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NSW Y8 Maths Volume Composite Shapes for Volume

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Composite Shapes for Volume Theory

\begin{multicols}{2} \textbf{Example 1}\\ Find the volume of the composite solid.\\ \begin{center} \begin{tikzpicture}[scale=0.55,line width=1pt] \coordinate[label=above:] (O) at (0,0); \coordinate[label=right:] (B) at (90:3); \coordinate[label=left:] (A) at (right:5); \begin{scope}[shift=(B)] \coordinate[label=left:] (C) at (right:5); \end{scope} \begin{scope}[shift=(B)] \coordinate[label=left:] (D) at (60:2); \end{scope} \begin{scope}[shift=(D)] \coordinate[label=left:] (F) at (right:3); \end{scope} \begin{scope}[shift=(C)] \coordinate[label=left:] (E) at (60:4); \end{scope} \begin{scope}[shift=(F)] \coordinate[label=left:] (G) at (60:2); \end{scope} \begin{scope}[shift=(G)] \coordinate[label=left:] (H) at (right:2); \end{scope} \begin{scope}[shift=(H)] \coordinate[label=left:] (I) at (-90:3); \end{scope} \draw (O)--(B) node[left,midway]{3 cm}--(C)--(A)--node[below,midway]{5 cm} cycle; \draw (B)--(D) node[left,midway]{2 cm} --(F)node[above,midway]{3 cm}--(G)--(H) edge (C); \draw (H)--(I)--(A) node[right,midway]{4 cm}; \end{tikzpicture} \end{center} \textbf{Example 1 solution}\\ \(\begin{aligned} \text { Area of top } & =3 \times 2+4 \times 2 \\ & =6+8 \\ & =14 \text{~cm}^2 \end{aligned}\)\\ \(\begin{aligned} \therefore \text { Volume of solid } & =14 \times 3 \\ & =42 \text{~cm}^3 \end{aligned}\) \columnbreak \textbf{Example 2}\\ Find the volume of the composite solid.\\ \begin{center} \begin{tikzpicture}[scale=0.45] \def\l{6} \def\w{6} \def\h{8} \coordinate (A) at (0,0,0); \coordinate (B) at (\l,0,0) ; \coordinate (C) at (\l,\w,0); \coordinate (D) at (0,\w,0); \coordinate (E) at (0,0,\h); \coordinate (F) at (\l,0,\h); \coordinate (G) at (\l,\w,\h); \coordinate (H) at (0,\w,\h); %Ecken \node[left= 1pt of A]{}; \node[right= 1pt of B]{}; \node[right= 1pt of C]{}; \node[left= 1pt of D]{}; \node[left= 1pt of E]{}; \node[right= 1pt of F]{}; \node[right= 1pt of G]{}; \node[left= 1pt of H]{}; %Kanten \draw[line width=1pt] (B) node[midway, below]{} -- (C) node[right,midway]{} -- (D) node[midway, above]{}; \draw[line width=1pt] (B) -- (F)node[midway,right,yshift=-2pt]{8 cm} -- (G) -- (C); \draw[line width=1pt] (G) -- (H) -- (D); \draw[line width=1pt] (E) -- (F) node[midway, below]{6 cm}; \draw[line width=1pt] (E) -- (H)node[left,midway]{6 cm}; \begin{scope}[shift=(230:3.2),yshift=0.5cm] \draw[line width=1pt] (0,0)--(4,0) node[below=-2pt,midway] {4 cm}--(2,4)--cycle; \draw[line width=1pt] (0,0) edge (30:2); \draw[|-|] (-0.4,0)--(-0.4,4) node[rectangle, fill=gray!10!white,pos=0.7,inner sep=1pt,right=-5pt] {\small 4 cm}; \end{scope} \end{tikzpicture} \end{center} \textbf{Example 2 solution}\\ For the triangular prism\\ \(\begin{aligned} \text { base area } & =\frac{1}{2} \times 4 \times 4 \\ & =8 \text{~cm}^2 \\ \therefore \text { Volume } & =8 \times 8 \\ & =64 \text{~cm}^3 \end{aligned}\)\\ For the rectangular prism.\\ \(\begin{aligned} \text { base area } & =6 \times 6 \\ & =36 \text{~cm}^2 \\ \therefore \text { Volume } & =36 \times 8 \\ & =288 \text{~cm}^3 \end{aligned}\)\\ \(\begin{aligned} \therefore \text { Volume of solid } & =288-64 \\ & =224 \text{~cm}^3 \end{aligned}\) \end{multicols}

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  • Composite Shapes for Volume - Video - Composite Shapes for Volume 1

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