Geometry Chap 11 The Geometry of Three Dimensions

Geometry Chap 11 The Geometry of Three Dimensions Eleanor Roosevelt High School Chin-Sung Lin

ERHS Math Geometry The Geometry of Three Dimensions The geometry of three dimensions is called solid geometry Mr. Chin-Sung Lin

ERHS Math Geometry Points, Lines, and Planes Mr. Chin-Sung Lin

ERHS Math Geometry Postulates of the Solid Geometry There is one and only one plane containing three non-collinear points B C A Mr. Chin-Sung Lin

ERHS Math Geometry Postulates of the Solid Geometry A plane containing any two points contains all of the points on the line determined by those two points B m A Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of the Points, Lines & Planes There is exactly one plane containing a line and a point not on the line B P m A Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of the Points, Lines & Planes If two lines intersect, then there is exactly one plane containing them Two intersecting lines determine a plane B P m n A Mr. Chin-Sung Lin

ERHS Math Geometry Parallel Lines in Space Lines in the same plane that have no points in common Two lines are parallel if and only if they are coplanar and have no points in common m n Mr. Chin-Sung Lin

ERHS Math Geometry Skew Lines in Space Skew lines are lines in space that are neither parallel nor intersecting n m Mr. Chin-Sung Lin

ERHS Math Geometry Example Both intersecting lines and parallel lines lie in a plane Skew lines do not lie in a plane H D Identify the parallel lines, intercepting lines, and skew lines in the cube A G C E F B Mr. Chin-Sung Lin

ERHS Math Geometry Perpendicular Lines and Planes Mr. Chin-Sung Lin

ERHS Math Geometry Postulates of the Solid Geometry If two planes intersect, then they intersect in exactly one line B A Mr. Chin-Sung Lin

ERHS Math Geometry Dihedral Angle A dihedral angle is the union of two half-planes with a common edge Mr. Chin-Sung Lin

ERHS Math Geometry The Measure of a Dihedral Angle The measure of the plane angle formed by two rays each in a different half-plane of the angle and each perpendicular to the common edge at the same point of the edge AC � AB and AD � AB The measure of the dihedral angle: m�CAD C A B D Mr. Chin-Sung Lin

ERHS Math Geometry Perpendicular Planes Perpendicular planes are two planes that intersect to form a right dihedral angle AC � AB, AD � AB, and AC � AD (m�CAD = 90) then C m B m �n A D n Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Perpendicular Lines & Planes If a line not in a plane intersects the plane, then it intersects in exactly one point k B P n A Mr. Chin-Sung Lin

ERHS Math Geometry A Line is Perpendicular to a Plane A line is perpendicular to a plane if and only if it is perpendicular to each line in the plane through the intersection of the line and the plane A plane is perpendicular to a line if the line is perpendicular to the plane k n s k �m, and k �n, n m p then k �s Mr. Chin-Sung Lin

ERHS Math Geometry Postulates of the Solid Geometry At a given point on a line, there are infinitely many lines perpendicular to the given line m n p k q r A Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Perpendicular Lines & Planes If a line is perpendicular to each of two intersecting lines at their point of intersection, then the line is perpendicular to the plane determined by k these lines m A B P Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Perpendicular Lines & Planes Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k � AP and k � BP k Prove: k �m m A B P Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Perpendicular Lines & Planes Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k � AP and k � BP k R Prove: k �m m A Connect AB Q Connect PT and T P B intersects AB at Q Make PR = PS S Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Perpendicular Lines & Planes Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k � AP and k � BP k R Prove: k �m m A Connect RA, SA Q SAS T P B ΔRAP = ΔSAP S Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Perpendicular Lines & Planes Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k � AP and k � BP k R Prove: k �m m A CPCTC Q AR = AS T P B S Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Perpendicular Lines & Planes Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k � AP and k � BP k R Prove: k �m m A Connect RB, SB Q SAS T P B ΔRBP = ΔSBP S Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Perpendicular Lines & Planes Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k � AP and k � BP k R Prove: k �m m A CPCTC Q T BR = BS P B S Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Perpendicular Lines & Planes Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k � AP and k � BP k R Prove: k �m m A SSS Q ΔRAB = ΔSAB T P B S Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Perpendicular Lines & Planes Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k � AP and k � BP k R Prove: k �m m A CPCTC Q �RAB = �SAB T P B S Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Perpendicular Lines & Planes Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k � AP and k � BP k R Prove: k �m m A Connect RQ, SQ Q SAS T P B ΔRAQ = ΔSAQ S Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Perpendicular Lines & Planes Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k � AP and k � BP k R Prove: k �m m A CPCTC Q QR = QS T P B S Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Perpendicular Lines & Planes Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k � AP and k � BP k R Prove: k �m m A SSS Q ΔRPQ = ΔSPQ T P B S Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Perpendicular Lines & Planes Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k � AP and k � BP k R Prove: k �m m A CPCTC Q m�RPQ = m�SPQ T B m�RPQ + m�SPQ = 180 P m�RPQ = m�SPQ = 90 S Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Perpendicular Lines & Planes If two planes are perpendicular to each other, one plane contains a line perpendicular to the other plane Given: Plane p � plane q C Prove: A line in p is perpendicular to q p and a line in q is perpendicular to p q D A B Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Perpendicular Lines & Planes If a plane contains a line perpendicular to another plane, then the planes are perpendicular Given: AC in plane p and AC � q C Prove: p � q q D A p B Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Perpendicular Lines & Planes Two planes are perpendicular if and only if one plane contains a line perpendicular to the other C q D A p B Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Perpendicular Lines & Planes Through a given point on a plane, there is only one line perpendicular to the given plane Given: Plane p and AB �p at A Prove: AB is the only line perpendicular to p at A p B A Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Perpendicular Lines & Planes Through a given point on a plane, there is only one line perpendicular to the given plane Given: Plane p and AB �p at A Prove: AB is the only line perpendicular to p at A C p D B q A Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Perpendicular Lines & Planes Through a given point on a line, there can be only one plane perpendicular to the given line Given: Any point P on AB Prove: There is only one plane perpendicular to AB A P B Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Perpendicular Lines & Planes Through a given point on a line, there can be only one plane perpendicular to the given line A Given: Any point P on AB Prove: There is only one plane perpendicular to AB P Q m n R B Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Perpendicular Lines & Planes If a line is perpendicular to a plane, then any line perpendicular to the given line at its point of intersection with the given plane is in the plane B Given: AB �p at A and AB �AC q Prove: AC is in plane p p D A C Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Perpendicular Lines & Planes If a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane Given: Plane p with AB �p at A, and B C any point not on p Prove: Plane q determined by A, B, and C is perpendicular to p p q C A Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Perpendicular Lines & Planes If a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane Given: Plane p with AB �p at A, and B C any point not on p Prove: Plane q determined by A, B, and C is perpendicular to p p D A q C E Mr. Chin-Sung Lin

ERHS Math Geometry Parallel Lines and Planes Mr. Chin-Sung Lin

ERHS Math Geometry Parallel Planes Parallel planes are planes that have no points in common m n Mr. Chin-Sung Lin

ERHS Math Geometry A Line is Parallel to a Plane A line is parallel to a plane if it has no points in common with the plane k m Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Parallel Lines & Planes If a plane intersects two parallel planes, then the intersection is two parallel lines p m n Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Parallel Lines & Planes If a plane intersects two parallel planes, then the intersection is two parallel lines Given: Plane p intersects plane m at AB and plane n at CD, m//n Prove: AB//CD m n A C p B D Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Parallel Lines & Planes Two lines perpendicular to the same plane are parallel Given: Plane p, LA⊥p at A, and MB⊥p at B q Prove: LA//MB p L A M B Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Parallel Lines & Planes Two lines perpendicular to the same plane are parallel Given: Plane p, LA⊥p at A, and MB⊥p at B q Prove: LA//MB p N L A M B C D Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Parallel Lines & Planes Two lines perpendicular to the same plane are coplanar Given: Plane p, LA⊥p at A, and MB⊥p at B q Prove: LA and MB are coplanar p L A M B Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Parallel Lines & Planes If two planes are perpendicular to the same line, then they are parallel Given: Plane p⊥AB at A and q⊥AB at B Prove: p//q p q A B Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Parallel Lines & Planes If two planes are perpendicular to the same line, then they are parallel Given: Plane p⊥AB at A and q⊥AB at B Prove: p//q p q s A B R Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Parallel Lines & Planes If two planes are parallel, then a line perpendicular to one of the planes is perpendicular to the other Given: Plane p parallel to plane q, and AB⊥p and intersecting plane q at B p Prove: q⊥AB q A B Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Parallel Lines & Planes If two planes are parallel, then a line perpendicular to one of the planes is perpendicular to the other Given: Plane p parallel to plane q, and AB⊥p and intersecting plane q at B p Prove: q⊥AB q A C B E Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Parallel Lines & Planes If two planes are parallel, then a line perpendicular to one of the planes is perpendicular to the other Given: Plane p parallel to plane q, and AB⊥p and intersecting plane q at B p Prove: q⊥AB q A B C D E F Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Parallel Lines & Planes Two planes are perpendicular to the same line if and only if the planes are parallel p q A B Mr. Chin-Sung Lin

ERHS Math Geometry Distance between Two Planes The distance between two planes is the length of the line segment perpendicular to both planes with an endpoint on each plane p q A B Mr. Chin-Sung Lin

ERHS Math Geometry Theorems of Parallel Lines & Planes Parallel planes are everywhere equidistant Given: Parallel planes p and q, with AC and BD each perpendicular to p and q with an endpoint on each plane Prove: AC = BD p q A C B D Mr. Chin-Sung Lin

ERHS Math Geometry Surface Area of a Prism Mr. Chin-Sung Lin

ERHS Math Geometry Polyhedron A polyhedron is a three-dimensional figure formed by the union of the surfaces enclosed by plane figures A polyhedron is a figure that is the union of polygons Mr. Chin-Sung Lin

ERHS Math Geometry Polyhedron: Faces, Edges & Vertices Faces: the portions of the planes enclosed by a plane figure Edges: The intersections of the faces Vertices: the intersections of the edges Vertex Edge Face Mr. Chin-Sung Lin

ERHS Math Geometry Prism A prism is a polyhedron in which two of the faces, called the bases of the prism, are congruent polygons in parallel planes Mr. Chin-Sung Lin

ERHS Math Geometry Prism: Lateral Sides, Lateral Edges, Altitude & Height Lateral sides: the surfaces between corresponding sides of the bases Lateral edges: the common edges of the lateral sides Altitude: a line segment perpendicular to each of the bases with an endpoint on each base Base Height: the length of an altitude Lateral Side Lateral Edge Altitude/Height Mr. Chin-Sung Lin

ERHS Math Geometry Prism: Lateral Edges The lateral edges of a prism are congruent and parallel Lateral Edges Mr. Chin-Sung Lin

ERHS Math Geometry Right Prism A right prism is a prism in which the lateral sides are all perpendicular to the bases All of the lateral sides of a right prism are rectangles Lateral Sides Mr. Chin-Sung Lin

ERHS Math Geometry Parallelepiped A parallelepiped is a prism that has parallelograms as bases Mr. Chin-Sung Lin

ERHS Math Geometry Rectangular Parallelepiped A rectangular parallelepiped is a parallelepiped that has rectangular bases and lateral edges perpendicular to the bases Mr. Chin-Sung Lin

ERHS Math Geometry Rectangular Solid A rectangular parallelepiped is also called a rectangular solid, and it is the union of six rectangles. Any two parallel rectangles of a rectangular solid can be the bases Mr. Chin-Sung Lin

ERHS Math Geometry Area of a Prism The lateral area of the prism is the sum of the areas of the lateral faces The total surface area is the sum of the lateral area and the areas of the bases Mr. Chin-Sung Lin

ERHS Math Geometry Area of a Prism Example Calculate the lateral area of the prism Calculate the total surface area of the prism 4 7 5 Mr. Chin-Sung Lin

ERHS Math Geometry Area of a Prism Example Area of the bases: 7 x 5 x 2 = 70 Lateral area: 2 x (4 x 5 + 4 x 7) = 96 Total surface area: 70 + 96 = 166 4 7 5 Mr. Chin-Sung Lin

ERHS Math Geometry Area of a Prism Example The bases of a right prism are equilateral triangles Calculate the lateral area of the prism Calculate the total surface area of the prism 5 4 Mr. Chin-Sung Lin

ERHS Math Geometry Area of a Prism Example Area of the bases: ½ x (4 x 2√ 3) x 2= 8√ 3 Lateral area: 3 x (4 x 5) = 60 4 2√ 3 Total surface area: 60 + 8√ 3 ≈ 73. 86 5 2 4 Mr. Chin-Sung Lin

ERHS Math Geometry Volume of a Prism Mr. Chin-Sung Lin

ERHS Math Geometry Volume of a Prism The volume (V) of a prism is equal to the area of the base (B) times the height (h) V=Bxh Height (h) Base (B) Mr. Chin-Sung Lin

ERHS Math Geometry Volume of a Prism Example A right prism is shown in the diagram 2 Calculate the Volume of the prism 5 4 Mr. Chin-Sung Lin

ERHS Math Geometry Volume of a Prism Example A right prism is shown in the diagram 2 Calculate the Volume of the prism B=½x 4 x 2=4 h=5 V = Bh = 4 x 5 = 20 5 4 Mr. Chin-Sung Lin

ERHS Math Geometry Volume of a Prism Example A right prism is shown in the diagram Calculate the Volume of the prism 4 3 5 Mr. Chin-Sung Lin

ERHS Math Geometry Volume of a Prism Example A right prism is shown in the diagram Calculate the Volume of the prism 4 B = 5 x 4 = 20 h=3 3 V = Bh = 20 x 3 = 60 5 Mr. Chin-Sung Lin

ERHS Math Geometry Pyramids Mr. Chin-Sung Lin

ERHS Math Geometry Pyramids A pyramid is a solid figure with a base that is a polygon and lateral faces that are triangles Mr. Chin-Sung Lin

ERHS Math Geometry Pyramids: Vertex & Altitude Vertex: All lateral edges meet in a point Altitude: the perpendicular line segment from the vertex to thebase Vertex Altitude Mr. Chin-Sung Lin

ERHS Math Geometry Regular Pyramids Slant Height Altitude A pyramid whose base is a regular polygon and whose altitude is perpendicular to the base at its center The lateral edges of a regular polygon are congruent The lateral faces of a regular pyramid are isosceles triangles The length of the altitude of a triangular lateral face is the slant height of the pyramid Mr. Chin-Sung Lin

ERHS Math Geometry Surface Area of Pyramids. Slant Height The lateral area of a pyramid is the sum of the areas of the faces (isosceles triangles) The total surface area is the lateral area plus the area of the base Mr. Chin-Sung Lin

ERHS Math Geometry Volume of Pyramids The volume (V) of a pyramid is equal to one third of the area of the base (B) times the height (h) Height V = (1/3) x B x h Base Area Mr. Chin-Sung Lin

ERHS Math Geometry Volume of Pyramids Example A regular pyramid has a square base. The length of an edge of the base is 10 centimeters and the length of the altitude to the base of each lateral side is 13 centimeters a. What is the total surface area of the pyramid? b. What is the volume of the pyramid? 13 10 Mr. Chin-Sung Lin

ERHS Math Geometry Volume of Pyramids Example A regular pyramid has a square base. The length of an edge of the base is 10 centimeters and the length of the altitude to the base of each lateral side is 13 centimeters a. What is the total surface area of the pyramid? b. What is the volume of the pyramid? 13 10 Mr. Chin-Sung Lin

ERHS Math Geometry Volume of Pyramids Example A regular pyramid has a square base. The 12 length of an edge of the base is 10 centimeters and the length of the altitude to the base of each lateral side is 13 centimeters a. What is the total surface area of the pyramid? b. What is the volume of the pyramid? 13 5 10 Mr. Chin-Sung Lin

ERHS Math Geometry Volume of Pyramids Example a. Total surface area: Lateral Area: ½ x 10 x 13 x 4 = 260 12 13 Base Area: 10 x 10 = 100 Total Area = 260 + 100 = 360 cm 2 b. Volume: B = 100 h = 12 V = (1/3) x 100 x 12 = 400 cm 3 5 10 Mr. Chin-Sung Lin

ERHS Math Geometry Properties of Regular Pyramids The base of a regular pyramid is a regular polygon and the altitude is perpendicular to the base at its center The center of a regular polygon is defined as the point that is equidistant to its vertices The lateral faces of a regular pyramid are isosceles triangles The lateral faces of a regular pyramid are congruent Mr. Chin-Sung Lin

ERHS Math Geometry Cylinders Mr. Chin-Sung Lin

ERHS Math Geometry Cylinders The solid figure formed by the congruent parallel curves and the surface that joins them is called a cylinder Mr. Chin-Sung Lin

ERHS Math Geometry Cylinders Altitude Bases: the closed curves Lateral surface: the surface that joins the bases Altitude: a line segment perpendicular to the bases with endpoints on the bases Lateral Bases Surface Height: the length of an altitude Mr. Chin-Sung Lin

ERHS Math Geometry Circular Cylinders A cylinder whose bases are congruent circles Mr. Chin-Sung Lin

ERHS Math Geometry Right Circular Cylinders If the line segment joining the centers of the circular bases is perpendicular to the bases, the cylinder is a right circular cylinder Mr. Chin-Sung Lin

ERHS Math Geometry Surface Area of Right Circular Cylinders Base Area: 2πr 2 r Lateral Area: 2πrh Total Surface Area: 2πrh + 2πr 2 h Mr. Chin-Sung Lin

ERHS Math Geometry Volume of Circular Cylinders Volume: B x h = πr 2 h Mr. Chin-Sung Lin

ERHS Math Geometry Right Circular Cylinders Example A right cylinder as shown in the diagram. 6 Calculate the total Surface Area Calculate the volume 14 Mr. Chin-Sung Lin

ERHS Math Geometry Right Circular Cylinders Example Base Area: 6 2πr 2 = 2π62 ≈ 226. 19 Lateral Area: 2πrh = 2π (6)(14) ≈ 527. 79 14 Total Surface Area: 226. 19 + 527. 79 = 754. 58 Volume: B x h = πr 2 h = π(62)(14) = 1583. 36 Mr. Chin-Sung Lin

ERHS Math Geometry Cones Mr. Chin-Sung Lin

ERHS Math Geometry Right Circular Conical Surface Line OQ is perpendicular to plane p at O, and a point P is on plane p Keeping point Q fixed, move P through a circle on p with center at O. The surface generated by PQ is a right QA circular conical surface * A conical surface extends infinitely p O P C Mr. Chin-Sung Lin

A ERHS Math Geometry Right Circular Cone The part of the conical surface generated by PQ from plane p to Q is called a right circular cone QA Q: vertex of the cone Circle O: base of the cone OQ: altitude of the cone OQ: height of the cone, and p O P C PQ: slant height of the cone Mr. Chin-Sung Lin

A ERHS Math Geometry Surface Area of a Cone Base Area: B = πr 2 Lateral Area: L = ½ Chs= ½ (2πr)hs = πrhs A Total Surface Area: πrhs + πr 2 * hs: * h c: * r: * B: * C: slant height radius base area circumference hs hc p B r C C Mr. Chin-Sung Lin

A ERHS Math Geometry Volume of a Cone Base Area: B = πr 2 A Volume: V = ⅓ Bhc= ⅓ πr 2 hc * hs: * h c: * r: * B: * C: slant height radius base area circumference hs hc p B r C C Mr. Chin-Sung Lin

A ERHS Math Geometry Surface Area of a Cone Example Calculate the base area, lateral area, and total area A 24 26 10 p C Mr. Chin-Sung Lin

A ERHS Math Geometry Surface Area of a Cone Example Calculate the base area, lateral area, and total area A Base Area: B = π(10)2 = 100π 24 Lateral Area: L = π(10)(26) = 260π Total Surface Area: 100π + 260π = 360π 26 10 p C Mr. Chin-Sung Lin

A ERHS Math Geometry Volume of a Cone Example r A cone and a cylinder have equal volumes h and equal heights. If the radius of the base of the cone is 3 centimeters, what is the radius of the base of the cylinder? A Volume of Cylinder: V = h = πr 2 h h Volume of Cone: V = ⅓ π32 h = 3πh πr 2 h = 3πh, r 2 = 3, r = √ 3 cm p C Mr. Chin-Sung Lin

ERHS Math Geometry Spheres Mr. Chin-Sung Lin

ERHS Math Geometry Spheres A sphere is the set of all points equidistant from a fixed point called the center The radius of a sphere is the length of the line segment from the center of the sphere to any point on the sphere O r Mr. Chin-Sung Lin

ERHS Math Geometry Sphere and Plane If the distance of a plane from the center of a sphere is d and the radius of the sphere is r r < d no points in common r = d one points in common O r p r d P p O P r > d infinite points in common (circle) d r. O p d P Mr. Chin-Sung Lin

ERHS Math Geometry Circles A circle is the set of all points in a plane equidistant from a fixed point in the plane called the center p O r Mr. Chin-Sung Lin

ERHS Math Geometry Theorem about Circles The intersection of a sphere and a plane through the center of the sphere is a circle whose radius is equal to the radius of the sphere O p r r Mr. Chin-Sung Lin

ERHS Math Geometry Great Circle of a Sphere A great circle of a sphere is the intersection of a sphere and a plane through the center of the sphere O p r r Mr. Chin-Sung Lin

ERHS Math Geometry Theorem about Circles If the intersection of a sphere and a plane does not contain the center of the sphere, then the intersection is a circle Given: A sphere with center at O plane p intersecting the sphere at A and B O Prove: The intersection is a circle p A C B Mr. Chin-Sung Lin

ERHS Math Geometry Theorem about Circles If the intersection of a sphere and a plane does not contain the center of the sphere, then the intersection is a circle Given: A sphere with center at O plane p intersecting the sphere at A and B O Prove: The intersection is a circle p A C r B Mr. Chin-Sung Lin

ERHS Math Geometry Theorem about Circles O A p Statements C r B Reasons 1. Draw a line OC, point C on plane p 1. Given, create two triangles 2. 3. 4. 5. 6. OC AC, OC BC �OCA and �OCB are right angles OA OB OC OAC OBC CA CB 7. The intersection is a circle Definition of perpendicular Radius of a sphere Reflexive postulate HL postulate CPCTC 7. Definition of circles Mr. Chin-Sung Lin

ERHS Math Geometry Theorem about Circles The intersection of a plane and a sphere is a circle A great circle is the largest circle that can be drawn on a sphere O p’ p Mr. Chin-Sung Lin

ERHS Math Geometry Theorem about Circles If two planes are equidistant from the center of a sphere and intersect the sphere, then the intersections are congruent circles C p q A O D B Mr. Chin-Sung Lin

ERHS Math Geometry A Surface Area of a Sphere Surface Area: S = 4πr 2 r: radius O r Mr. Chin-Sung Lin

ERHS Math Geometry A Volume of a Sphere Volume: V = 4/3 πr 3 r: radius O r Mr. Chin-Sung Lin

ERHS Math Geometry A Sphere Example Find the surface area and the volume of a sphere whose radius is 6 cm O r Mr. Chin-Sung Lin

A ERHS Math Geometry Sphere Example Find the surface area and the volume of a sphere whose radius is 6 cm Surface Area: S = 4π62 = 144π cm 2 O r Volume: V = 4/3 π63 = 288π cm 3 Mr. Chin-Sung Lin

ERHS Math Geometry Q&A Mr. Chin-Sung Lin

ERHS Math Geometry The End Mr. Chin-Sung Lin