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展開図

立体を、平面に展開した図を、展開図(てんかいず)と呼ぶ。

\begin{xy} (0,16)*{立方体}="A", {(0,0) \ar @{-}(9,6)}, {(0,0) \ar @{-}(0,-9)}, {(0,0) \ar @{-}(-9,6)}, {(0,12) \ar @{-}(9,6)}, {(0,12) \ar @{-}(-9,6)}, {(9,6) \ar @{-}(9,-3)}, {(-9,6) \ar @{-}(-9,-3)}, {(0,-9) \ar @{-}(9,-3)}, {(0,-9) \ar @{-}(-9,-3)}, {(0,3) \ar @{.}(9,-3)}, {(0,3) \ar @{.}(-9,-3)}, {(0,3) \ar @{.}(0,12)}, \end{xy} \begin{xy} (5,36)*{立方体の展開図}="A", (5,30)*{正方形が6枚}="A", {(0,-12) \ar @{-}(0,24)}, {(-12,0) \ar @{-}(36,0)}, {(12,-12) \ar @{-}(12,24)}, {(-12,12) \ar @{-}(36,12)}, {(0,24) \ar @{-}(12,24)}, {(0,-12) \ar @{-}(12,-12)}, {(-12,0) \ar @{-}(-12,12)}, {(24,0) \ar @{-}(24,12)}, {(36,0) \ar @{-}(36,12)}, \end{xy} \begin{xy} (10,30)*{円柱}="A", {(0,0) \ar @{-}(0,20)}, {(20,0) \ar @{-}(20,20)}, (0,-2)*+[o]{}="A";(20,-2)*+{}="B", "A";"B" **\crv{~**@{.} (0,3)&(10,6)&(20,3)}; (0,2)*+[o]{}="A";(20,2)*+{}="B", "A";"B" **\crv{~**@{-} (0,-3)&(10,-6)&(20,-3)}; (0,18)*+[o]{}="A";(20,18)*+{}="B", "A";"B" **\crv{~**@{-} (0,23)&(10,26)&(20,23)}; (0,22)*+[o]{}="A";(20,22)*+{}="B", "A";"B" **\crv{~**@{-} (0,17)&(10,14)&(20,17)}; \end{xy} \begin{xy} (15,51)*{円柱の展開図}="A", (15,45)*{円が2枚+4角形}="A", {(0,0) \ar @{-}(0,20)}, {(40,0) \ar @{-}(40,20)}, {(0,0) \ar @{-}(40,0)}, {(0,20) \ar @{-}(40,20)}, (10,30) *\cir<10mm>{}, (10,-10) *\cir<10mm>{}, \end{xy}

<例題 \( \Large 1 \) > 以下の3角柱の、展開図を書きなさい。

\begin{xy} (0,25)*{3角柱}="A", {(0,6) \ar @{.}(-6,0)}, {(0,6) \ar @{.}(6,0)}, {(6,0) \ar @{-}(-6,0)}, {(0,21) \ar @{-}(-6,15)}, {(0,21) \ar @{-}(6,15)}, {(6,15) \ar @{-}(-6,15)}, {(0,6) \ar @{.}(0,21)}, {(6,0) \ar @{-}(6,15)}, {(-6,0) \ar @{-}(-6,15)}, \end{xy}

<解答 \( \Large 1 \) >
\begin{xy} (10,30)*{3角柱の展開図}="A", {(0,-10) \ar @{-}(-6,0)}, {(0,-10) \ar @{-}(6,0)}, {(-6,0) \ar @{-}(30,0)}, {(0,25) \ar @{-}(-6,15)}, {(0,25) \ar @{-}(6,15)}, {(-6,15) \ar @{-}(30,15)}, {(6,0) \ar @{-}(6,15)}, {(-6,0) \ar @{-}(-6,15)}, {(18,0) \ar @{-}(18,15)}, {(30,0) \ar @{-}(30,15)}, \end{xy}

円錐の展開図

\begin{xy} (10,24)*{円錐}="A", {(0,0) \ar @{-}(10,15)}, {(20,0) \ar @{-}(10,15)}, (10,15)*{\bullet}="A", (20,0)*{\bullet}="A", (10,19)*{A}="A", (21,4)*{B}="A", (0,-2)*+[o]{}="A";(20,-2)*+{}="B", "A";"B" **\crv{~**@{.} (0,1)&(10,4)&(20,1)}; (0,2)*+[o]{}="A";(20,2)*+{}="B", "A";"B" **\crv{~**@{-} (0,-3)&(10,-5)&(20,-3)}; (10,-10)*{ABを母線と呼ぶ}="A", \end{xy} \begin{xy} (15,16)*{円錐の展開図}="A", (15,10)*{円+扇形}="A", (15,0) *\cir<15mm>{dl_ul}, (15,0)*{\bullet}="A", {(15,0) \ar @{-}(26,-11)}, (26,-11)*{\bullet}="A", {(15,0) \ar @{-}(4,-11)}, (4,-11)*{\bullet}="A", (15,4)*{A}="A", (29,-11)*{B}="A", (0,-11)*{B}="A", (15,-25) *\cir<10mm>{}, \end{xy}

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