例1 已知过球面上三点的截面和球心的距离为球半径的一半，且，求球的表面积

解：设截面圆心为，连结，设球半径为，

则，

在中，，

∴，∴，

∴．

例2．半球内有一个内接正方体，正方体的一个面在半球的底面圆内，若正方体棱长为，求球的表面积和体积

解：作轴截面如图所示，

，，

设球半径为，

则

∴，

∴，．

例3．表面积为的球，其内接正四棱柱的高是，求这个正四棱柱的表面积

解：设球半径为，正四棱柱底面边长为，

则作轴截面如图，，，

又∵，∴，

∴，∴，

∴．

例4　 正四面体ABCD的棱长为a，球O是内切球，球O₁是与正四面体的三个面和球O都相切的一个小球，求球O₁的体积．

分析：正四面体的内切球与各面的切点是面的中心，球心到各面的距离相等．

解：如图，设球O半径为R，球O₁的半径为r，E为CD中点，球O与平面ACD、BCD切于点F、G，球O₁与平面ACD切于点H．

由题设

　．

∵　△AOF∽△AEG　　 ∴　，得．

∵　△AO₁H∽△AOF　 ∴　 ，得．

∴　．

1 球的表面积：

设球的半径为，我们把球面任意分割为一些“小球面片”，它们的面积分别用表示，则球的表面积:

以这些“小球面片”为底，球心为顶点的“小锥体”的体积和等于求的体积，这些“小锥体”可近似地看成棱锥，“小锥体”的底面积可近似地等于“小锥体”的底面积，球的半径近似地等于小棱锥的高，因此，第个小棱锥的体积，当“小锥体”的底面非常小时，“小锥体”的底面几乎是“平的”，于是球的体积:

，

又∵，且

∴可得，

又∵，∴，

∴即为球的表面积公式

6．球的体积公式：

4．两点的球面距离：

球面上两点之间的最短距离，就是经过两点的大圆在这两点间的一段劣弧的长度，我们把这个弧长叫做两点的球面距离

5 半球的底面：

　　 已知半径为的球，用过球心的平面去截球，球被截面分成大小相等的两个半球，截面圆(包含它内部的点)，叫做所得半球的底面

3．经度、纬度：

经线：球面上从北极到南极的半个大圆；

纬线：与赤道平面平行的平面截球面所得的小圆；

经度：某地的经度就是经过这点的经线与地轴确定的半平面与经线及轴确定的半平面所成的二面角的度数；

纬度：某地的纬度就是指过这点的球半径与赤道平面所成角的度数

　　 与定点距离等于或小于定长的点的集合，叫做球体，简称球定点叫球心，定长叫球的半径与定点距离等于定长的点的集合叫做球面一个球或球面用表示它的球心的字母表示，例如球．

2．球的截面：

用一平面去截一个球，设是平面的垂线段，为垂足，且，所得的截面是以球心在截面内的射影为圆心，以为半径的一个圆，截面是一个圆面

球面被经过球心的平面截得的圆叫做大圆，被不经过球心的平面截得的圆叫做小圆

　根据课文内容用适当的词填空，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 品中考网

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

[连句成篇]　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.