圆台

一个圆台
一个圆台

圓台,又稱截頂圓錐圓亭,是几何学中研究的一类三维形体,指一个圆锥平行于它的底面的一个平面所截後,截面与底面之间的几何形体。截面也称为圆台的上底面,原来圆锥的底面称为下底面。随着圆锥形状不同,圆台的称呼也不相同。一般说到圆台都是指正圆台,也就是指正圆锥截出的圆台。正圆台和圆形有相同的对称结构。以下除非另作注明,“圆台”都指正圆台。

性质

体积

圆台的体积取决于两底面之间的距离(圆台的高),以及原来圆锥的体积。设為圆台的高,為棱台的上下底面半径, 為圆台的体积。由于圆台是由一个平面截去圆锥的一部分(也就是和原来圆锥相似的一个小圆锥)得到,所以计算体积的时候,可以先算出原来圆锥的体积,再减去和它相似的小圆锥的体积。圆锥被平行于底面的平面所截时,截面圆的半径与底面半径的比,等于小圆锥和原圆锥的高的比。假设原圆锥的高是,那么小圆锥的高是。也就是说:

所以:

圆台的体积等于原圆锥体积减去小圆锥的体积:

九章算術》記載的圓台體積公式:「上下周相乘,又各自乘,并之,以高乘之,三十六而一。」这是将圆周率的值取为3得到的。

表面积

圆台的侧面展开图是一个“扇面形”,也就是两个同心扇形的差。展开图的面积,就是两个扇形的面积差,其中是圆台的母线长度:

圆台的表面积St等于圆台的侧面积Sc加上两底的面积SuSd

参看

  • 棱台:平行于棱锥底面的平面截棱锥,截面和底面之间的部分;
  • 圆锥:圆的各个切线和圆外一点所成的平面包围得到的立体。
  • 平截头体:平行于锥体底面的平面截去锥体顶部后得到的几何体,分为棱台和圆台。

参考资料

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