长方体的三条棱长分别为 1 、 2 、 3 ，则其外接球的表面积为＿＿＿＿＿＿。 14 外接球与内切球外接球长方体体对角线新课标I卷 2023

14 π 长方体的体对角线就是外接球的直径： 2 R = 1 2 + 2 2 + 3 2 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"/> = 14 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"/> 。故 R = 2 14 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"/> ， R 2 = 4 14 。球表面积 S = 4 π R 2 = 4 π × 4 14 = 14 π 。

高考数学考点题库/空间几何体与线面关系/外接球与内切球

2023 年新课标I卷·第12题

填空题★★★☆☆ 中档改编题通法

长方体的三条棱长分别为

1

、

2

、

3

，则其外接球的表面积为＿＿＿＿＿＿。

答案

14 π

分步解析

1长方体的体对角线就是外接球的直径： $2 R = 1^{2} + 2^{2} + 3^{2} = 14$ 。
2故 $R = \frac{14}{2}$ ， $R^{2} = \frac{14}{4}$ 。
3球表面积 $S = 4 π R^{2} = 4 π \times \frac{14}{4} = 14 π$ 。

点评「墙角模型」：三条两两垂直的棱可以补成长方体，外接球直径等于体对角线，这是外接球问题的第一联想。

#外接球#长方体#体对角线

2023 年真题新课标I卷

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