CBSE Class 9 Answered

Give the derivation of the volume of the sphere without using calculas.

Asked by Ayush | 09 Jan, 2016, 10:32: PM

Expert Answer

T o space d e r i v e space t h e space v o l u m e space o f space a space s p h e r e space w i t h o u t space c a l c u l u s comma w e space h a v e space t o space u s e space t h e space r e s u l t space f o r space v o l u m e space o f space a space c o n e space a n d space a space c y l i n d e r. W e space w i l l space f i r s t space f i n d space v o l u m e space o f space a space h e m i s p h e r e space a n d space m u l t i p l y space i t space b y space 2 space t o space g e t space v o l u m e space o f space a space s p h e r e. I n space t h e space f i g u r e space a b o v e comma space i n space t h e space h e m i s p h e r e space a space c r o s s space s e c t i o n space a t space a space d i s tan c e space apostrophe a apostrophe f r o m space t h e space c e n t e r space o f space h e m i s p h e r e space i s space s h o w n. space apostrophe r apostrophe space i s space t h e space r a d i u s space o f space t h e space h e m i s p h e r e. F r o m space t h e space f i g u r e comma space t h e space r a d i u s space o f space t h e space c i r c u l a r space c o r s s space s e c t i o n space u sin g space p y t h a g o r a s space t h e o r e m space i s space square root of r squared minus a squared end root H e n c e comma space t h e space a r e a space o f space t h e space c r o s s space s e c t i o n space i s space straight pi open parentheses straight r squared minus straight a squared close parentheses.... left parenthesis 1 right parenthesis T h e space s e c o n d space f i g u r e space s h o w s space a space c y l i n d e r space o f space h e i g h t space apostrophe r apostrophe space a n d space b a s e space r a d i u s space apostrophe r apostrophe. W e space a l s o space h a v e space a space i n v e r t e d space c o n e space o f space r a d i u s space apostrophe r apostrophe space a n d space h e i g h t space apostrophe h apostrophe. T h e space a r e a space o f space t h e space c r o s s space s e c t i o n a l space r e g i o n space b e t w e e n space t h e space c o n e space a n d space t h e space c y l i n d e r comma w h i c h space i s space s h a d e d space i n space b l u e comma space c a n space b e space f o u n d space b y space s u b t r a c t i n g space a r e a s space o f space t w o space c i r c l e s. H e n c e comma space t h e space s h a d e d space c r o s s space s e c t i o n a l space a r e a space i s space straight pi open parentheses straight r squared minus straight a squared close parentheses.... left parenthesis 2 right parenthesis From space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis comma space the space area space of space the space cross space section space of space the space sphere space is space equal space to space the space difference space of space cross space sectional space areas space of space the space cylinder space and space the space cone space at space any space distance space apostrophe straight a apostrophe from space the space center. space Since comma space the space heights space are space equal space for space both space and space areas space of space each space slice space at space same space distance space from space the space straight c are space equal comma space we space can space say space the space volume space of space hemisphere space will space be space equal space to space the space diffference space of space volume space of space the space cylinder space and space cone. Volume space of space the space cylinder equals πr squared cross times straight r equals πr cubed Volume space of space the space cone equals 1 third πr squared cross times straight r equals 1 third πr cubed Hence comma space volume space of space the space sphere equals πr cubed minus 1 third πr cubed equals 2 over 3 πr cubed So space volume space of space the space sphere equals 2 cross times 2 over 3 πr cubed equals 4 over 3 πr cubed

Answered by satyajit samal | 13 Jan, 2016, 02:37: PM

CBSE Class 9 Answered

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Volume: Sphere and Hemisphere

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CBSE Class 9 Answered

Concept Videos

Volume: Sphere and Hemisphere ...

Volume: Sphere and Hemisphere

Volume: Sphere and Hemisphere ...

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