Question

The lateral surface area of a cube is 324 cm ² . Find its volume and total surface area​

Answer 1

Given :

• Lateral Surface Area of Cube = 324 cm²

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To Find :

• Volume of the Cube = ?
• Total Surface Area of the Cube = ?

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Solution :

~ Formula Used :

• Lateral Surface Area :

$\large{\red{\dashrightarrow}} \: \: {\underline{\boxed{\purple{\sf{ Lateral \: Surface \: Area {\small_{(Cube)}} = 4 a² }}}}}$

• Volume :

$\large{\red{\dashrightarrow}} \: \: {\underline{\boxed{\purple{\sf{ Volume{\small_{(Cube)}} = a³ }}}}}$

• Total Surface Area :

$\large{\red{\dashrightarrow}} \: \: {\underline{\boxed{\purple{\sf{ Total \: Surface \: Area {\small_{(Cube)}} = 6a² }}}}}$

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~ Calculating the Side :

${\longmapsto{\qquad{\sf{ LSA = 4a² }}}} \\ \\ \ {\longmapsto{\qquad{\sf{ 324 = 4 \times a² }}}} \\ \\ \ {\longmapsto{\qquad{\sf{ \cancel\dfrac{324}{4} = a² }}}} \\ \\ \ {\longmapsto{\qquad{\sf{ 81 = a² }}}} \\ \\ \ {\longmapsto{\qquad{\sf{ \sqrt{81} = a }}}} \\ \\ \ {\qquad{\textsf{ Side of the Cube = {\pink{\sf{ 9 \: cm}}}}}}$

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~ Calculating the Volume :

${\longmapsto{\qquad{\sf{ Volume = a³ }}}} \\ \\ \ {\longmapsto{\qquad{\sf{ Volume = 9³ }}}} \\ \\ \ {\longmapsto{\qquad{\sf{ Volume = 9 \times 9 \times 9 }}}} \\ \\ \ {\qquad{\textsf{ Volume of the Cube = {\green{\sf{ 729 \: cm³ }}}}}}$

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~ Calculating the Total Surface Area :

${\longmapsto{\qquad{\sf{ TSA = 6a² }}}} \\ \\ \ {\longmapsto{\qquad{\sf{ TSA = 6 \times 9² }}}} \\ \\ \ {\longmapsto{\qquad{\sf{ TSA = 6 \times 9 \times 9 }}}} \\ \\ \ {\longmapsto{\qquad{\sf{ TSA = 6 \times 81 }}}} \\ \\ \ {\qquad{\textsf{ Total Surface Area of the Cube = {\red{\sf{ 486 \: cm² }}}}}}$

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Therefore :

❝ Volume of the Cube is 729 cm³ and it's Total Surface Area is 486 cm² .❞

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Answer 2

Answer:

Diagram :

$\setlength{\unitlength}{4mm}\begin{picture}(10,6)\thicklines\put(0,1){\line(0,1){10}}\put(0,1){\line(1,0){10}}\put(10,1){\line(0,1){10}}\put(0,11){\line(1,0){10}}\put(0,11){\line(1,1){5}}\put(10,11){\line(1,1){5}}\put(10,1){\line(1,1){5}}\put(0,1){\line(1,1){5}}\put(5,6){\line(1,0){10}}\put(5,6){\line(0,1){10}}\put(5,16){\line(1,0){10}}\put(15,6){\line(0,1){10}}\put(4.6,-0.5){\bf\large{9\ cm}}\put(13.5,3){\bf\large{9\ cm}}\put(-4,5.8){\bf\large{9\ cm}}\end{picture}$

↝ The diagram of cube is given above. See this latex diagram on website Brainly.in.

$\begin{gathered}\end{gathered}$

Given :

• ➠ The lateral surface area of a cube is 324 cm².

$\begin{gathered}\end{gathered}$

To Find :

• ➠ Side
• ➠ Volume
• ➠ Lateral surface area

$\begin{gathered}\end{gathered}$

Concept :

↝ Here the concept of lateral surface area of cube has been used. We've been given that the lateral surface area of cube is 324 cm². With this information, we've been asked to find out the side, LSA and volume of the cube.

$\begin{gathered}\end{gathered}$

Using Formulas :

${\longrightarrow{\underline{\boxed{\sf{ LSA \: of \: cube= 4{a}^{2} }}}}}$

${\longrightarrow{\underline{\boxed{\sf{ Volume = {a}^{3}}}}}}$

${ \longrightarrow{\underline{\boxed{\sf{TSA \: of \: cube= 6 {a}^{2}}}}}}$

• → a = side
• → V = Volume
• → LSA = Lateral surface area
• → TSA = Total surface area

$\begin{gathered}\end{gathered}$

Solution :

Finding, the side of cube by substituting the values in the formula :

$\longrightarrow \: \:{\sf{LSA \: of \: cube = 4(a)^{2}}}$

$\longrightarrow \: \:{\sf{324 = 4(a)^{2}}}$

$\longrightarrow \: \:{\sf{(a)^{2} = {\dfrac{324}{4}}}}$

$\longrightarrow \: \:{\sf{(a)^{2} = \cancel{\dfrac{324}{4}}}}$

$\longrightarrow \: \:{\sf{(a)^{2} = 81}}$

$\longrightarrow \: \:{\sf{a = \sqrt{81} }}$

$\longrightarrow \: \:{\sf{a = \sqrt{9 \times 9}}}$

$\longrightarrow \: \:{\sf{a = 9 \: cm}}$

$\bigstar \: {\red{\underline{\boxed{\sf{Side \: of \: cube = 9 \: cm}}}}}$

∴ The side of cube is 9 cm.

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Finding the volume of cube by substituting the values in the formula :

$\longrightarrow \: \:{\sf{ Volume \: of \: cube = {a}^{3}}}$

$\longrightarrow \: \:{\sf{ Volume \: of \: cube= {9}^{3}}}$

$\longrightarrow \: \:{\sf{ Volume \: of \: cube= {9 \times 9 \times 9}}}$

$\longrightarrow \: \:{\sf{ Volume \: of \: cube= {81 \times 9}}}$

$\longrightarrow \: \:{\sf{ Volume \: of \: cube= 729\: {cm}^{3}}}$

$\bigstar \: {\red{\underline{\boxed{\sf{Volume \: of \: cube = 729 \: {cm}^{3}}}}}}$

∴ The volume of cube is 729 cm³.

$\rule{300}{1.5}$

Finding the total surface area of cube by substituting the values in the formula :

$\longrightarrow \: \: {\sf{TSA \: of \: cube = 6(a)^{2}}}$

$\longrightarrow \: \: {\sf{TSA \: of \: cube = 6(9)^{2}}}$

$\longrightarrow \: \: {\sf{TSA \: of \: cube = 6(9 \times 9)}}$

$\longrightarrow \: \: {\sf{TSA \: of \: cube = 6(81)}}$

$\longrightarrow \: \: {\sf{TSA \: of \: cube = 6 \times 81}}$

$\longrightarrow \: \: {\sf{TSA \: of \: cube = 486 \: {cm}^{2}}}$

$\bigstar \: {\red{\underline{\boxed{\sf{TSA \: of \: cube = 486\: {cm}^{2}}}}}}$

∴ The total surface area of cube is 486 cm².

$\begin{gathered}\end{gathered}$

Learn More :

$\begin{gathered}\begin{array}{|c|c|c|}\cline{1-3}\bf Shape&\bf Volume\ formula&\bf Surface\ area formula\\\cline{1-3}\sf Cube&\tt l^3}&\tt 6l^2\\\cline{1-3}\sf Cuboid&\tt lbh&\tt 2(lb+bh+lh)\\\cline{1-3}\sf Cylinder&\tt {\pi}r^2h&\tt 2\pi{r}(r+h)\\\cline{1-3}\sf Hollow\ cylinder&\tt \pi{h}(R^2-r^2)&\tt 2\pi{rh}+2\pi{Rh}+2\pi(R^2-r^2)\\\cline{1-3}\sf Cone&\tt 1/3\ \pi{r^2}h&\tt \pi{r}(r+s)\\\cline{1-3}\sf Sphere&\tt 4/3\ \pi{r}^3&\tt 4\pi{r}^2\\\cline{1-3}\sf Hemisphere&\tt 2/3\ \pi{r^3}&\tt 3\pi{r}^2\\\cline{1-3}\end{array}\end{gathered}$

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