Cube and Cuboid: Surface Area & Volume

Cube and Cuboid: Lateral Surface Area, Total Surface Area, Volume

A cube and a cuboid are both three-dimensional shapes (solids). They are both examples of rectangular prisms.

A cube is a special type of cuboid where all six faces are squares, and all edges have the same length.

A cuboid (also called a rectangular prism) is a solid with six rectangular faces. It has three pairs of identical faces. These faces meet at right angles. A cuboid is defined by its length ($l$), width ($w$), and height ($h$).

Formulae

Cube (with side length $a$):

• Lateral Surface Area (LSA): $4a^2$
• Total Surface Area (TSA): $6a^2$
• Volume (V): $a^3$

Cuboid (with length $l$, width $w$, and height $h$):

• Lateral Surface Area (LSA): $2h(l + w)$
• Total Surface Area (TSA): $2(lw + lh + wh)$
• Volume (V): $lwh$

Examples

Example-1: A cube has a side length of 5 cm. Calculate its:

• Lateral Surface Area: $4 \times 5^2 = 4 \times 25 = 100 \, cm^2$
• Total Surface Area: $6 \times 5^2 = 6 \times 25 = 150 \, cm^2$
• Volume: $5^3 = 125 \, cm^3$

Example-2: A cuboid has a length of 6 cm, a width of 4 cm, and a height of 3 cm. Calculate its:

• Lateral Surface Area: $2 \times 3 \times (6 + 4) = 6 \times 10 = 60 \, cm^2$
• Total Surface Area: $2 \times (6 \times 4 + 6 \times 3 + 4 \times 3) = 2 \times (24 + 18 + 12) = 2 \times 54 = 108 \, cm^2$
• Volume: $6 \times 4 \times 3 = 72 \, cm^3$

Common mistakes by students

• Confusing the formulas for LSA and TSA.
• Using the wrong units (e.g., forgetting to square the units for area and cube the units for volume).
• Miscalculating the arithmetic, especially when dealing with multiple steps.
• Not understanding the difference between LSA and TSA (LSA only considers the sides, TSA includes all faces).

Real Life Application

These concepts are widely used in real-life scenarios, such as:

• Construction: Calculating the amount of paint needed to cover the walls of a room (LSA) or the entire surface (TSA), or calculating the volume of concrete needed for a foundation.
• Packaging: Determining the amount of cardboard needed to make a box (TSA) or the volume a box can hold.
• Architecture/Interior Design: Planning room layouts, calculating the amount of materials (tiles, flooring, etc.) needed for projects.
• Shipping: Calculating the volume of packages to determine shipping costs.

Fun Fact

The Rubik’s Cube is a classic example of a cube puzzle. It has a volume of 27 smaller cubes ($3 \times 3 \times 3$)!

Recommended YouTube Videos for Deeper Understanding

Q.1 The volume of a cube is $1331 cm^3$. Find the length of its side.

Check Solution

Ans: B

The volume of a cube is $s^3$, where s is the side length. So, $s = \sqrt[3]{1331} = 11$.

Q.2 The dimensions of a cuboid are $10 cm$, $8 cm$, and $5 cm$. Find the lateral surface area.

Check Solution

Ans: A

Lateral Surface Area of a cuboid is $2h(l + b) = 2 \times 5 \times (10 + 8) = 10 \times 18 = 180 cm^2$

Q.3 A cuboid has a total surface area of $1900 cm^2$. The length and breadth of the cuboid are $20 cm$ and $15 cm$ respectively. Find the height of the cuboid.

Check Solution

Ans: A

Total Surface Area of a cuboid = $2(lb + bh + hl)$. $1900 = 2(20 \times 15 + 15h + 20h)$. $1900 = 2(300 + 35h)$. $950 = 300 + 35h$. $650 = 35h$. $h = 10 cm$

Q.4 The length, breadth, and height of a cuboid are in the ratio $5:3:2$. If the volume of the cuboid is $240 cm^3$, then the length of the cuboid is:

Check Solution

Ans: A

Let the length be $5x$, breadth be $3x$, and height be $2x$. Volume = $l \times b \times h = 5x \times 3x \times 2x = 30x^3$. $30x^3 = 240$, $x^3 = 8$, $x = 2$. Length = $5x = 5 \times 2 = 10 cm$

Q.5 If the length of each edge of a cube is doubled, then the ratio of the volume of the new cube to the volume of the original cube is:

Check Solution

Ans: C

Let the original edge be $s$. Original volume is $s^3$. New edge is $2s$. New volume is $(2s)^3 = 8s^3$. Ratio is $8s^3/s^3 = 8:1$

Next Topic: Right Circular Cylinder: Surface Area & Volume