Para esta demonstração iremos utilizar apenas conceitos básicos de Geometria Podemos decompor a esfera em uma infinidade de pirâmides cujas bases compõem a superfície esférica e os vértices se encontram no centro da esfera.[br][br][br][br]                                    [url=http://lh4.ggpht.com/-9f2I0IM-Chc/TnynxUMWIZI/AAAAAAAAOo0/4urBMqmgUYQ/s1600-h/clip_image002%25255B2%25255D.jpg][img width=437,height=434]http://lh4.ggpht.com/-ByOe6LZ8Tl4/TnxqG7t0YSI/AAAAAAAAOo4/-Z0Wje-N3dM/clip_image002_thumb%25255B1%25255D.jpg?imgmax=800[/img][br][br][/url][br]Desta forma, a superfície da esfera fica dividida em [i]N[/i] polígonos e a área da superfície esférica [i]ASE[/i] é dada por:[br][url=http://lh6.ggpht.com/-aZaEOf-F7ew/TnxqHQfU65I/AAAAAAAAOo8/0lWof8c9lZk/s1600-h/clip_image004%25255B2%25255D.gif][img width=242,height=22]http://lh3.ggpht.com/-pUPbKJX_s3w/TnxqH4rbSuI/AAAAAAAAOpA/7zc5DWEgQBA/clip_image004_thumb%25255B2%25255D.gif?imgmax=800[/img][br][/url][br]Para o Volume da esfera, podemos dizer que é igual à soma dos volumes dessas[i]N[/i] por:[br][url=http://lh3.ggpht.com/-_XT2P1367wE/TnxqIbC9G2I/AAAAAAAAOpE/0paXof9Ku5Y/s1600-h/clip_image006%25255B2%25255D.gif][img width=225,height=22]http://lh6.ggpht.com/-nji89dL9s8s/TnxqIxXGT6I/AAAAAAAAOpI/TalvTq7L6dU/clip_image006_thumb%25255B2%25255D.gif?imgmax=800[/img][br][/url][br]Sabemos que o volume de uma pirâmide é dado pela fórmula:[br][url=http://lh4.ggpht.com/-MySjhHoQwM0/TnxqJYgOA3I/AAAAAAAAOpM/nj4Wi5XyG6I/s1600-h/clip_image008%25255B2%25255D.gif][img width=123,height=39]http://lh4.ggpht.com/-y0uEz2IfW8I/TnxqJ5Fk4aI/AAAAAAAAOpQ/1z2d4WrvGHY/clip_image008_thumb%25255B2%25255D.gif?imgmax=800[/img][br][/url][br]No caso destas pirâmides que compõem a esfera, suas alturas são exatamente o raio [i]R[/i] da esfera. Assim, a relação (3) fica:[br][url=http://lh6.ggpht.com/-VkVDtKVP1tQ/TnxqKK91Q8I/AAAAAAAAOpU/ZkM5kGiYRhA/s1600-h/clip_image010%25255B2%25255D.gif][img width=125,height=39]http://lh5.ggpht.com/-F9DXVsn6fnA/TnxqKeoT1vI/AAAAAAAAOpY/rfB8LEkEYRM/clip_image010_thumb%25255B2%25255D.gif?imgmax=800[/img][br][/url][br]e o volume da esfera será a soma dos volumes destas pirâmides:[br][url=http://lh5.ggpht.com/-RdkKfX0-9TU/TnxqKukSaNI/AAAAAAAAOpc/2-8isS887qU/s1600-h/clip_image012%25255B2%25255D.gif][img width=287,height=39]http://lh3.ggpht.com/-RlDfuLoFsgo/TnxqLP7qD5I/AAAAAAAAOpg/xb39B2w2XvU/clip_image012_thumb%25255B2%25255D.gif?imgmax=800[/img][/url][br][url=http://lh4.ggpht.com/-0egsRMUe4C8/TnxqLroFygI/AAAAAAAAOpk/sxSzQFDgOEA/s1600-h/clip_image014%25255B2%25255D.gif][img width=264,height=39]http://lh3.ggpht.com/-L-RLPZIdeBE/TnxqL-RfStI/AAAAAAAAOpo/hT1czRospl0/clip_image014_thumb%25255B2%25255D.gif?imgmax=800[/img][br][/url][br]Vejam que a soma das áreas da relação (5) é igual à superfície esférica dada na relação (1). Assim, temos que:[br][url=http://lh5.ggpht.com/-lTjME7aM7uc/TnxqMM6sE6I/AAAAAAAAOps/92NICf8dEZM/s1600-h/clip_image016%25255B2%25255D.gif][img width=120,height=39]http://lh3.ggpht.com/-uk1ueS16ytE/TnxqMqCoPDI/AAAAAAAAOpw/9zPgIXlO7Xw/clip_image016_thumb%25255B2%25255D.gif?imgmax=800[/img][br][/url][br]Partindo do princípio em que já sabemos como calcular o volume da esfera:[br][url=http://lh6.ggpht.com/-rjgtWWVs5Vs/TnxqNA41fbI/AAAAAAAAOp0/uMnnQHvM5wM/s1600-h/clip_image018%25255B2%25255D.gif][img width=113,height=39]http://lh4.ggpht.com/-BUhDaKNkJpg/TnxqNeQK6cI/AAAAAAAAOp4/T6DitUrTQDM/clip_image018_thumb%25255B2%25255D.gif?imgmax=800[/img][br][/url][br]podemos determinar a superfície esférica substituindo a relação (7) em (6), obtendo:[br][url=http://lh4.ggpht.com/-i8xqWR1yGKg/TnxqNsXNzOI/AAAAAAAAOp8/jmDaTJ31X-U/s1600-h/clip_image020%25255B2%25255D.gif][img width=107,height=39]http://lh6.ggpht.com/-NgBmLQJaE6A/TnxqOPfpm-I/AAAAAAAAOqA/jtWKHWybT2w/clip_image020_thumb%25255B2%25255D.gif?imgmax=800[/img][/url][br][url=http://lh5.ggpht.com/-OuAdU19tXXQ/TnxqOQV3v1I/AAAAAAAAOqE/cFMv8ZU85ZY/s1600-h/clip_image022%25255B2%25255D.gif][img width=118,height=22]http://lh5.ggpht.com/-nBRfmQG3zmY/TnxqOv2CdkI/AAAAAAAAOqI/33ZMJiuXvpM/clip_image022_thumb%25255B2%25255D.gif?imgmax=800[/img][br][/url][br]Vejam que aqui só utilizamos conceitos básicos de Geometria, levando em conta que já sabíamos previamente a fórmula do volume da esfera.[br]