T\u1eadp h\u1ee3p c\u00e1c \u0111i\u1ec3m M trong kh\u00f4ng gian c\u00e1ch \u0111i\u1ec3m O c\u1ed1 \u0111\u1ecbnh m\u1ed9t kho\u1ea3ng R (R>0) kh\u00f4ng \u0111\u1ed5i g\u1ecdi l\u00e0 m\u1eb7t c\u1ea7u c\u00f3 t\u00e2m O v\u00e0 b\u00e1n kh\u00ednh b\u1eb1ng R.<\/p>\n

<\/p>\n

M\u1eb7t c\u1ea7u nh\u01b0 th\u1ebf th\u01b0\u1eddng \u0111\u01b0\u1ee3c k\u00ed hi\u1ec7u l\u00e0 S(O ; R).<\/p>\n

Cho m\u1eb7t c\u1ea7u t\u00e2m o b\u00e1n k\u00ednh r v\u00e0 M l\u00e0 m\u1ed9t \u0111i\u1ec3m b\u1ea5t k\u00ec trong kh\u00f4ng gian.<\/p>\n

– N\u1ebfu OM = r th\u00ec ta n\u00f3i \u0111i\u1ec3m M n\u1eb1m tr\u00ean m\u1eb7t c\u1ea7u S(O;r).
\n– N\u1ebfu OM<r th\u00ec ta n\u00f3i \u0111i\u1ec3m M n\u1eb1m trong m\u1eb7t c\u1ea7u S(O;r)
\n– N\u1ebfu OM > r th\u00ec ta n\u00f3i \u0111i\u1ec3m M n\u1eb1m ngo\u00e0i m\u1eb7t c\u1ea7u S(O;r)<\/p>\n

2. Giao c\u1ee7a m\u1eb7t c\u1ea7u v\u00e0 m\u1eb7t ph\u1eb3ng<\/strong><\/p>\n

Cho m\u1eb7t c\u1ea7u S (O;r) v\u00e0 m\u1eb7t ph\u1eb3ng (P). H l\u00e0 h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a O l\u00ean m\u1eb7t ph\u1eb3ng (P). Khi \u0111\u00f3 OH =h l\u00e0 kho\u1ea3ng c\u00e1ch t\u1eeb t\u00e2m O c\u1ee7a m\u1eb7t c\u1ea7u t\u1edbi m\u1eb7t ph\u1eb3ng (P).<\/p>\n

Ta c\u00f3 tr\u01b0\u1eddng h\u1ee3p:<\/p>\n

– N\u1ebfu h>r th\u00ec m\u1eb7t ph\u1eb3ng (P) kh\u00f4ng c\u1eaft m\u1eb7t c\u1ea7u;
\n– N\u1ebfu h=r th\u00ec m\u1eb7t ph\u1eb3ng (P) ti\u1ebfp x\u00fac v\u1edbi m\u1eb7t c\u1ea7u t\u1ea1i \u0111i\u1ec3m H. Ta c\u00f3 OH vu\u00f4ng g\u00f3c v\u1edbi (P).
\n– N\u1ebfu h<r th\u00ec m\u1eb7t ph\u1eb3ng (P) c\u1eaft m\u1eb7t c\u1ea7u theo \u0111\u01b0\u1eddng tr\u00f2n c\u00f3 b\u00e1n k\u00ednh … v\u00e0 c\u00f3 t\u00e2m l\u00e0 \u0111i\u1ec3m H.<\/p>\n

Khi h=0 th\u00ec m\u1eb7t ph\u1eb3ng (P) c\u1eaft m\u1eb7t c\u1ea7u theo m\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n l\u1edbn c\u00f3 b\u00e1n k\u00ednh.<\/p>\n

3. Giao c\u1ee7a m\u1eb7t c\u1ea7u v\u1edbi \u0111\u01b0\u1eddng th\u1eb3ng, ti\u1ebfp tuy\u1ebfn c\u1ee7a m\u1eb7t c\u1ea7u.<\/strong><\/p>\n

Cho m\u1eb7t c\u1ea7u S (O;r) v\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng<\/p>\n

1. Tr\u01b0\u1eddng h\u1ee3p\u0111i qua t\u00e2m O c\u1ee7a m\u1eb7t c\u1ea7u th\u00ecc\u1eaft m\u1eb7t c\u1ea7u t\u1ea1i hai \u0111i\u1ec3m A, B v\u1edbi AB =2r.<\/p>\n

2. Tr\u01b0\u1eddng h\u1ee3pkh\u00f4ng \u0111i qua t\u00e2m O c\u1ee7a m\u1eb7t c\u1ea7u, ta g\u1ecdi d l\u00e0 kho\u1ea3ng c\u00e1ch t\u1eeb t\u00e2m O \u0111\u1ebfn \u0111\u01b0\u1eddng th\u1eb3ng, khi \u0111\u00f3:<\/p>\n

– N\u1ebfu d<r, \u0111\u01b0\u1eddng th\u1eb3ngc\u1eaft m\u1eb7t c\u1ea7u t\u1ea1i hai \u0111i\u1ec3m M, N;
\n– N\u1ebfu d=r, \u0111\u01b0\u1eddnggi\u00e1c ti\u1ebfp x\u00fac v\u1edbi m\u1eb7t c\u1ea7u t\u1ea1i \u0111i\u1ec3m H, H g\u1ecdi l\u00e0 ti\u1ebfp \u0111i\u1ec3m v\u00e0 \u0111\u01b0\u1eddng th\u1eb3ngg\u1ecdi l\u00e0 ti\u1ebfp tuy\u1ebfn c\u1ee7a m\u1eb7t c\u1ea7u.
\n– N\u1ebfu d>r, \u0111\u01b0\u1eddng th\u1eb3ngkh\u00f4ng c\u1eaft m\u1eb7t c\u1ea7u.<\/p>\n

4. M\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp m\u1ed9t h\u00ecnh \u0111a di\u1ec7n<\/strong><\/p>\n

M\u1ed9t m\u1eb7t c\u1ea7u \u0111i qua m\u1ecdi \u0111\u1ec9nh c\u1ee7a m\u1ed9t h\u00ecnh \u0111a di\u1ec7n H, v\u00e0 h\u00ecnh \u0111a di\u1ec7n H g\u1ecdi l\u00e0 n\u1ed9i ti\u1ebfp trong m\u1eb7t c\u1ea7u \u0111\u00f3.<\/p>\n

a. B\u1ea5t k\u00ec h\u00ecnh t\u1ee9 di\u1ec7n n\u00e0o c\u0169ng c\u00f3 m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp.<\/p>\n

X\u00e9t h\u00ecnh t\u1ee9 di\u1ec7n ABCD. G\u1ecdi A l\u00e0 tr\u1ee5c c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2ng ngo\u1ea1i ti\u1ebfp. G\u1ecdil\u00e0 tr\u1ee5c c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfpBCD v\u00e0 (P) l\u00e0 m\u1eb7t trung tr\u1ef1c c\u1ee7a c\u1ea1nh AB, th\u00ec t\u00e2m O c\u1ee7a m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp ABCD l\u00e0 giao \u0111i\u1ec3m c\u1ee7a\u00a0v\u00e0 (P).<\/p>\n

b. H\u00ecnh ch\u00f3p S.A1A2…An c\u00f3 m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp khi v\u00e0 ch\u1ec9 khi \u0111\u00e1y A1A2…An c\u00f3 \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp. T\u00e2m c\u1ee7a m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp h\u00ecnh ch\u00f3p l\u00e0 giao \u0111i\u1ec3m tr\u1ee5c c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp \u0111a gi\u00e1c\u00a0A1A2…An v\u00e0 m\u1eb7t trung tr\u1ef1c m\u1ed9t c\u1ea1nh b\u00ean c\u1ee7a h\u00ecnh ch\u00f3p.<\/p>\n

c. H\u00ecnh l\u0103ng tr\u1ee5\u00a0A1A2…An c\u00f3 m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp khi v\u00e0 ch\u1ec9 khi l\u00e0 h\u00ecnh l\u0103ng tr\u1ee5 \u0111\u1ee9ng v\u00e0 d\u00e1y c\u00f3 \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp. G\u1ecdi I v\u00e0 I’ l\u1ea7n l\u01b0\u1ee3t l\u00e0 t\u00e2m hai \u0111\u00e1y (t\u00e2m \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp), th\u00ec t\u00e2m m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp c\u1ee7a h\u00ecnh l\u0103ng tr\u1ee5 l\u00e0 trung \u0111i\u1ec3m O c\u1ee7a \u0111o\u1ea1n th\u1eb3ng II’.<\/p>\n

5. M\u1eb7t c\u1ea7u n\u1ed9i ti\u1ebfp h\u00ecnh \u0111a di\u1ec7n<\/strong><\/p>\n

N\u1ebfu c\u00f3 m\u1ed9t m\u1eb7t c\u1ea7u (S) ti\u1ebfp x\u00fac v\u1edbi t\u1ea5t c\u1ea3 c\u00e1c m\u1eb7t c\u1ee7a m\u1ed9t h\u00ecnh \u0111a di\u1ec7n H th\u00ec ta n\u00f3i (S) l\u00e0 m\u1eb7t c\u1ea7u n\u1ed9i ti\u1ebfp trong h\u00ecnh \u0111a di\u1ec7n H v\u00e0 H g\u1ecdi l\u00e0 h\u00ecnh \u0111a di\u1ec7n ngo\u1ea1i ti\u1ebfp (S)<\/p>\n

6. Di\u1ec7n t\u00edch m\u1eb7t c\u1ea7u v\u00e0 th\u1ec3 t\u00edch kh\u1ed1i c\u1ea7u.<\/strong><\/p>\n

<\/p>\n

B\u00e0i t\u1eadp \u00e1p d\u1ee5ng\u00a0C\u00f4ng th\u1ee9c t\u00ednh di\u1ec7n t\u00edch m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp h\u00ecnh ch\u00f3p<\/strong><\/h3>\n

B\u00e0i t\u1eadp 1:<\/strong> Cho h\u00ecnh ch\u00f3p S.ABCD c\u00f3 \u0111\u00e1y l\u00e0 h\u00ecnh thang vu\u00f4ng t\u1ea1i A v\u00e0 D, AB = AD = a. CD=2a. C\u1ea1nh b\u00ean SD vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t \u0111\u00e1y v\u00e0 SD = a. G\u1ecdi E l\u00e0 trung \u0111i\u1ec3m c\u1ee7a DC. T\u00ednh di\u1ec7n t\u00edch m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp h\u00ecnh ch\u00f3p S.BCE.<\/p>\n

B\u00e0i gi\u1ea3i<\/strong><\/p>\n

T\u1eeb ABCD l\u00e0 h\u00ecnh thang vu\u00f4ng t\u1ea1i A v\u00e0 D, AB=AD=a, CD=2a suy ra tam gi\u00e1c BEC vu\u00f4ng c\u00e2n \u0111\u1ec9nh E.<\/p>\n

G\u1ecdi M l\u00e0 trung \u0111i\u1ec3m BC th\u00ec M l\u00e0 t\u00e2m \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp tam gi\u00e1c BEC. Do \u0111\u00f3, tr\u1ee5c \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp tam gi\u00e1c BEC l\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng\u00a0\u0111i qua M v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng (ABCD). Khi \u0111\u00f3 \/\/SD.<\/p>\n

G\u1ecdi (\u03b1)\u00a0l\u00e0 m\u1eb7t ph\u1eb3ng trung tr\u1ef1c c\u1ee7a SC v\u00e0 I l\u00e0 giao \u0111i\u1ec3m c\u1ee7a (\u03b1) v\u00e0th\u00ec I l\u00e0 t\u00e2m m\u1eb7t c\u1ea7u (S) ngo\u1ea1i ti\u1ebfp h\u00ecnh tr\u00f2n S.BCE
\nTa th\u1ea5y r\u1eb1ng tam gi\u00e1c DBC vu\u00f4ng c\u00e2n t\u1ea1i B v\u00e0 n\u00ean<\/p>\n

B\u00e0i t\u1eadp s\u1ed1 2:<\/strong> Cho h\u00ecnh ch\u00f3p t\u1ee9 gi\u00e1c \u0111\u1ec1u S.ABCD c\u00f3 t\u1ea5t c\u1ea3 c\u00e1c c\u1ea1nh b\u1eb1ng a. M\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp h\u00ecnh ch\u00f3p n\u00e0y c\u00f3 di\u1ec7n t\u00edch t\u00ednh theo a l\u00e0:<\/p>\n

A.\u00a0\u03c0a\u00b2\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0B. 2\u03c0a\u00b2\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 C. 3.\u00a0\u03c0a\u00b2\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0D. 4\u03c0a\u00b2<\/p>\n

G\u1ee3i \u00fd gi\u1ea3i<\/strong><\/p>\n

K\u1ebb \u0111\u01b0\u1eddng sao cho SH c\u1ee7a h\u00ecnh ch\u00f3p th\u00ec H l\u00e0 t\u00e2m c\u1ee7a \u0111\u00e1y ABCD.<\/p>\n

Ta c\u00f3 SA=SC=a. AC =\u00a0.Suy ra, AC\u00b2=SA\u00b2+SC\u00b2=2a\u00b2.<\/p>\n

Do \u0111\u00f3,\u00a0SAC vu\u00f4ng t\u1ea1i S.<\/p>\n

Ch\u1ee9ng minh t\u01b0\u01a1ng t\u1ef1,\u00a0SBD vu\u00f4ng t\u1ea1i S.<\/p>\n

Ta \u0111\u01b0\u1ee3c HA=HB=HC=HD=HS=<\/p>\n

Do v\u1eady, m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp h\u00ecnh ch\u00f3p S.ABCD c\u00f3 t\u00e2m l\u00e0 H v\u00e0 b\u00e1n k\u00ednh R=<\/p>\n

Di\u1ec7n t\u00edch c\u1ee7a m\u1eb7t c\u1ea7u n\u00e0y l\u00e0 S=4\u03c0R\u00b2 = 4\u03c0=2\u03c0a\u00b2.<\/p>\n

B\u00e0i t\u1eadp s\u1ed1 3.\u00a0. Cho h\u00ecnh ch\u00f3p S.ABC c\u00f3 \u0111\u00e1y ABC l\u00e0 tam gi\u00e1c vu\u00f4ng c\u00e2n t\u1ea1i B, SA vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng (ABC) v\u00e0 c\u1ea1nh SA=AB=10cm. Di\u1ec7n t\u00edch m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp h\u00ecnh ch\u00f3p b\u1eb1ng:
\nA. 12\u03c0dm\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 B.\u00a0 1200\u03c0dm\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 C.1200\u03c0dm\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0D. 12\u03c0dm\u00b2<\/p>\n

B\u00e0i gi\u1ea3i<\/strong><\/p>\n

<\/p>\n

 <\/p>\n

<\/p>\n

 <\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

\u0110\u1ecbnh ngh\u0129a m\u1eb7t c\u1ea7u T\u1eadp h\u1ee3p c\u00e1c \u0111i\u1ec3m M trong kh\u00f4ng gian c\u00e1ch \u0111i\u1ec3m O c\u1ed1 \u0111\u1ecbnh m\u1ed9t kho\u1ea3ng R (R>0) kh\u00f4ng \u0111\u1ed5i g\u1ecdi l\u00e0 m\u1eb7t c\u1ea7u c\u00f3 t\u00e2m O… <\/p>\n","protected":false},"author":9,"featured_media":10915,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[450,469],"tags":[],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/onthitot.com\/wp-json\/wp\/v2\/posts\/10897"}],"collection":[{"href":"https:\/\/onthitot.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/onthitot.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/onthitot.com\/wp-json\/wp\/v2\/users\/9"}],"replies":[{"embeddable":true,"href":"https:\/\/onthitot.com\/wp-json\/wp\/v2\/comments?post=10897"}],"version-history":[{"count":0,"href":"https:\/\/onthitot.com\/wp-json\/wp\/v2\/posts\/10897\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/onthitot.com\/wp-json\/wp\/v2\/media\/10915"}],"wp:attachment":[{"href":"https:\/\/onthitot.com\/wp-json\/wp\/v2\/media?parent=10897"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/onthitot.com\/wp-json\/wp\/v2\/categories?post=10897"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/onthitot.com\/wp-json\/wp\/v2\/tags?post=10897"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}