Finding the Total Surface Area of a Cuboid

The question asks us to find the total surface area of a cuboid given its dimensions. A cuboid is a three-dimensional shape with six rectangular faces. The dimensions provided are 18 m, 6 m, and 3 m.

Let's identify the dimensions:

• Length (l) = 18 m
• Breadth (b) = 6 m
• Height (h) = 3 m

The total surface area (TSA) of a cuboid is the sum of the areas of all six faces. The formula for the total surface area of a cuboid is:

$$\text{TSA} = 2(lb + bh + hl)$$

Where:

• \(l\) is the length
• \(b\) is the breadth
• \(h\) is the height

Now, we substitute the given dimensions into the formula:

$$\text{TSA} = 2((18 \text{ m} \times 6 \text{ m}) + (6 \text{ m} \times 3 \text{ m}) + (3 \text{ m} \times 18 \text{ m}))$$

First, calculate the area of each pair of opposite faces:

• Area of two faces with dimensions l and b: \(lb = 18 \text{ m} \times 6 \text{ m} = 108 \text{ m}^2\)
• Area of two faces with dimensions b and h: \(bh = 6 \text{ m} \times 3 \text{ m} = 18 \text{ m}^2\)
• Area of two faces with dimensions h and l: \(hl = 3 \text{ m} \times 18 \text{ m} = 54 \text{ m}^2\)

Now, sum these areas and multiply by 2 (since there are two identical faces for each pair of dimensions):

$$\text{TSA} = 2(108 \text{ m}^2 + 18 \text{ m}^2 + 54 \text{ m}^2)$$

Add the areas inside the parenthesis:

$$108 + 18 + 54 = 180 \text{ m}^2$$

Finally, multiply by 2:

$$\text{TSA} = 2 \times 180 \text{ m}^2 = 360 \text{ m}^2$$

The total surface area of the cuboid is 360 m².

Let's check this result against the provided options:

• 90 m²
• 180 m²
• 324 m²
• 360 m²

Our calculated value, 360 m², matches the fourth option.

Revision Table: Cuboid Surface Area Calculation

Additional Information: Understanding Cuboids and Areas

A cuboid is a rectangular prism. All its angles are right angles, and opposite faces are identical rectangles. Understanding the different types of surface area is important in geometry.

• Total Surface Area (TSA): This is the sum of the areas of all six faces (top, bottom, front, back, left side, right side). It's calculated as \(2(lb + bh + hl)\).
• Lateral Surface Area (LSA): This is the sum of the areas of the four side faces, excluding the top and bottom faces. The formula is \(2(lh + bh)\) or \(2h(l+b)\). This is also sometimes called the area of the walls if the cuboid represents a room.
• Volume: The space occupied by the cuboid is its volume, calculated by multiplying its three dimensions: \(V = l \times b \times h\).

Surface area is always measured in square units (like m², cm², in²), while volume is measured in cubic units (like m³, cm³, in³).

Calculating surface area is useful in real-world applications such as determining the amount of paint needed to cover a box, the material required to build a container, or the heat transfer rate of a building.