Eleanor Roosevelt High School Chin-Sung Lin

The geometry of three dimensions is called The geometry of three dimensions is called

solid geometrysolid geometry

Mr. Chin-Sung Lin

ERHS Math Geometry

Mr. Chin-Sung Lin

ERHS Math Geometry

There is one and only one plane containing There is one and only one plane containing three non-collinear pointsthree non-collinear points

Mr. Chin-Sung Lin

ERHS Math Geometry

A

A plane containing any A plane containing any two points two points contains contains all of the points on the line determined by all of the points on the line determined by those two pointsthose two points

Mr. Chin-Sung Lin

ERHS Math Geometry

A

There is exactly one plane containing a line There is exactly one plane containing a line and a point not on the lineand a point not on the line

Mr. Chin-Sung Lin

ERHS Math Geometry

A

If two lines intersect, then there is exactly If two lines intersect, then there is exactly one plane containing themone plane containing them

Two Two intersecting lines intersecting lines determine a planedetermine a plane

Mr. Chin-Sung Lin

ERHS Math Geometry

A

Lines in the same Lines in the same planeplane that have that have no points no points in commonin common

Two lines are parallel if and only if they are Two lines are parallel if and only if they are coplanarcoplanar and have and have no points in no points in commoncommon

Mr. Chin-Sung Lin

ERHS Math Geometry

Skew lines are lines in space that are Skew lines are lines in space that are neither neither parallel nor intersectingparallel nor intersecting

Mr. Chin-Sung Lin

ERHS Math Geometry

Both intersecting lines and parallel lines lie in a Both intersecting lines and parallel lines lie in a planeplane

Skew lines do not lie in a planeSkew lines do not lie in a plane

Identify the parallel lines, Identify the parallel lines,

intercepting lines, and skew linesintercepting lines, and skew lines

in the cubein the cube

Mr. Chin-Sung Lin

ERHS Math Geometry

A

E

Mr. Chin-Sung Lin

ERHS Math Geometry

If two planes intersect, then they intersect in If two planes intersect, then they intersect in exactly exactly one lineone line

Mr. Chin-Sung Lin

ERHS Math Geometry

A

A A dihedral angle dihedral angle is the union of two is the union of two half-half-planes planes with with a common edgea common edge

Mr. Chin-Sung Lin

ERHS Math Geometry

The measure of the plane angle formed by two rays The measure of the plane angle formed by two rays each in a different half-plane of the angle and each in a different half-plane of the angle and each perpendicular to the common edge at the each perpendicular to the common edge at the same point of the edgesame point of the edge

AC AB and AD AB AC AB and AD AB

The measure of the dihedral angle:The measure of the dihedral angle:

mCAD mCAD

Mr. Chin-Sung Lin

ERHS Math Geometry

A

Perpendicular planes are two planes that intersect to Perpendicular planes are two planes that intersect to form a form a right dihedral angleright dihedral angle

AC ABAC AB,, AD AB AD AB, and

AC AD AC AD (mCAD = 90mCAD = 90)then

m nm n

Mr. Chin-Sung Lin

ERHS Math Geometry

A

If a line not in a plane intersects the plane, If a line not in a plane intersects the plane, then it intersects in exactly then it intersects in exactly one pointone point

Mr. Chin-Sung Lin

ERHS Math Geometry

A

A line is perpendicular to a plane if and only if it is A line is perpendicular to a plane if and only if it is perpendicular to each line in the plane perpendicular to each line in the plane through the intersection of the line and the through the intersection of the line and the planeplane

A plane is perpendicular to a line if the line is A plane is perpendicular to a line if the line is perpendicular to the planeperpendicular to the plane

k mk m, , and k n k n,

then k sk s

Mr. Chin-Sung Lin

ERHS Math Geometry

At a given point on a line, there are At a given point on a line, there are infinitely infinitely many linesmany lines perpendicular to the given perpendicular to the given lineline

Mr. Chin-Sung Lin

ERHS Math Geometry

If a line is perpendicular to each of two If a line is perpendicular to each of two intersecting lines at their point of intersection, intersecting lines at their point of intersection, then the line is perpendicular to the plane then the line is perpendicular to the plane determined by these linesdetermined by these lines

Mr. Chin-Sung Lin

ERHS Math Geometry

Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP

Prove: k mk m

Mr. Chin-Sung Lin

ERHS Math Geometry

Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP

Prove: k mk m

Connect ABConnect AB

Connect PT and intersects AB at QMake PR = PS

Mr. Chin-Sung Lin

ERHS Math Geometry

Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP

Prove: k mk m

Connect RA, SAConnect RA, SA

SASΔRAP = ΔSAP

Mr. Chin-Sung Lin

ERHS Math Geometry

Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP

Prove: k mk m

CPCTC

AR = AS

Mr. Chin-Sung Lin

ERHS Math Geometry

Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP

Prove: k mk m

Connect RB, SBConnect RB, SB

SASΔRBP = ΔSBP

Mr. Chin-Sung Lin

ERHS Math Geometry

Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP

Prove: k mk m

CPCTC

BR = BS

Mr. Chin-Sung Lin

ERHS Math Geometry

Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP

Prove: k mk m

SSS

ΔRAB = ΔSAB

Mr. Chin-Sung Lin

ERHS Math Geometry

Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP

Prove: k mk m

CPCTC

RAB = SAB

Mr. Chin-Sung Lin

ERHS Math Geometry

Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP

Prove: k mk m

Connect RQ, SQConnect RQ, SQ

SASΔRAQ = ΔSAQ

Mr. Chin-Sung Lin

ERHS Math Geometry

Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP

Prove: k mk m

CPCTC

QR = QS

Mr. Chin-Sung Lin

ERHS Math Geometry

Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP

Prove: k mk mSSSΔRPQ = ΔSPQ

Mr. Chin-Sung Lin

ERHS Math Geometry

Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP

Prove: k mk mCPCTCmRPQ = mSPQmRPQ + mSPQ = 180mRPQ = mSPQ = 90

Mr. Chin-Sung Lin

ERHS Math Geometry

If two planes are perpendicular to each other, one plane contains a line perpendicular to the other plane

Given: Plane p plane q

Prove: A line in p is perpendicular to q

and a line in q is perpendicular to p

Mr. Chin-Sung Lin

ERHS Math Geometry

A

If a plane contains a line perpendicular to another plane, then the planes are perpendicular

Given: AC in plane p and AC q

Prove: p q

Mr. Chin-Sung Lin

ERHS Math Geometry

A

Two planes are perpendicular if and only if one plane contains a line perpendicular to the other

Mr. Chin-Sung Lin

ERHS Math Geometry

A

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

Mr. Chin-Sung Lin

ERHS Math Geometry

A

Mr. Chin-Sung Lin

ERHS Math Geometry

A

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

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

Mr. Chin-Sung Lin

ERHS Math Geometry

P

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

Mr. Chin-Sung Lin

ERHS Math Geometry

P

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

Given: AB p at A and AB AC

Prove: AC is in plane p

Mr. Chin-Sung Lin

ERHS Math Geometry

A

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

C any point not on p

Prove: Plane q determined by A, B, and C

is perpendicular to p

Mr. Chin-Sung Lin

ERHS Math Geometry

A

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

C any point not on p

Prove: Plane q determined by A, B, and C

is perpendicular to p

Mr. Chin-Sung Lin

ERHS Math Geometry

A

Mr. Chin-Sung Lin

ERHS Math Geometry

Parallel planes are planes that have no points Parallel planes are planes that have no points in commonin common

Mr. Chin-Sung Lin

ERHS Math Geometry

m

n

A line is parallel to a plane if it has no points A line is parallel to a plane if it has no points in common with the planein common with the plane

Mr. Chin-Sung Lin

ERHS Math Geometry

m

If a plane intersects two parallel planes, then the If a plane intersects two parallel planes, then the intersection is two parallel linesintersection is two parallel lines

Mr. Chin-Sung Lin

ERHS Math Geometry

n

m

p

If a plane intersects two parallel planes, then the If a plane intersects two parallel planes, then the intersection is two parallel linesintersection is two parallel lines

Given: Given: Plane p intersects plane m at AB Plane p intersects plane m at AB

and plane n at CD, m//nand plane n at CD, m//n

Prove: AB//CDProve: AB//CD

Mr. Chin-Sung Lin

ERHS Math Geometry

n

m

p

Two lines perpendicular to the same plane are parallelTwo lines perpendicular to the same plane are parallel

Given: Given: Plane p, LA⊥p at A, and MB⊥p at B Plane p, LA⊥p at A, and MB⊥p at B

Prove: LA//MBProve: LA//MB

Mr. Chin-Sung Lin

ERHS Math Geometry

p

q

Two lines perpendicular to the same plane are parallelTwo lines perpendicular to the same plane are parallel

Given: Given: Plane p, LA⊥p at A, and MB⊥p at B Plane p, LA⊥p at A, and MB⊥p at B

Prove: LA//MBProve: LA//MB

Mr. Chin-Sung Lin

ERHS Math Geometry

p

q

Two lines perpendicular to the same plane are Two lines perpendicular to the same plane are coplanarcoplanar

Given: Given: Plane p, LA⊥p at A, and MB⊥p at B Plane p, LA⊥p at A, and MB⊥p at B

Prove: LA and MB are coplanarProve: LA and MB are coplanar

Mr. Chin-Sung Lin

ERHS Math Geometry

p

q

If two planes are perpendicular to the same line, then If two planes are perpendicular to the same line, then they are parallelthey are parallel

Given: Given: Plane p⊥AB at A and q⊥AB at B Plane p⊥AB at A and q⊥AB at B

Prove: p//qProve: p//q

Mr. Chin-Sung Lin

ERHS Math Geometry

q

p

If two planes are perpendicular to the same line, then If two planes are perpendicular to the same line, then they are parallelthey are parallel

Given: Given: Plane p⊥AB at A and q⊥AB at B Plane p⊥AB at A and q⊥AB at B

Prove: p//qProve: p//q

Mr. Chin-Sung Lin

ERHS Math Geometry

q

p

s

If two planes are parallel, then a line perpendicular to If two planes are parallel, then a line perpendicular to one of the planes is perpendicular to the otherone of the planes is perpendicular to the other

Given: Given: Plane p parallel to plane q, and Plane p parallel to plane q, and

AB⊥p and intersectingAB⊥p and intersecting

plane q at Bplane q at B

Prove: Prove: q⊥AB q⊥AB

Mr. Chin-Sung Lin

ERHS Math Geometry

q

p

If two planes are parallel, then a line perpendicular to If two planes are parallel, then a line perpendicular to one of the planes is perpendicular to the otherone of the planes is perpendicular to the other

Given: Given: Plane p parallel to plane q, and Plane p parallel to plane q, and

AB⊥p and intersectingAB⊥p and intersecting

plane q at Bplane q at B

Prove: Prove: q⊥AB q⊥AB

Mr. Chin-Sung Lin

ERHS Math Geometry

q

p

If two planes are parallel, then a line perpendicular to If two planes are parallel, then a line perpendicular to one of the planes is perpendicular to the otherone of the planes is perpendicular to the other

Given: Given: Plane p parallel to plane q, and Plane p parallel to plane q, and

AB⊥p and intersectingAB⊥p and intersecting

plane q at Bplane q at B

Prove: Prove: q⊥AB q⊥AB

Mr. Chin-Sung Lin

ERHS Math Geometry

q

p

Two planes are perpendicular to the same line if and Two planes are perpendicular to the same line if and only if the planes are parallelonly if the planes are parallel

Mr. Chin-Sung Lin

ERHS Math Geometry

q

p

The distance between two planes is the length of the The distance between two planes is the length of the line segment perpendicular to both planes with an line segment perpendicular to both planes with an endpoint on each planeendpoint on each plane

Mr. Chin-Sung Lin

ERHS Math Geometry

q

p

Parallel planes are everywhere equidistantParallel planes are everywhere equidistant

Given: Given: Parallel planes p and q, Parallel planes p and q,

with AC and BD each with AC and BD each

perpendicular to p and q perpendicular to p and q

with an endpoint on each with an endpoint on each

planeplane

Prove: Prove: AC = BDAC = BD

Mr. Chin-Sung Lin

ERHS Math Geometry

q

p

Mr. Chin-Sung Lin

ERHS Math Geometry

A polyhedron is a three-dimensional figure formed by A polyhedron is a three-dimensional figure formed by the union of the surfaces enclosed by plane the union of the surfaces enclosed by plane figuresfigures

A polyhedron is a figure that is the union of polygonsA polyhedron is a figure that is the union of polygons

Mr. Chin-Sung Lin

ERHS Math Geometry

Faces: the portions of the planes enclosed by a plane Faces: the portions of the planes enclosed by a plane figurefigure

Edges: The intersections of the facesEdges: The intersections of the faces

Vertices: the intersections of the edgesVertices: the intersections of the edges

Mr. Chin-Sung Lin

ERHS Math Geometry

Vertex

Edge

Face

A prism is a polyhedron in which two of the faces, A prism is a polyhedron in which two of the faces, called the bases of the prism, are congruent called the bases of the prism, are congruent polygons in parallel planespolygons in parallel planes

Mr. Chin-Sung Lin

ERHS Math Geometry

Lateral sides: the surfaces between corresponding sides of the Lateral sides: the surfaces between corresponding sides of the basesbases

Lateral edges: the common edges of the lateral sidesLateral edges: the common edges of the lateral sides

Altitude: a line segment perpendicular to each of the bases with an Altitude: a line segment perpendicular to each of the bases with an endpoint on each baseendpoint on each base

Height: the length of an altitudeHeight: the length of an altitude

Mr. Chin-Sung Lin

ERHS Math Geometry

Lateral Side

Lateral Edge

Altitude/Height

Base

The lateral edges of a prism are congruent and parallelThe lateral edges of a prism are congruent and parallel

Mr. Chin-Sung Lin

ERHS Math Geometry

Lateral Edges

A right prism is a prism in which the lateral sides are all A right prism is a prism in which the lateral sides are all perpendicular to the basesperpendicular to the bases

All of the lateral sides of a right prism are rectanglesAll of the lateral sides of a right prism are rectangles

Mr. Chin-Sung Lin

ERHS Math Geometry

Lateral Sides

A A parallelepipedparallelepiped is a prism that has is a prism that has parallelogramsparallelograms as basesas bases

Mr. Chin-Sung Lin

ERHS Math Geometry

A A rectangular parallelepiped rectangular parallelepiped is a parallelepiped that has is a parallelepiped that has rectangular bases rectangular bases and lateral edges and lateral edges perpendicularperpendicular to to the basesthe bases

Mr. Chin-Sung Lin

ERHS Math Geometry

A rectangular parallelepiped is also called a A rectangular parallelepiped is also called a rectangular solidrectangular solid, and it is the union of , and it is the union of six six rectanglesrectangles. Any two parallel rectangles of a . Any two parallel rectangles of a rectangular solid can be the basesrectangular solid can be the bases

Mr. Chin-Sung Lin

ERHS Math Geometry

The The lateral area lateral area of the prism is the sum of the areas of of the prism is the sum of the areas of the the lateral faceslateral faces

The The total surface area total surface area is the sum of the is the sum of the lateral area lateral area and the and the areas of the basesareas of the bases

Mr. Chin-Sung Lin

ERHS Math Geometry

Calculate the Calculate the lateral area lateral area of the prismof the prism

Calculate the Calculate the total surface area total surface area of the prismof the prism

Mr. Chin-Sung Lin

ERHS Math Geometry

4

7

5

Area of the bases:Area of the bases: 7 x 5 x 2 = 707 x 5 x 2 = 70

Lateral area:Lateral area: 2 x (4 x 5 + 4 x 7) = 962 x (4 x 5 + 4 x 7) = 96

Total surface area:Total surface area: 70 + 96 = 16670 + 96 = 166

Mr. Chin-Sung Lin

ERHS Math Geometry

4

75

The bases of a right prism are equilateral triangles The bases of a right prism are equilateral triangles

Calculate the Calculate the lateral area lateral area of the prismof the prism

Calculate the Calculate the total surface area total surface area of the prismof the prism

Mr. Chin-Sung Lin

ERHS Math Geometry

5

4

Area of the bases:Area of the bases: ½ x (4 x 2√3) x 2= 8√3½ x (4 x 2√3) x 2= 8√3

Lateral area:Lateral area: 3 x (4 x 5) = 603 x (4 x 5) = 60

Total surface area:Total surface area: 60 + 8√3 ≈ 73.8660 + 8√3 ≈ 73.86

Mr. Chin-Sung Lin

ERHS Math Geometry

5

4

2

2√34

Mr. Chin-Sung Lin

ERHS Math Geometry

The The volume (V)volume (V) of a prism is equal to the of a prism is equal to the area of the area of the base (B) times the height (h)base (B) times the height (h)

V = B x hV = B x h

Mr. Chin-Sung Lin

ERHS Math Geometry

Base (B)

Height (h)

A right prism is shown in the diagramA right prism is shown in the diagram

Calculate the Calculate the Volume Volume of the prismof the prism

Mr. Chin-Sung Lin

ERHS Math Geometry

5

4

2

A right prism is shown in the diagramA right prism is shown in the diagram

Calculate the Calculate the Volume Volume of the prismof the prism

B = ½ x 4 x 2 = 4

h = 5

V = Bh = 4 x 5 = 20

Mr. Chin-Sung Lin

ERHS Math Geometry

5

4

2

A right prism is shown in the diagramA right prism is shown in the diagram

Calculate the Calculate the Volume Volume of the prismof the prism

Mr. Chin-Sung Lin

ERHS Math Geometry

3

5

4

A right prism is shown in the diagramA right prism is shown in the diagram

Calculate the Calculate the Volume Volume of the prismof the prism

B = 5 x 4 = 20

h = 3

V = Bh = 20 x 3 = 60

Mr. Chin-Sung Lin

ERHS Math Geometry

3

5

4

Mr. Chin-Sung Lin

ERHS Math Geometry

A pyramid is a solid figure with a base that is a A pyramid is a solid figure with a base that is a polygon and lateral faces that are trianglespolygon and lateral faces that are triangles

Mr. Chin-Sung Lin

ERHS Math Geometry

Vertex: All lateral edges meet in a pointVertex: All lateral edges meet in a point

Altitude: the perpendicular line segment from the Altitude: the perpendicular line segment from the vertex to thebasevertex to thebase

Mr. Chin-Sung Lin

ERHS Math Geometry

Vertex

Altitude

Vertex

Altitude

A pyramid whose A pyramid whose base is a is a regular polygon regular polygon and whose and whose altitudealtitude is perpendicular to is perpendicular to the base at its the base at its centercenter

The The lateral edges lateral edges of a regular polygon are of a regular polygon are congruentcongruent

The The lateral faces lateral faces of a regular pyramid are of a regular pyramid are isosceles trianglesisosceles triangles

The The length of the altitude length of the altitude of a triangular of a triangular lateral facelateral face is the is the slant height slant height of the of the pyramidpyramid

Mr. Chin-Sung Lin

ERHS Math Geometry

Slant HeightAltitud

e

The lateral area of a pyramid is the sum of The lateral area of a pyramid is the sum of the areas of the faces (isosceles the areas of the faces (isosceles triangles)triangles)

The total surface area is the lateral area plus The total surface area is the lateral area plus the area of the basethe area of the base

Mr. Chin-Sung Lin

ERHS Math Geometry

Slant Height

The The volume (V)volume (V) of a pyramid is equal to of a pyramid is equal to one third of the of the area of the base (B) times area of the base (B) times the height (h)the height (h)

V = (1/3) x B x hV = (1/3) x B x h

Mr. Chin-Sung Lin

ERHS Math Geometry

Base Area

Height

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?

Mr. Chin-Sung Lin

ERHS Math Geometry

13

10

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?

Mr. Chin-Sung Lin

ERHS Math Geometry

13

10

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?

Mr. Chin-Sung Lin

ERHS Math Geometry

13

10

5

12

a. Total surface area:

Lateral Area: ½ x 10 x 13 x 4 = 260

Base Area: 10 x 10 = 100

Total Area = 260 + 100 = 360 cm2

b. Volume:

B = 100

h = 12

V = (1/3) x 100 x 12 = 400 cm3

Mr. Chin-Sung Lin

ERHS Math Geometry

13

10

5

12

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

Mr. Chin-Sung Lin

ERHS Math Geometry

The solid figure formed by the The solid figure formed by the congruent parallel congruent parallel curves curves and the and the surface surface that joins them is called a that joins them is called a cylindercylinder

Mr. Chin-Sung Lin

ERHS Math Geometry

Bases: the closed curvesBases: the closed curves

Lateral surface: the surface that joins Lateral surface: the surface that joins the basesthe bases

Altitude: a line segment perpendicular Altitude: a line segment perpendicular to the bases with endpoints on the to the bases with endpoints on the basesbases

Height: the length of an altitudeHeight: the length of an altitude

Mr. Chin-Sung Lin

ERHS Math Geometry

BasesLateral Surface

Altitude

A cylinder whose bases are A cylinder whose bases are congruent circles

Mr. Chin-Sung Lin

ERHS Math Geometry

If the line segment joining the centers of the circular If the line segment joining the centers of the circular bases is bases is perpendicularperpendicular to the bases, the cylinder to the bases, the cylinder is a is a right circular cylinder

Mr. Chin-Sung Lin

ERHS Math Geometry

Base Area: 2πrBase Area: 2πr22

Lateral Area: 2πrh Lateral Area: 2πrh

Total Surface Area: 2πrh + 2πrTotal Surface Area: 2πrh + 2πr22

Mr. Chin-Sung Lin

ERHS Math Geometry

r

h

Volume: B x h = πrVolume: B x h = πr22hh

Mr. Chin-Sung Lin

ERHS Math Geometry

A right cylinder as shown in the diagram.A right cylinder as shown in the diagram.

Calculate the total Surface Area Calculate the total Surface Area

Calculate the volumeCalculate the volume

Mr. Chin-Sung Lin

ERHS Math Geometry

6

14

Base Area: Base Area:

2πr2πr22 = 2π6 = 2π622 ≈ 226.19 ≈ 226.19

Lateral Area: Lateral Area:

2πrh = 2π (6)(14) ≈ 527.792πrh = 2π (6)(14) ≈ 527.79

Total Surface Area: Total Surface Area:

226.19 + 527.79 = 754.58226.19 + 527.79 = 754.58

Volume:Volume:

B x h = πrB x h = πr22h = π(6h = π(62)2)(14) = 1583.36(14) = 1583.36

Mr. Chin-Sung Lin

ERHS Math Geometry

6

14

Mr. Chin-Sung Lin

ERHS Math Geometry

Line OQ is perpendicular to plane p at Line OQ is perpendicular to plane p at O, and a point P is on plane pO, and a point P is on plane p

Keeping point Q fixed, move P through Keeping point Q fixed, move P through a circle on p with center at O. The a circle on p with center at O. The surface generated by PQ is a surface generated by PQ is a right right circular conical surfacecircular conical surface

* A conical surface extends infinitely* A conical surface extends infinitely

Mr. Chin-Sung Lin

ERHS Math Geometry

A

CP

O

Q

p

The part of the conical surface The part of the conical surface generated by PQ from plane p to Q generated by PQ from plane p to Q is called a is called a right circular coneright circular cone

Q:Q: vertex of the conevertex of the cone

Circle O: base of the coneCircle O: base of the cone

OQ: altitude of the coneOQ: altitude of the cone

OQ: height of the cone, and OQ: height of the cone, and

PQ: slant height of the conePQ: slant height of the cone

Mr. Chin-Sung Lin

ERHS Math GeometryA

A

CP

O

Q

p

Base Area: B = πrBase Area: B = πr22

Lateral Area: L = ½ ChLateral Area: L = ½ Chss= ½ (2πr)h= ½ (2πr)hss = πrh = πrhss

Total Surface Area: πrhTotal Surface Area: πrhss + πr + πr22

* h* hss:: slant heightslant height

* h* hcc:: heightheight

* r:* r: radiusradius

* B:* B: base areabase area

* C:* C: circumferencecircumference

Mr. Chin-Sung Lin

ERHS Math GeometryA

A

C

hs

C

r

p

hc

B

Base Area: B = πrBase Area: B = πr22

Volume: V = ⅓ BhVolume: V = ⅓ Bhcc= ⅓ πr= ⅓ πr22hhcc

* h* hss:: slant heightslant height

* h* hcc:: heightheight

* r:* r: radiusradius

* B:* B: base areabase area

* C:* C: circumferencecircumference

Mr. Chin-Sung Lin

ERHS Math GeometryA

A

C

hs

B r

p

hc

C

Calculate the base area, lateral area, and Calculate the base area, lateral area, and total areatotal area

Mr. Chin-Sung Lin

ERHS Math GeometryA

A

C

26

10

p

24

Calculate the base area, lateral area, and Calculate the base area, lateral area, and total areatotal area

Base Area: B = π(10)Base Area: B = π(10)22 = 100π = 100π

Lateral Area: L = π(10)(26) = 260π Lateral Area: L = π(10)(26) = 260π

Total Surface Area: 100π + 260π Total Surface Area: 100π + 260π

= 360π = 360π

Mr. Chin-Sung Lin

ERHS Math GeometryA

A

C

26

10

p

24

A cone and a cylinder have equal volumes A cone and a cylinder have equal volumes and equal heights. If the radius of the and equal heights. If the radius of the base of the cone is 3 centimeters, what base of the cone is 3 centimeters, what is the radius of the base of the is the radius of the base of the cylinder?cylinder?

Volume of Cylinder: V = h = πrVolume of Cylinder: V = h = πr22hh

Volume of Cone: V = ⅓ π3Volume of Cone: V = ⅓ π322h = 3πhh = 3πh

πrπr22h = 3πh, rh = 3πh, r22 = 3, r = √3 cm = 3, r = √3 cm

Mr. Chin-Sung Lin

ERHS Math GeometryA

A

C3 cm

p

h

r

h

Mr. Chin-Sung Lin

ERHS Math Geometry

A sphere is the set of all points A sphere is the set of all points equidistant equidistant from a fixed point from a fixed point called the called the centercenter

The The radiusradius of a sphere is the length of of a sphere is the length of the line segment from the center of the line segment from the center of the sphere to any point on the the sphere to any point on the spheresphere

Mr. Chin-Sung Lin

ERHS Math Geometry

rO

If the If the distancdistancee of a plane from the center of a sphere is of a plane from the center of a sphere is dd

and the and the radiusradius of the sphere is of the sphere is rr

Mr. Chin-Sung Lin

ERHS Math Geometry

P

O

p

dr

P

O

p

dr

P

O

p

dr

r < d no points in common

r = d one points in common

r > d infinite points

in common (circle)

A A circle circle is the set of all points in a plane is the set of all points in a plane equidistantequidistant from a fixed point in the plane called the from a fixed point in the plane called the centercenter

Mr. Chin-Sung Lin

ERHS Math Geometry

Op

r

The intersection of a The intersection of a sphere sphere and a and a planeplane through the through the center center of the sphere is a of the sphere is a circlecircle whose whose radiusradius is is equal to the equal to the radius of the sphereradius of the sphere

Mr. Chin-Sung Lin

ERHS Math Geometry

O

p

r

r

A A great circle of a sphere great circle of a sphere is the intersection of is the intersection of a a sphere sphere and and a plane a plane through the through the center of the center of the spheresphere

Mr. Chin-Sung Lin

ERHS Math Geometry

O

p

r

r

If the intersection of If the intersection of a sphere a sphere and and a plane a plane does does notnot contain contain the the center of the spherecenter of the sphere, then the intersection is a , then the intersection is a circlecircle

Given: A sphere with center at O Given: A sphere with center at O

plane p intersecting plane p intersecting

the sphere at A and Bthe sphere at A and B

Prove: The intersection is a circleProve: The intersection is a circle

Mr. Chin-Sung Lin

ERHS Math Geometry

O

pCA

B

If the intersection of If the intersection of a sphere a sphere and and a plane a plane does does notnot contain contain the the center of the spherecenter of the sphere, then the intersection is a , then the intersection is a circlecircle

Given: A sphere with center at O Given: A sphere with center at O

plane p intersecting plane p intersecting

the sphere at A and Bthe sphere at A and B

Prove: The intersection is a circleProve: The intersection is a circle

Mr. Chin-Sung Lin

ERHS Math Geometry

O

p

rCA

B

Mr. Chin-Sung Lin

Statements Reasons

1. Draw a line OC, point C on plane p 1. Given, create two triangles

OCAC, OCBC

2. OCA and OCB are right angles 2. Definition of perpendicular

3. OA OB 3. Radius of a sphere

4. OC OC 4. Reflexive postulate

5. OAC OBC 5. HL postulate

6. CA CB 6. CPCTC

7. The intersection is a circle 7. Definition of circles

ERHS Math Geometry

O

p

rCA B

The intersection of The intersection of a plane a plane and and a sphere a sphere is is a circlea circle

A A great circle great circle is the is the largest cilargest circle that can be drawn on a rcle that can be drawn on a spheresphere

Mr. Chin-Sung Lin

ERHS Math Geometry

O

p

p’

If two planes are equidistant from the center of a sphere and If two planes are equidistant from the center of a sphere and intersect the sphere, then the intersections are congruent intersect the sphere, then the intersections are congruent circlescircles

Mr. Chin-Sung Lin

ERHS Math Geometry

O

q

p

A

B

C

D

Surface Area: S = 4πrSurface Area: S = 4πr22

r:r: radiusradius

Mr. Chin-Sung Lin

ERHS Math GeometryA

rO

Volume: V = Volume: V = 44//33 πr πr33

r:r: radiusradius

Mr. Chin-Sung Lin

ERHS Math GeometryA

rO

Find the surface area and the volume of a Find the surface area and the volume of a sphere whose radius is 6 cmsphere whose radius is 6 cm

Mr. Chin-Sung Lin

ERHS Math GeometryA

rO

Find the surface area and the volume of a Find the surface area and the volume of a sphere whose radius is 6 cmsphere whose radius is 6 cm

Surface Area: S = 4π6Surface Area: S = 4π622 = 144π cm = 144π cm22

Volume: V = Volume: V = 44//33 π6 π633 = 288π cm = 288π cm33

Mr. Chin-Sung Lin

ERHS Math GeometryA

rO

Mr. Chin-Sung Lin

ERHS Math Geometry

Mr. Chin-Sung Lin

ERHS Math Geometry