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例1 已知過(guò)球面上三點(diǎn)的截面和球心的距離為球半徑的一半，且，求球的表面積

解：設(shè)截面圓心為，連結(jié)，設(shè)球半徑為，

則，

在中，，

∴，∴，

∴．

例2．半球內(nèi)有一個(gè)內(nèi)接正方體，正方體的一個(gè)面在半球的底面圓內(nèi)，若正方體棱長(zhǎng)為，求球的表面積和體積

解：作軸截面如圖所示，

，，

設(shè)球半徑為，

則

∴，

∴，．

例3．表面積為的球，其內(nèi)接正四棱柱的高是，求這個(gè)正四棱柱的表面積

解：設(shè)球半徑為，正四棱柱底面邊長(zhǎng)為，

則作軸截面如圖，，，

又∵，∴，

∴，∴，

∴．

例4　 正四面體ABCD的棱長(zhǎng)為a，球O是內(nèi)切球，球O₁是與正四面體的三個(gè)面和球O都相切的一個(gè)小球，求球O₁的體積．

分析：正四面體的內(nèi)切球與各面的切點(diǎn)是面的中心，球心到各面的距離相等．

解：如圖，設(shè)球O半徑為R，球O₁的半徑為r，E為CD中點(diǎn)，球O與平面ACD、BCD切于點(diǎn)F、G，球O₁與平面ACD切于點(diǎn)H．

由題設(shè)

　．

∵　△AOF∽△AEG　　 ∴　，得．

∵　△AO₁H∽△AOF　 ∴　 ，得．

∴　．

1 球的表面積：

設(shè)球的半徑為，我們把球面任意分割為一些“小球面片”，它們的面積分別用表示，則球的表面積:

以這些“小球面片”為底，球心為頂點(diǎn)的“小錐體”的體積和等于求的體積，這些“小錐體”可近似地看成棱錐，“小錐體”的底面積可近似地等于“小錐體”的底面積，球的半徑近似地等于小棱錐的高，因此，第個(gè)小棱錐的體積，當(dāng)“小錐體”的底面非常小時(shí)，“小錐體”的底面幾乎是“平的”，于是球的體積:

，

又∵，且

∴可得，

又∵，∴，

∴即為球的表面積公式

6．球的體積公式：

4．兩點(diǎn)的球面距離：

球面上兩點(diǎn)之間的最短距離，就是經(jīng)過(guò)兩點(diǎn)的大圓在這兩點(diǎn)間的一段劣弧的長(zhǎng)度，我們把這個(gè)弧長(zhǎng)叫做兩點(diǎn)的球面距離

5 半球的底面：

　　 已知半徑為的球，用過(guò)球心的平面去截球，球被截面分成大小相等的兩個(gè)半球，截面圓(包含它內(nèi)部的點(diǎn))，叫做所得半球的底面

3．經(jīng)度、緯度：

經(jīng)線：球面上從北極到南極的半個(gè)大圓；

緯線：與赤道平面平行的平面截球面所得的小圓；

經(jīng)度：某地的經(jīng)度就是經(jīng)過(guò)這點(diǎn)的經(jīng)線與地軸確定的半平面與經(jīng)線及軸確定的半平面所成的二面角的度數(shù)；

緯度：某地的緯度就是指過(guò)這點(diǎn)的球半徑與赤道平面所成角的度數(shù)

　　 與定點(diǎn)距離等于或小于定長(zhǎng)的點(diǎn)的集合，叫做球體，簡(jiǎn)稱球定點(diǎn)叫球心，定長(zhǎng)叫球的半徑與定點(diǎn)距離等于定長(zhǎng)的點(diǎn)的集合叫做球面一個(gè)球或球面用表示它的球心的字母表示，例如球．

2．球的截面：

用一平面去截一個(gè)球，設(shè)是平面的垂線段，為垂足，且，所得的截面是以球心在截面內(nèi)的射影為圓心，以為半徑的一個(gè)圓，截面是一個(gè)圓面

球面被經(jīng)過(guò)球心的平面截得的圓叫做大圓，被不經(jīng)過(guò)球心的平面截得的圓叫做小圓

　根據(jù)課文內(nèi)容用適當(dāng)?shù)脑~填空，使短文完整。

It was September.Two Chinese girls，Li Daiyu and Liu Qian were 1.________a trip to Canada to visit their cousins who live 2.________the Atlantic coast.They decided to 3.________the whole continent from east to west by train.

They first arrived in Vancouver，the most beautiful city in Canada，as it is 4.________by mountains on the north and east and the Pacific Ocean on the west.Then they took “The True North，”the cross­Canada train.Earlier that day 5.________ the train took them across the Rocky Mountains，they were excited to see mountain goats and even a grizzly bear 6.________the window.The next stop was Calgary，a city 7.________is famous for a Western festival，the Calgary Stampede.

After two days 8.________traveling，they found Canada is quite 9.________ with a very small population，but it has huge farms with thousands of square kilometres 10.________.Now they came to the Thunder Bay，which is at the top of the Great Lakes.After that，the train rushed across the top of the Lake Superior，11.________the great forests and southward to wards Toronto.In Toronto，they went up the tall CN Tower，from 12.________they saw the misty cloud 13.________from the great Niagara Falls.Later they were 14.________ to have dinner in downtown Chinatown and enjoyed good Cantonese food.Then the train took them to Montreal，where they found people everywhere were speaking French.They 15.________(tell) Montreal was a city with wonderful restaurants and clubs.That night the train took them to the east coast，their final destination.

答案：1.on　2.on　3.cross　4.surrounded　5.as　6.out of

7．which/that　8.of　9.empty　10.in size　11.through　12.where　13.rising　14.invited　15.were told

全 品中考網(wǎng)

8. Ottawa is also a good place for people to settle down.

[連句成篇]　Covering an area of 9,976,000 square kilometres, Canada is the second largest country in the world with a population of slightly over 30 million. Bordering the USA on the north, it is situated in the northern part of North America, lying on the Atlantic on the east and on the west of the Pacific. Canada is rich in natural resources. English and French are official languages of the country. Ottawa is one of the most beautiful capitals in the world which is also a good place for people to settle down.

7. Ottawa is one of the most beautiful capitals in the world.