【数学I】例題4.3.9：円錐に内接する球（One More）★★★

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右の図のように，底面の半径$3$，高さが4の直円錐があり，球Oが円錐の側面および底面に接し，底面との接点は底面の中心Mである．このとき，次の問いに答えよ．

(1) 円錐の母線の長さを求めよ．

(2) 球Oの半径を求めよ．

(3) 球Oの体積$V$と表面積$S$を求めよ．

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AとMを通る平面で円錐を切った切断面は，右の図のようになる．

(1) 求める母線の長さは，

\[
\sqrt{\mathrm{BM}^2+\mathrm{AM}^2}=\sqrt{3^2+4^2}=5
\]

(2) 球Oの半径を$r$，$\triangle\mathrm{ABC}$の面積を$S^\prime$とすると，

\[
S^\prime=\frac{1}{2}r(\mathrm{AB}+\mathrm{BC}+\mathrm{CA})=\frac{1}{2}r(6+2\cdot 5)=8r
\]

$S^\prime=\frac{1}{2}\cdot 6\cdot 4=12$であるから，$8r=12$

よって，$r=\frac{3}{2}$

(3) (2)より，$V=\frac{4}{3}\pi(\frac{3}{2})^3=\frac{9}{2}\pi,S=4\pi(\frac{3}{2})^2=9\pi$

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% 問題I4.3.9：円錐に内接する球（One More）★★★
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右の図のように，底面の半径$5$，高さが$12$の直円錐があり，球Oが円錐の側面および底面に接し，底面との接点は底面の中心Mである．このとき，次の問いに答えよ．

(1) 円錐の母線の長さを求めよ．

(2) 球Oの半径を求めよ．

(3) 球Oの体積$V$と表面積$S$を求めよ．

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% 問題I4.3.9の解答（One More）★★★
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AとMを通る平面で円錐を切った切断面は，右の図のようになる．

(1) 求める母線の長さは，$\sqrt{\mathrm{BM}^2+\mathrm{AM}^2}=\sqrt{5^2+12^2}=13$

(2) 球Oの半径を$r$，$\triangle\mathrm{ABC}$の面積を$S^\prime$とすると，

\[
S^\prime=\frac{1}{2}r(\mathrm{AB}+\mathrm{BC}+\mathrm{CA})=\frac{1}{2}r(10+2\cdot 13)=18r
\]

$S^\prime=\frac{1}{2}\cdot 10\cdot 12=60$であるから，$18r=60$

よって，$r=\frac{10}{3}$

(3) (2)より，$V=\frac{4}{3}\pi(\frac{10}{3})^3=\frac{4000}{81}\pi,S=4\pi(\frac{10}{3})^2=\frac{400}{9}\pi$

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% 例題I4.3.9の側注：円錐に内接する球（One More）★★★
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% 側注1
三平方の定理を用いる．

% 側注2
$S^\prime=\frac{1}{2}r(a+b+c)$

% 側注3
$S^\prime=\frac{1}{2}\mathrm{BC}\cdot\mathrm{AM}$

% 側注4
球の体積は$V=\frac{4}{3}\pi r^3$，表面積は$S=4\pi r^2$

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% 問題I4.3.9の側注：円錐に内接する球（One More）★★★
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% 側注1
三平方の定理を用いる．

% 側注2
$S^\prime=\frac{1}{2}r(a+b+c)$

% 側注3
$S^\prime=\frac{1}{2}\mathrm{BC}\cdot\mathrm{AM}$

% 側注4
球の体積は$V=\frac{4}{3}\pi r^3$，表面積は$S=4\pi r^2$

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% 例題I4.3.9の考え方・補足など：円錐に内接する球（One More）★★★
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% 考え方
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円錐の頂点Aと底面の円の中心Mを通るように，円錐を切った平面図形の切断面を考える（平面図形を取り出す）．

(1) 右の図のような切断面の二等辺三角形ABCにおいて，辺ABのような回転体の側面をつくる線分を母線という．

(2) 球の半径は$\triangle\mathrm{ABC}$の内接円の半径に一致することを利用する．

(3) 球の体積と表面積の公式$V=\frac{4}{3}\pi r^3,S=4\pi r^2$と(2)の結果を利用する．

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扱いやすい平面図形となるように，適切な平面で切断する．