内切球与外接球习题讲义教师版

1、position-related consumption of civil servants has been swept by finance, consumer, regardless of cost, extravagance and waste in the civil service position-related consumption, abuse, corruption and embezzlement, corruption is important. Then, under the conditions of market economy, how to reform t

2、he existing civil duty consumption management, explores a source to prevent and curb the post consumption corruption way, is currently a major issue faced by honest work. Recently, I conducted research on this issue, this problem on some humble opinions. First, the existing public servants ' dut

3、y consumption the main problems seen from the investigation and reasons, in recent years, public servants ' duty consumption caused by the abuses and not a person of integrity, is one of the major problems in the party in Government, its operation order have a negative effect on the party and Go

4、vernment organs, seriously damaging the image of the party and the Government, undermining the relationship between party and the masses, effect, opening up and economic construction. From I County in recent years of governance situation see, positions consumption in the produced of "two not&qu

5、ot; phenomenon rendering four a features: a is positions consumption system lost has due of binding, right is greater than rules, and right is greater than method of phenomenon more highlight; II is in positions consumption in the Camera Obscura operation, using terms, will positions consumption int

6、o has personal consumption, will corporate points to into personal points to, makes positions consumption in some aspects has into positions enjoy and self-dealing of means; three is to positions consumption for name, fraud, false impersonator, Trend of negative corruption phenomena such as corrupti

7、on and misappropriation; four palaces, follow the fashion, rivalries, wasteful, and post consumption became a symbol of showing off their individual capacities. Caused by public servants ' duty consumption of many "two" phenomenon in which people reflect the biggest problems are: (a) t

8、he official car problems. Mainly in three aspects: one is the larger buses cost expenditure. According to statistics, until November 2003, XX County township Department bus 159 cars, which department owns the bus 145 vehicles, and showed an increasing trend. Financial expenses cost per bus per year

9、to 35,000 yuan, and in fact every cost up to 50,000 yuan. Some units also hiring temporary drivers and expenditure on wages and subsidies. Necessary to "keep a car", but also "dependants", leading to larger expenses. Second, gongchesiyong breed unhealthy tendencies. Some people b

10、elieve that now some bus drivers use one-third, one-third leading private one-third used for official purposes. Some public servants, especially leading officials motoring, cars for private purposes, violating the self-discipline regulations, and even lead to traffic accidents. According to statisti

11、cs from related departments, since 2004, the correct investigation in our County serves nearly 30 cars for private purposes, only the first half of this year, cars for private purposes or立体几何中的“内切”与“外接”问题的探究1 球与柱体规则的柱体，如正方体、长方体、正棱柱等能够和球进行充分的组合，以外接和内切两种形态进行结合，通过球的半径和棱柱的棱产生联系，然后考查几何体的体积或者表面积等相关问题.1.1

12、球与正方体如图1所示，正方体,设正方体的棱长为，为棱的中点，为球的球心。常见组合方式有三类：一是球为正方体的内切球，截面图为正方形和其内切圆，则；二是与正方体各棱相切的球，截面图为正方形和其外接圆，则；三是球为正方体的外接球，截面图为长方形和其外接圆，则.通过这三种类型可以发现，解决正方体与球的组合问题，常用工具是截面图，即根据组合的形式找到两个几何体的轴截面，通过两个截面图的位置关系，确定好正方体的棱与球的半径的关系，进而将空间问题转化为平面问题 。例 1 棱长为1的正方体的8个顶点都在球的表面上，分别是棱，的中点，则直线被球截得的线段长为（ ）A B CD1.2 球与长方体长方体各顶点可在

13、一个球面上，故长方体存在外切球.但是不一定存在内切球.设长方体的棱长为其体对角线为.当球为长方体的外接球时，截面图为长方体的对角面和其外接圆，和正方体的外接球的道理是一样的，故球的半径例 2 在长、宽、高分别为2，2，4的长方体内有一个半径为1的球，任意摆动此长方体，则球经过的空间部分的体积为( )A.B.4C.D.1.3 球与正棱柱球与一般的正棱柱的组合体，常以外接形态居多。下面以正三棱柱为例，介绍本类题目的解法构造直角三角形法。设正三棱柱的高为，底面边长为，如图2所示，和分别为上下底面的中心。根据几何体的特点，球心必落在高的中点，借助直角三角形的勾股定理，可求。例3 正四棱柱的各顶点都在半

14、径为的球面上，则正四棱柱的侧面积有最 值，为 .2 球与锥体规则的锥体，如正四面体、正棱锥、特殊的一些棱锥等能够和球进行充分的组合，以外接和内切两种形态进行结合，通过球的半径和棱锥的棱和高产生联系，然后考查几何体的体积或者表面积等相关问题.2.1 球与正四面体正四面体作为一个规则的几何体，它既存在外接球，也存在内切球，并且两心合一，利用这点可顺利解决球的半径与正四面体的棱长关系。如图4，设正四面体的棱长为，内切球半径为，外接球的半径为，取的中点为，为在底面的射影，连接为正四面体的高。在截面三角形,作一个与边和相切，圆心在高上的圆，即为内切球的截面。因为正四面体本身的对称性可知，外接球和内切球的

15、球心同为。此时， 则有解得：这个解法是通过利用两心合一的思路，建立含有两个球的半径的等量关系进行求解.同时我们可以发现，球心为正四面体高的四等分点.如果我们牢记这些数量关系，可为解题带来极大的方便.例4 将半径都为的四个钢球完全装入形状为正四面体的容器里，这个正四面体的高的最小值为 ( )A. B. 2+ C. 4+ D. 球的外切正四面体，这个小球球心与外切正四面体的中心重合，而正四面体的中心到顶点的距离是中心到地面距离的3倍.2.2 球与三条侧棱互相垂直的三棱锥球与三条侧棱互相垂直的三棱锥组合问题，主要是体现在球为三棱锥的外接球.解决的基本方法是补形法，即把三棱柱补形成正方体或者长方体。常

16、见两种形式：一是三棱锥的三条棱互相垂直且相等，则可以补形为一个正方体，它的外接球的球心就是三棱锥的外接球的球心。如图5，三棱锥的外接球的球心和正方体的外接球的球心重合，设，则。二是如果三棱锥的三条侧棱互相垂直且不相等，则可以补形为一个长方体，它的外接球的球心就是三棱锥的外接球的球心，（为长方体的体对角线长）。例5 在正三棱锥中，分别是棱的中点，且,若侧棱,则正三棱锥外接球的表面积是 。2.3 球与正棱锥球与正棱锥的组合，常见的有两类，一是球为三棱锥的外接球，此时三棱锥的各个顶点在球面上，根据截面图的特点，可以构造直角三角形进行求解.二是球为正棱锥的内切球，例如正三棱锥的内切球，球与正三棱锥四个

17、面相切，球心到四个面的距离相等，都为球半径这样求球的半径可转化为球球心到三棱锥面的距离，故可采用等体积法解决，即四个小三棱锥的体积和为正三棱锥的体积.例6 在三棱锥PABC中，PAPB=PC=,侧棱PA与底面ABC所成的角为60°，则该三棱锥外接球的体积为（ ） A B. C. 4D.2.4 球与特殊的棱锥球与一些特殊的棱锥进行组合，一定要抓住棱锥的几何性质，可综合利用截面法、补形法、等进行求解。例如，四面体都是直角三角形的三棱锥，可利用直角三角形斜边中点几何特征，巧定球心位置。如图8，三棱锥,满足面，取的中点为，由直角三角形的性质可得：,所以点为三棱锥的外接球的球心,则.例7 矩形

18、中，沿将矩形折成一个直二面角,则四面体的外接球的体积是( )A. B. C. D.3 球与球对个多个小球结合在一起，组合成复杂的几何体问题，要求有丰富的空间想象能力，解决本类问题需掌握恰当的处理手段，如准确确定各个小球的球心的位置关系，或者巧借截面图等方法，将空间问题转化平面问题求解.例8 在半径为的球内放入大小相等的4个小球，则小球的半径的最大值为（）4 球与几何体的各条棱相切球与几何体的各条棱相切问题，关键要抓住棱与球相切的几何性质，达到明确球心的位置为目的，然后通过构造直角三角形进行转换和求解.如与正四面体各棱都相切的球的半径为相对棱的一半：.例8 把一个皮球放入如图10所示的由8根长均

19、为20 cm的铁丝接成的四棱锥形骨架内，使皮球的表面与8根铁丝都有接触点，则皮球的半径为（）A. B. C. D. 综合上面的四种类型，解决与球的外切问题主要是指球外切多面体与旋转体，解答时首先要找准切点，通过作截面来解决.如果外切的是多面体，则作截面时主要抓住多面体过球心的对角面来作；把一个多面体的几个顶点放在球面上即为球的内接问题解决这类问题的关键是抓住内接的特点，即球心到多面体的顶点的距离等于球的半径发挥好空间想象力，借助于数形结合进行转化，问题即可得解如果是一些特殊的几何体，如正方体、正四面体等可以借助结论直接求解,此时结论的记忆必须准确.外接球内切球问题1. （陕西理）一个正三棱锥的

20、四个顶点都在半径为1的球面上，其中底面的三个顶点在该球的一个大圆上，则该正三棱锥的体积是（ ）A B C D 答案B2. 直三棱柱的各顶点都在同一球面上，若,，则此球的表面积等于 。解:在中,可得,由正弦定理,可得外接圆半径r=2,设此圆圆心为，球心为，在中，易得球半径，故此球的表面积为.3正三棱柱内接于半径为的球，若两点的球面距离为，则正三棱柱的体积为 答案 84.表面积为 的正八面体的各个顶点都在同一个球面上，则此球的体积为A B C D答案 A【解析】此正八面体是每个面的边长均为的正三角形，所以由知，则此球的直径为，故选A。5.已知正方体外接球的体积是，那么正方体的棱长等于（ ）A.2

21、B. C. D.答案 D6.（山东卷）正方体的内切球与其外接球的体积之比为 ( )A. 1 B. 13 C. 13 D. 19答案 C7.（海南、宁夏理科）一个六棱柱的底面是正六边形，其侧棱垂直底面已知该六棱柱的顶点都在同一个球面上，且该六棱柱的体积为，底面周长为3，则这个球的体积为答案 8. （天津理）一个长方体的各顶点均在同一球的球面上，且一个顶点上的三条棱的长分别为1，2，3，则此球的表面积为答案 9.（全国理）一个正四棱柱的各个顶点在一个直径为2 cm的球面上。如果正四棱柱的底面边长为1 cm，那么该棱柱的表面积为 cm2.答案 ABCPDEF10.（辽宁）如图，半径为2的半球内有一内

22、接正六棱锥，则此正六棱锥的侧面积是_答案 11.（辽宁省抚顺一中）棱长为2的正四面体的四个顶点都在同一个球面上，若过该球球心的一个截面如图，则图中三角形(正四面体的截面)的面积是 .答案 12.（枣庄一模）一个几何体的三视图如右图所示，则该几何体外接球的表面积为（ ）ABCD以上都不对答案C13.(吉林省吉林市)设正方体的棱长为，则它的外接球的表面积为（ ）A B2 C4D答案C14（新课标理）已知三棱锥的所有顶点都在球的求面上,是边长为的正三角形,为球的直径,且;则此棱锥的体积为（）ABCD15（辽宁文）已知点P,A,B,C,D是球O表面上的点,PA平面ABCD,四边形ABCD是边长为2正方

23、形.若PA=2,则OAB的面积为_.leaders driving a vehicle accident caused by road accidents, 1, 1 people killed and direct economic losses amounting to more than 100,000 yuan. Third, high efficiency and low cost of the bus. Surveys show that, the operating costs of taxis for the 8200/. Is a fundamental priority o

24、f the reform, it is a difficult problem that must be solved in the reform process. Clearly, the post consumption averages three years before as a base and fine-tuned on the basis of this single practices must be improved. Improvements to adhere to three principles: first, under the existing policy p

25、rovisions approved for public servants ' duty consumption standards, calibration, is not contrary to policy. Second, according to the local financial situation and people's sustainability, public servants ' duty consumption standards approved, both financial reach, and people passing thr

26、ough. Third, according to the operational needs of civil servants responsible for authorized public servants ' duty consumption standards, both high and low positions, but also the nature of the work and the workload. In reform of method Shang, approved civil servants positions consumption stand

27、ard to "big unified, and small dispersed" suitable, that most positions consumption project should according to policy provides proposed unified standard, consider to ranks, and units and the work task of differences sex, unified of standard should has elastic of and dynamic of, makes regi

28、ons, and units in implementation unified standard Shi has must of flexible disposal right; but since set of standard must after financial, and audit, sector audit approved Hou to implementation. (C) reform package. Public servants ' duty consumption elasticity of consumption to limit consumption

29、, turning mess into kitchen after dinner, follow the civil servant with a how to use personal title subsidies, or how to manage a personal duty consumption use of subsidies. Therefore, we must establish and perfect with public servants ' duty consumption monetization reform support of the series of governance systems such as the civil service examination of responsibility, system, low fault investigat