If the length of a diagonal of a cube is 12 root 3 cm find its surface area and volume

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1. Answer:

   Given: Length of the diagonal of a cube is [tex]\sf 12 \sqrt{3}[/tex] cm.

   To find: Surface area & Volume of cube?

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   ☯ Let side of cube be a cm.

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   Now,

   [tex]\dag\;{\underline{\frak{As\;we\;know\;that,}}}\\ \\[/tex]

   Diagonal of cube is given by,

   [tex]\star\;{\boxed{\sf{\pink{Diagonal_{\:(cube)} = \sqrt{3} \times side}}}}\\ \\[/tex]

   [tex]:\implies\sf 12 \sqrt{3} = \sqrt{3} \times \times a\\ \\ \\ :\implies\sf a = \dfrac{12 \cancel{\sqrt{3}}}{ \cancel{\sqrt{3}}}\\ \\ \\ :\implies{\underline{\boxed{\frak{\purple{a = 12\:cm}}}}}\;\bigstar\\ \\[/tex]

   [tex]\therefore\:{\underline{\sf{Length\:of\:Side\:of\:cube\:is\: {\textsf{\textbf{12\:cm}}}.}}}[/tex]

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   ⋆ Now, Finding Surface Area of cube,

   [tex]\star\;{\boxed{\sf{\pink{TSA_{\:(cube)} = 6 \times (side)^2}}}}\\ \\[/tex]

   [tex]:\implies\sf TSA_{\:(cube)} = 6 \times (12)^2\\ \\ \\:\implies\sf TSA_{\:(cube)} = 6 \times 12 \times 12\\ \\ \\ :\implies\sf TSA_{\:(cube)} = 6 \times 144\\ \\ \\:\implies{\underline{\boxed{\frak{\purple{TSA_{\:(cube)} = 864\:cm^2}}}}}\;\bigstar\\ \\[/tex]

   [tex]\therefore\:{\underline{\sf{Surface\:area\:of\:cube\:is\: \bf{864\:cm^2}.}}}[/tex]

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   ⋆ Now, Volume of cube,

   [tex]\star\;{\boxed{\sf{\pink{Volume_{\:(cube)} = (side)^3}}}}\\ \\[/tex]

   [tex]:\implies\sf Volume_{\:(cube)} = (12)^3\\ \\ \\ :\implies\sf Volume_{\:(cube)} = 12 \times 12 \times 12\\ \\ \\ :\implies{\underline{\boxed{\frak{\purple{Volume_{\:(cube)} = 1728\:cm^3}}}}}\;\bigstar\\ \\[/tex]

   [tex]\therefore\:{\underline{\sf{Volume\:of\:cube\:is\: \bf{1728\:cm^3}.}}}[/tex]

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2. Answer :–

   • The surface area of cube = 864cm²
   • The volume of cube = 1728cm³

   Step-by-step explanation:

   To Find:–

   • The surface area of cube
   • The volume of cube

   Solution:

   Given that,

   • The length of a diagonal of cube is 12√3 cm.

   We know that,

   Diagonal of cube = a√3 units,

   Where,

   • a = side of cube.

   ∴ The side of cube :-

   ⟶ a√3 = Diagonal of cube

   ⟶ a√3 = 12√3

   ⟶ a = 12cm.

   According the question,

   • The surface area of cube

   We know that,

   Surface area of cube = 6(a)² sq. units,

   ∴ The surface area is :-

   ⟶ 6(a)²

   ⟶ 6(12)²

   ⟶ 6 × 144

   ⟶ 864cm²

   • The volume of cube

   We know that,

   Volume of cube = (a)³ cubic units,

   ∴ The volume is :-

   ⟶ (a)³

   ⟶ (12)³

   ⟶ ( 12 × 12 × 12 )

   ⟶ ( 144 × 12 )

   ⟶ 1728cm³

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