #11, p183: The radii of parallel circular sections of a sphere are 4 in and 4.8 in, while the radius of the sphere is 5 in. Find the volume of the portion of the sphere included between these sections. ANS: 314.4 in 3 Reference: Solid Mensuration: Understanding the 3-D Space by

7.2 EXERCISES #18, p184: The radius of a sphere is 30 cm while the volume of its spherical wedge is 3600π cm3. Find the area of the lune base and the angle of lune in degrees. #19, p184: The area of a lune with an angle of 12° is 120π cm2. Find the volume of the spherical wedge. ANS: 1200π cm3 ≈ 3770 cm3 Reference: Solid Mensuration: Understanding the 3-D Space by

7.2 EXERCISES #20, p184: The area of a spherical lune is equal to one-third the area of the corresponding sphere. Find the volume of the spherical wedge if the radius of the sphere is 15 cm.

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SPHERICAL SECTOR A solid generated by revolving a sector of a great circle about a diameter of that circle. If AOB revolves about the diameter CD, Spherical sector ABOFE will be formed. Zone ABFE is considered as base.

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SPHERICAL SECTOR

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EXAMPLES #12, p185: Find the volume of a spherical sector cut out of a sphere whose surface area is 400π in2 if the altitude of the zone of the base is 3 in. ANS: 200π in3

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SPHERICAL CONE A solid generated by revolving a sector of a great circle about a diameter of the circle which coincides with one of the radii of the sector. The sector AOC generates the spherical cone CAOB.

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SPHERICAL CONE

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EXAMPLES #13, p187: A sector of a great circle of a sphere is rotated about one of its radii. Find the volume of the solid generated if the central angle of the sector is 60° and the area of the sector is 24π cm2. ANS: 576π cm3

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7.3 EXERCISES #1, p200: Find the volume and total surface area of a spherical cone which consists of a right circular cone with a slant height of 15 inches and a spherical segment with a base radius of 9 inches. ANS: 450π in3, 225π in2 #2, p200: A circular sector with an angle of 60° and an area of 36π cm2 rotates about one of its radii. Find the volume of the solid generated. Reference: Solid Mensuration: Understanding the 3-D Space by

7.3 EXERCISES #3, p200: The zone altitude of a spherical cone is 6 ft, while the radius of the sphere is 10 ft. Find the surface area and the volume of the spherical cone. ANS: 665 ft 2, 400π ft3 #6, p201: Find the volume of a spherical cone if the radius of the circle corresponding segment is 12 cm and radius of the corresponding sphere is 15 cm. Reference: Solid Mensuration: Understanding the 3-D Space by

HOMEWORK 7 7.1 EXERCISES: #’s 11, 12, 15, 19, & 27 pp. 174-175 7.2 EXERCISES: #’s 3, 7, 10, 12, 15, & 16 pp. 183-184 7.3 EXERCISES: #’s 4, 5, & 7 pp. 200201 Reference: Solid Mensuration: Understanding the 3-D Space by