已知正四棱柱的侧面积为24,体积为12,其8个顶点在球O的表面上,则该球的表面积等于 .
【答案】
分析:设正四棱柱的底面边长为a,高为h,根据题意建立关于a、h的方程组,解之得a=2且h=3,由此算出正四棱柱的对角线长为
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125533113262342/SYS201310251255331132623012_DA/0.png)
,得外接球的半径R=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125533113262342/SYS201310251255331132623012_DA/1.png)
,再用球的表面积公式即可算出正四棱柱的外接球表面积.
解答:解:设正四棱柱的底面边长为a,高为h,可得
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125533113262342/SYS201310251255331132623012_DA/images2.png)
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125533113262342/SYS201310251255331132623012_DA/2.png)
,解之得a=2且h=3
∴该正四棱柱的对角线长为
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125533113262342/SYS201310251255331132623012_DA/3.png)
=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125533113262342/SYS201310251255331132623012_DA/4.png)
由此可得外接球的半径R=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125533113262342/SYS201310251255331132623012_DA/5.png)
∴该正四棱柱的外接球表面积S=4πR
2=17π
故答案为:17π
点评:本题给出正四棱柱的侧面积和体积,求它的外接球表面积,着重考查了正四棱柱的性质、棱柱的外接球表面积的求法等知识,属于基础题.