7/25/2019 Lesson 7 MATH13-1

1/51

Lesson 7 SPHERES

Week 9

MATH13-1Solid Mensuration

7/25/2019 Lesson 7 MATH13-1

2/51

Defnitions Relating toS!ere

A sphereis a t!ree-di"ensional solid#ounded #$ a sur%a&e &onsisting o% all ointse'uidistant %ro" an interior oint &alled t!ecenter( T!e &losed sur%a&e is &alled

spherical surface) *!ile t!e sa&e &o+ered#$ t!e sur%a&e is &alled t!e interior( As!ere is usuall$ designated #$ its &enter(Hen&e) s!ere Ois t!e s!ere *!ose &enteris at oint O(

Re%eren&e, Solid Mensuration, nderstanding t!e 3-D Sa&e #$

7/25/2019 Lesson 7 MATH13-1

3/51

Defnitions Relating toS!ere

Radius- Line %ro" t!e &enter to a oint on t!es!eri&al sur%a&e(

Diameter.AB/ 0R

- Line assing t!roug! t!e &enter and 2oiningt*o oints on t!e s!eri&al sur%a&e(

Re%eren&e, Solid Mensuration, nderstanding t!e 3-D Sa&e #$

7/25/2019 Lesson 7 MATH13-1

4/51

Defnitions Relating toS!ere

An$ se&tion o% a s!ere is a &ir&le(Great Circle.&ir&le C

- or"ed *!en a lane asses t!roug! t!e&enter(

Small Circle.&ir&le F

- or"ed *!en a lane does not asst!roug! t!e &enter(

Re%eren&e, Solid Mensuration, nderstanding t!e 3-D Sa&e #$

7/25/2019 Lesson 7 MATH13-1

5/51

Defnitions Relating toS!ere

Axis of a circle of a sphere.AB- A dia"eter o% t!e s!ere erendi&ular tot!e lane o% t!e &ir&le o% t!e s!ere(

Poles of the circle.ointsAand B

- Endoints o% t!e a4is(

Re%eren&e, Solid Mensuration, nderstanding t!e 3-D Sa&e #$

7/25/2019 Lesson 7 MATH13-1

6/51

Defnitions Relating toS!ere

Spherical distance- Measure o% t!e "inor ar& o% a great &ir&le&ontaining t!e t*o oints( 5t is t!e s!ortestdistan&e #et*een t*o oints on t!e sur%a&e o% t!e

s!ere(Polar Distance

- S!eri&al distan&e %ro" t!e nearest ole(

Quadrant

- A olar distan&e along t!e ar& o% a great &ir&le*!i&! is e'ual to one %ourt! o% t!e &ir&u"%eren&eo% a great &ir&le .i(e( S!eri&al distan&e %ro" nort!ole 6or sout! ole to t!e e'uator(

Re%eren&e, Solid Mensuration, nderstanding t!e 3-D Sa&e #$

7/25/2019 Lesson 7 MATH13-1

7/51

Defnitions Relating toS!ere

Tangent line- 5nterse&ts t!e s!ere at e4a&tl$ one oint(

T!e oint o% interse&tion is &alled t!e pointof tangency(

Internally tangent spheres

- 5% t!e$ lie on one side o% t!e lane(

Externally tangent spheres

- 5% t!e$ lie on oosite sides o% t!e lane(

Re%eren&e, Solid Mensuration, nderstanding t!e 3-D Sa&e #$

7/25/2019 Lesson 7 MATH13-1

8/51

Defnitions Relating toS!ere

Spherical angle .8 / s!eri&al CPD/APB- Di!edral angle %or"ed #$ t!e lanes o% t*ogreat &ir&les( 5t !as t!e sa"e "easure as t!elane angle %or"ed #$ t!e tangents to t!e

ar&s dra*n t!roug! t!eir oint o%interse&tion(

Re%eren&e, Solid Mensuration, nderstanding t!e 3-D Sa&e #$

7/25/2019 Lesson 7 MATH13-1

9/51

Defnitions Relating toS!ere

Arc length .ar& CD/ 8) less t!an 1:;

7/25/2019 Lesson 7 MATH13-1

10/51

Sur%a&e Area and >olu"e

S!ereS/ ?@R0

V/ @R3

He"is!ere

S/ 0@R0

V/ @R3

Re%eren&e, Solid Mensuration, nderstanding t!e 3-D Sa&e #$

7/25/2019 Lesson 7 MATH13-1

11/51

5ns&ri#ed Solids 5n+ol+ingS!eres

Sphere inscried in a polyhedron- T!e ol$!edron is &ir&u"s&ri#eda#out t!e s!ere(

- T!e s!ere is tangent t!e all t!e%a&es o% t!e ol$!edron(

Re%eren&e, Solid Mensuration, nderstanding t!e 3-D Sa&e #$

7/25/2019 Lesson 7 MATH13-1

12/51

5ns&ri#ed Solids 5n+ol+ingS!eres

Sphere circumscried aout apolyhedron

- T!e ol$!edron is ins&ri#ed in t!e

s!ere(-All T!e +erti&es o% a ol$!edron lie ont!e sur%a&e o% t!e s!ere(

Re%eren&e, Solid Mensuration, nderstanding t!e 3-D Sa&e #$

7/25/2019 Lesson 7 MATH13-1

13/51

5ns&ri#ed Solids 5n+ol+ingS!eres

T!e radius o% t!e s!ere &ir&u"s&ri#eda#out a regular tetra!edron is e'ual to

t!ree-%ourt!s o% t!e altitude o% t!etetra!edron *!ile t!e radius o% t!e ins&ri#ed

s!ere is one-%ourt! o% t!e altitude(

R/ h R/ Bh

Re%eren&e, Solid Mensuration, nderstanding t!e 3-D Sa&e #$

7/25/2019 Lesson 7 MATH13-1

14/51

5ns&ri#ed Solids 5n+ol+ingS!eres

5% a s!ere is ins&ri#ed in a &$linder)t!e +olu"e o% t!e s!ere is t*o-t!irdst!at o% t!e &$linder) and t!e sur%a&earea o% t!e s!ere is e'ual to t!elateral area o% t!e &$linder) or t*o-t!irds o% t!e total area o% t!e &$linder(

Vs!ere/ CV&$linder

S/ LSA

S/ CTSA

Re%eren&e, Solid Mensuration, nderstanding t!e 3-D Sa&e #$

7/25/2019 Lesson 7 MATH13-1

15/51

5ns&ri#ed Solids 5n+ol+ingS!eres

5% a &one !as its #ase and !eig!t e'ualrese&ti+el$ to t!e #ase and !eig!t o% a

&$linder) its +olu"e is one-t!ird t!at o%t!e &$linder or one-!al% t!at o% a s!ere

ins&ri#ed in t!e &$linder(V&one/ V&$linder/ Vs!ere

Re%eren&e, Solid Mensuration, nderstanding t!e 3-D Sa&e #$

7/25/2019 Lesson 7 MATH13-1

16/51

5ns&ri#ed Solids 5n+ol+ingS!eres

5% a s!ere is ins&ri#ed in a &u#e) t!e +olu"eo% t!e s!ere is a#out one-!al% t!e +olu"e o%

t!e &u#e) or Vs!ere/ .V&u#e to #e e4a&t(

Vs!ere/ .V&u#e F V&u#e

Re%eren&e, Solid Mensuration, nderstanding t!e 3-D Sa&e #$

7/25/2019 Lesson 7 MATH13-1

17/51

EGAMPLES

1) 1I:, An e'uilateral triangle *it!an area o% 9J3 s'( units is ins&ri#ed ina &ir&ular se&tion o% t!e s!ere( Ho*%ar %ro" t!e &enter o% t!e s!ere is t!elane o% t!e triangle i% t!e radius o% t!es!ere is 1? unitsK AS, 0J?I units

3) 17;, T!e radii o% t*o s!eres are? %t( and : %t( ind t!e +olu"e o% as!ere *!ose sur%a&e area is e'ual to

t!e su" o% t!e sur%a&e areas o% t!eseRe%eren&e, Solid Mensuration, nderstanding t!e 3-D Sa&e #$

7/25/2019 Lesson 7 MATH13-1

18/51

EGAMPLES

) 171, ind t!e +olu"e o% a!e"is!eri&al s!ell in *!i&! t!e outerand inner sur%a&e areas are 10:@ in0and ;@ in0) rese&ti+el$(

AS, 0:@ in3

Re%eren&e, Solid Mensuration, nderstanding t!e 3-D Sa&e #$

7/25/2019 Lesson 7 MATH13-1

19/51

7(1 EGERN5SES

9) 17?, T!e lane o% a s"all &ir&le in as!ere o% radius 1: in&!es is I in&!es %ro"t!e &enter o% t!e s!ere( ind t!e area o% t!es"all &ir&le and t!e olar distan&e o% a oint

on t!e &ir&u"%eren&e o% t!e s"all &ir&le(AS, 0::@ in0) 00(0 in or 7;(LME A SPHER5NAL SE=MET

Re%eren&e, Solid Mensuration, nderstanding t!e 3-D Sa&e #$

7/25/2019 Lesson 7 MATH13-1

25/51

E

Portion o% t!e sur%a&e o% a s!ere#ounded #$ t*o arallel lanes(

Parallel &ir&ular se&tions are &alled

ases of the !one(Distan&e #et*een t!e #ases is t!ealtitude of the !one(

Ma$ #e t!oug!t o% as t!e lateral areao% a s!eri&al seg"ent(

Re%eren&e, Solid Mensuration, nderstanding t!e 3-D Sa&e #$

7/25/2019 Lesson 7 MATH13-1

26/51

E

Re%eren&e, Solid Mensuration, nderstanding t!e 3-D Sa&e #$

7/25/2019 Lesson 7 MATH13-1

27/51

AREA A E

Z/ 0@Rh

T!is %or"ula alies to all t!ree t$es

o% a Uone(t*o #ases

one #ase *!ere hV R

one #ase *!ere h R

Re%eren&e, Solid Mensuration, nderstanding t!e 3-D Sa&e #$

7/25/2019 Lesson 7 MATH13-1

28/51

TSA A SPHER5NALSE=MET

or one-#ase seg"ent,TSA/Z B/ 0@Rh @r0

or t*o-#ase seg"ent,TSA/Z B1 B0/ 0@Rh @ @

Re%eren&e, Solid Mensuration, nderstanding t!e 3-D Sa&e #$

7/25/2019 Lesson 7 MATH13-1

29/51

EGAMPLES

9) 179, A lane &uts a s!ere o%radius %t( at a distan&e o% 3 %t( %ro"t!e &enter o% t!e s!ere( ind t!e su"o% t!e areas o% t!e t*o Uones %or"ed(

AS, 1;;@ %t0

Re%eren&e, Solid Mensuration, nderstanding t!e 3-D Sa&e #$

7/25/2019 Lesson 7 MATH13-1

30/51

LE

Portion o% t!e sur%a&e o% a s!ere #ounded#$ t*o interse&ting ar&s o% great &ir&les(

Partial re+olution o% a se"i&ir&le a#out itsdia"eter as an a4is(

T!e angle of the lune .8 is t!e s!eri&alangel %or"ed #$ t!e se"i&ir&les #oundingt!e lune(

8 / 1:;< %or a !e"is!ere(

8 / 3I;< %or t!e sur%a&e o% a s!ere(

Re%eren&e, Solid Mensuration, nderstanding t!e 3-D Sa&e #$

7/25/2019 Lesson 7 MATH13-1

31/51

LE

CADis t!e angle o% t!e luneACBD(

Re%eren&e, Solid Mensuration, nderstanding t!e 3-D Sa&e #$

7/25/2019 Lesson 7 MATH13-1

32/51

AREA LE

L/ 0R0

8

Re%eren&e, Solid Mensuration, nderstanding t!e 3-D Sa&e #$

7/25/2019 Lesson 7 MATH13-1

33/51

EGAMPLES

1;) 1:1, T!e area o% a lune is 30@&"0and t!e area o% its s!ere is 0I@&"0( ind t!e angle o% t!e lune indegrees(

AS, 8 / @X? / ?