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数学の楽園
＞中学１年
＞立体図形
＞Ｂ辺と面

辺と面の関係

立体図形では、辺と面は、３つの関係になれる。交わるか、平行か、含まれるかだ。

例えば、直方体ＡＢＣＤ－ＥＦＧＨで、辺と面の関係を、考えていこう。

\begin{xy} (-3,27)*{A}="A", (23,15)*{B}="A", (-23,15)*{D}="A", (-3,-3)*{C}="A", (-3,9)*{E}="A", (23,-9)*{F}="A", (-23,-9)*{H}="A", (-3,-21)*{G}="A", (0,0)*{\bullet}="A", (20,12)*{\bullet}="A", (-20,12)*{\bullet}="A", (20,-6)*{\bullet}="A", (-20,-6)*{\bullet}="A", (0,-18)*{\bullet}="A", (0,6)*{\bullet}="A", (0,24)*{\bullet}="A", {(0,0) \ar @{-}(20,12)}, {(0,0) \ar @{-}(0,-18)}, {(0,0) \ar @{-}(-20,12)}, {(0,24) \ar @{-}(20,12)}, {(0,24) \ar @{-}(-20,12)}, {(20,12) \ar @{-}(20,-6)}, {(-20,12) \ar @{-}(-20,-6)}, {(0,-18) \ar @{-}(20,-6)}, {(0,-18) \ar @{-}(-20,-6)}, {(0,6) \ar @{.}(20,-6)}, {(0,6) \ar @{.}(-20,-6)}, {(0,6) \ar @{.}(0,24)}, \end{xy}

辺と面が交わる

辺ＡＢが交わるのは、面辺ＡＤＨＥと面ＢＣＧＨだ。

交わるとは、１点を共有する、ともいえる。

\begin{xy} (-3,27)*{A}="A", (23,15)*{B}="A", (-23,15)*{D}="A", (-3,-3)*{C}="A", (-3,9)*{E}="A", (23,-9)*{F}="A", (-23,-9)*{H}="A", (-3,-21)*{G}="A", (0,0)*{\bullet}="A", (20,12)*{\bullet}="A", (-20,12)*{\bullet}="A", (20,-6)*{\bullet}="A", (-20,-6)*{\bullet}="A", (0,-18)*{\bullet}="A", (0,6)*{\bullet}="A", (0,24)*{\bullet}="A", {(0,0) \ar @{-}(20,12)}, {(0,0) \ar @{-}(0,-18)}, {(0,24) \ar @{-}(20,12)}, {(0,24) \ar @{-}(-20,12)}, {(20,12) \ar @{-}(20,-6)}, {(-20,12) \ar @{-}(-20,-6)}, {(0,-18) \ar @{-}(20,-6)}, {(0,6) \ar @{.}(-20,-6)}, {(0,6) \ar @{.}(0,24)}, \end{xy}

＜例題 \( \Large 1 \) ＞直方体ＡＢＣＤ－ＥＦＧＨで、辺ＢＦと交わる面を、すべて書きなさい。

\begin{xy} (-3,27)*{A}="A", (23,15)*{B}="A", (-23,15)*{D}="A", (-3,-3)*{C}="A", (-3,9)*{E}="A", (23,-9)*{F}="A", (-23,-9)*{H}="A", (-3,-21)*{G}="A", (0,0)*{\bullet}="A", (20,12)*{\bullet}="A", (-20,12)*{\bullet}="A", (20,-6)*{\bullet}="A", (-20,-6)*{\bullet}="A", (0,-18)*{\bullet}="A", (0,6)*{\bullet}="A", (0,24)*{\bullet}="A", {(0,0) \ar @{-}(20,12)}, {(0,0) \ar @{-}(0,-18)}, {(0,0) \ar @{-}(-20,12)}, {(0,24) \ar @{-}(20,12)}, {(0,24) \ar @{-}(-20,12)}, {(20,12) \ar @{-}(20,-6)}, {(-20,12) \ar @{-}(-20,-6)}, {(0,-18) \ar @{-}(20,-6)}, {(0,-18) \ar @{-}(-20,-6)}, {(0,6) \ar @{.}(20,-6)}, {(0,6) \ar @{.}(-20,-6)}, {(0,6) \ar @{.}(0,24)}, \end{xy}

＜解答 \( \Large 1 \) ＞
    面ＡＢＣＤ    面ＥＦＧＨ

辺と面が平行

辺ＡＢが平行なのは、面辺ＣＤＨＧと面ＥＦＧＨだ。

平行な辺と平行な面は、どこまで延長しても、交わらない。

\begin{xy} (-3,27)*{A}="A", (23,15)*{B}="A", (-23,15)*{D}="A", (-3,-3)*{C}="A", (-3,9)*{E}="A", (23,-9)*{F}="A", (-23,-9)*{H}="A", (-3,-21)*{G}="A", (0,0)*{\bullet}="A", (20,12)*{\bullet}="A", (-20,12)*{\bullet}="A", (20,-6)*{\bullet}="A", (-20,-6)*{\bullet}="A", (0,-18)*{\bullet}="A", (0,6)*{\bullet}="A", (0,24)*{\bullet}="A", {(0,0) \ar @{-}(-20,12)}, {(0,0) \ar @{-}(0,-18)}, {(0,24) \ar @{-}(20,12)}, {(-20,12) \ar @{-}(-20,-6)}, {(-20,-6) \ar @{-}(0,-18)}, {(-20,-6) \ar @{.}(0,6)}, {(0,6) \ar @{.}(20,-6)}, {(0,-18) \ar @{-}(20,-6)}, \end{xy}

＜例題 \( \Large 2 \) ＞直方体ＡＢＣＤ－ＥＦＧＨで、辺ＢＦと平行な面を、すべて書きなさい。

\begin{xy} (-3,27)*{A}="A", (23,15)*{B}="A", (-23,15)*{D}="A", (-3,-3)*{C}="A", (-3,9)*{E}="A", (23,-9)*{F}="A", (-23,-9)*{H}="A", (-3,-21)*{G}="A", (0,0)*{\bullet}="A", (20,12)*{\bullet}="A", (-20,12)*{\bullet}="A", (20,-6)*{\bullet}="A", (-20,-6)*{\bullet}="A", (0,-18)*{\bullet}="A", (0,6)*{\bullet}="A", (0,24)*{\bullet}="A", {(0,0) \ar @{-}(20,12)}, {(0,0) \ar @{-}(0,-18)}, {(0,0) \ar @{-}(-20,12)}, {(0,24) \ar @{-}(20,12)}, {(0,24) \ar @{-}(-20,12)}, {(20,12) \ar @{-}(20,-6)}, {(-20,12) \ar @{-}(-20,-6)}, {(0,-18) \ar @{-}(20,-6)}, {(0,-18) \ar @{-}(-20,-6)}, {(0,6) \ar @{.}(20,-6)}, {(0,6) \ar @{.}(-20,-6)}, {(0,6) \ar @{.}(0,24)}, \end{xy}

＜解答 \( \Large 2 \) ＞
    面ＡＤＨＥ    面ＣＤＨＧ

辺が面に含まれる

辺ＡＢが含まれるのは、面辺ＡＢＣＤと面ＡＢＦＥだ。

\begin{xy} (-3,27)*{A}="A", (23,15)*{B}="A", (-23,15)*{D}="A", (-3,-3)*{C}="A", (-3,9)*{E}="A", (23,-9)*{F}="A", (0,0)*{\bullet}="A", (20,12)*{\bullet}="A", (-20,12)*{\bullet}="A", (20,-6)*{\bullet}="A", (0,6)*{\bullet}="A", (0,24)*{\bullet}="A", {(0,0) \ar @{-}(20,12)}, {(0,0) \ar @{-}(-20,12)}, {(0,24) \ar @{-}(20,12)}, {(0,24) \ar @{-}(-20,12)}, {(20,12) \ar @{-}(20,-6)}, {(0,6) \ar @{.}(20,-6)}, {(0,6) \ar @{.}(0,24)}, \end{xy}

＜例題 \( \Large 3 \) ＞直方体ＡＢＣＤ－ＥＦＧＨで、辺ＢＦが含まれる面を、すべて書きなさい。

\begin{xy} (-3,27)*{A}="A", (23,15)*{B}="A", (-23,15)*{D}="A", (-3,-3)*{C}="A", (-3,9)*{E}="A", (23,-9)*{F}="A", (-23,-9)*{H}="A", (-3,-21)*{G}="A", (0,0)*{\bullet}="A", (20,12)*{\bullet}="A", (-20,12)*{\bullet}="A", (20,-6)*{\bullet}="A", (-20,-6)*{\bullet}="A", (0,-18)*{\bullet}="A", (0,6)*{\bullet}="A", (0,24)*{\bullet}="A", {(0,0) \ar @{-}(20,12)}, {(0,0) \ar @{-}(0,-18)}, {(0,0) \ar @{-}(-20,12)}, {(0,24) \ar @{-}(20,12)}, {(0,24) \ar @{-}(-20,12)}, {(20,12) \ar @{-}(20,-6)}, {(-20,12) \ar @{-}(-20,-6)}, {(0,-18) \ar @{-}(20,-6)}, {(0,-18) \ar @{-}(-20,-6)}, {(0,6) \ar @{.}(20,-6)}, {(0,6) \ar @{.}(-20,-6)}, {(0,6) \ar @{.}(0,24)}, \end{xy}

＜解答 \( \Large 3 \) ＞
    面ＡＢＦＥ    面ＢＣＧＦ

次へ

Ｃ面と面
Ｄ柱体と錐体
Ｅ正多面体
Ｆ投影図
Ｇ展開図
Ｈ立体の表面積
Ｉ立体の体積
Ｊ球
Ｋ回転体
Ｌ立体の切断

前へ

Ａ辺と辺
講座一覧に戻る