**How do you find the Surface Area and Volume of a Cube**

If the length of each edge of a cube is ‘a’ units, then

- Total surface area of the cube = 6a
^{2}sq. units. - Lateral surface area = 4a
^{2} - Volume of the cube = a
^{3}cubic units - Diagonal of the cube = a units.
- Length of its diagonals = a√3
- Total length of its edges = 12a

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**Surface Area and Volume of a Cube Example Problems with Solutions**

**Example 1:** If each edge (side) of a cube is 8 cm ; find its surface area and lateral surface area.

**Solution: **Given each side of the cube (a) = 8 cm

∴ Its surface area = 6a^{2} = 6 × 8^{2} sq. cm

= 6 × 64 cm^{2} = 384 cm^{2}

Lateral surface area = 4a^{2} = 4 × 8^{2} sq. cm

= 4 × 64 cm^{2} = 256 cm^{2}

**Example 2:** A cubical box has each edge 10 cm and another cuboidal box is 12.5 cm long, 10 cm wide and 8 cm high.

(i) Which box has the greater lateral surface area and by how much ?

(ii) Which box has the smaller total surface area and by how much ?

**Solution: **(i) For the cubical box :

Each edge = 10 cm i.e., a = 10 cm

∴ Lateral surface area of the cubical box

= 4a^{2} = 4 × 10^{2} cm^{2} = 400 cm^{2}

For the cuboidal box :

ℓ = 12.5 cm, b = 10 cm & h = 8 cm

∴ Lateral surface area of the cuboidal box

= 2(ℓ + b) × h

= 2(12.5 + 10) × 8 cm^{2}

= 2 × 22.5 × 8 cm2 = 360 cm^{2}

Clearly, cubical box has greater lateral surface area by

400 cm^{2} – 360 cm^{2} = 40 cm^{2}

(ii) Total surface area of the cubical box:

= 6 a^{2} = 6 × 10^{2} sq. cm = 600 cm^{2}

Total surface area of the cuboidal box

= 2(ℓ × b + b × h + h × ℓ)

= 2(12.5 × 10 +10 × 8 + 8 × 12.5) cm^{2} = 2(125 + 80 + 100) cm^{2} = 610 cm^{2}

Clearly, cubical box has smaller surface area by

610 cm^{2} – 600 cm^{2} = 10 cm^{2}

**Example 3:** Find the volume of a solid cube of side 12 cm. If this cube is cut into 8 identical cubes, find :

(i) Volume of each small cube.

(ii) Side of each small cube.

(iii) Surface area of each small cube.

**Solution: **Since, side (edge) of the given solid cube = 12 cm.

∴ Volume of given solid cube = (edge)^{3}

= (12 cm)^{3} = 1728 cm^{3} Ans.

(i) As the given cube is cut into 8 identical cubes.

⇒ Vol. of 8 small cubes obtained

= Vol. of given cube = 1728 cm^{3}

⇒ Volume of each small cube

= \(\frac{{1728\,\,c{m^3}}}{8}\) = 216 cm^{3}

(ii) If edge (side) of each small cube = x cm

(edge)^{3} = Volume

⇒ x^{3} = 216 = 6 × 6 × 6 = 6^{3 }⇒ x = 6 cm

∴ Side of each small cube = 6 cm

(iii) Surface area of each small cube

= 6 × (edge)^{2}

= 6 × (6 cm)^{2} = 216 cm^{2}

**Example 4:** A river 3 m deep and 40 m wide is flowing at the rate of 2 km per hour. How much water will fall into the sea in a minute ? ** **

**Solution: **Volume of water that flows through a river, canal or pipe, etc., in unit time

= Area of cross-section × Speed of water through it.

x km/hr = x × \(\frac{5}{{18}}m/s\)

Reason : 1 km/hr = \(\frac{{1000m}}{{60 \times 60\,\,\,\sec }} = \frac{5}{{18}}m/s\)

Since, area of cross-section of the river

= Its depth × its width

= 3m × 40m = 120 m^{2}

And, speed of flow of water through the river

= 2 km/hr = 2 × \(\frac{5}{{18}}m/s = \frac{5}{9}m/s\)

∴ Vol. of water that flows through it in 1 sec.

= Area of cross-section × speed of water through it.

= 120 × \(\frac{5}{9}{m^3} = \frac{{200}}{3}{m^3}\)

⇒ Vol. of water that flows through it in

1 min. (60 sec.)

= \(\frac{{200}}{3} \times 60\,\,{m^3}\) = 4000 m^{3}

⇒ Vol. of water that will fall into the sea in a minute. = 4000 m^{3}

**Example 5:** The volume of a cube is numerically equal to its surface area. Find the length of its one side.

**Solution: **Let length of each side is a unit.

Given: Volume of the cube = Surface area of the cube.

⇒ a^{3} = 6a^{2} ⇒ a = 6

∴ The length of one side of the cube = 6 cm

**Example 6:** A solid cuboid has square base and height

12 cm. If its volume is 768 cm^{3}, find :

(i) side of its square base.

(ii) surface area.

**Solution: **(i) Let side of the square base be x cm

i.e., ℓ = b = x cm

ℓ × b × h = volume

⇒ x × x × 12 = 768

[Given, height = 12 cm]

⇒ x^{2} = \(\frac{{768}}{{12}} = 64\) ⇒ x = √64cm = 8 cm.

∴ Side of the square base = 8 cm

(ii) Now, ℓ = 8 cm, b = 8 cm and h = 12 cm

∴ Surface area = 2(ℓ × b + b × h + h × ℓ)

= 2(8 × 8 + 8 × 12 + 12 × 8) cm^{2} = 512 cm^{2}