Question

If the length of the diagonal of a cube is 12$\sqrt{3}$ cm, find it's surface area and volume.

Answer 1

Answer :

›»› The surface area of cube is 864 cm².

›»› The volume of cube is 1728 cm³.

Step-by-step explanation :

Given :

• Length of the diagonal of a cube = 12√3 cm.

To Find :

• Surface area of cube = ?
• Volume of cube = ?

Formula required :

Formula of diagonal of cube to calculate the side of cube is given by,

→ Diagonal of cube = a√3.

Here,

• a is the edge or side of cube.

Units,

• The unit of side is centimetre (cm).

Formula to calculate the surface area of cube is given by,

→ Surface area of cube = 6a².

Here,

• a is the edge or side of cube.

Units,

• The unit of side is centimetre (cm).

Formula to calculate the volume of cube is given by,

→ Volume of cube = a³.

Here,

• a is the edge or side of cube.

Units,

• The unit of side is centimetre (cm).

Solution :

As we are given with the diagonal of a cube then we know the required formula, that is,

→ Diagonal of cube = a√3.

By using the formula of diagonal of cube to calculate the side of cube and substituting the given values in the formula, we get :

→ 12√3 = a√3

→ 12 = a

→ a = 12

∴ The side of cube is 12 cm.

Now,

We know that, if we are given with the side of cube then we have the required formula, that is,

→ Surface area of cube = 6a².

By using the formula to calculate the surface area of cube and substituting the given values in the formula, we get :

→ Surface area of cube = 6 * (12)²

→ Surface area of cube = 6 * 12 * 12

→ Surface area of cube = 72 * 12

→ Surface area of cube = 864.

Hence, the surface area of cube is 864 cm².

We know that, if we are given with the side of cube then we have the required formula, that is,

→ Volume of cube = a³.

By using the formula to calculate the volume of cube and substituting the given values in the formula, we get :

→ Volume of cube = (12)³

→ Volume of cube = 12 * 12 * 12

→ Volume of cube = 144 * 12

→ Volume of cube = 1728.

Hence, the volume of cube is 1728 cm³.

Answer 2

Given: Length of the diagonal of a cube is $\sf 12 \sqrt{3}$ cm.

To find: Surface area & Volume of cube?

⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━

☯ Let side of cube be a cm.

⠀⠀⠀⠀

Now,

$\dag\;{\underline{\frak{As\;we\;know\;that,}}}$

Diagonal of cube is given by,

$\star\;{\boxed{\sf{\pink{Diagonal_{\:(cube)} = \sqrt{3} \times side}}}}$

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

$\therefore\:{\underline{\sf{Length\:of\:Side\:of\:cube\:is\: {\textsf{\textbf{12\:cm}}}.}}}$

⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━

⋆ Now, Finding Surface Area of cube,

$\star\;{\boxed{\sf{\pink{TSA_{\:(cube)} = 6 \times (side)^2}}}}$

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

$\therefore\:{\underline{\sf{Surface\:area\:of\:cube\:is\: \bf{864\:cm^2}.}}}$

⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━

⋆ Now, Volume of cube,

$\star\;{\boxed{\sf{\pink{Volume_{\:(cube)} = (side)^3}}}}$

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

$\therefore\:{\underline{\sf{Volume\:of\:cube\:is\: \bf{1728\:cm^3}.}}}$