Calculating Total Surface Area of a Cuboid

The problem asks us to find the total surface area of a cuboid given its dimensions: length, breadth, and height.

Understanding Cuboid Dimensions

A cuboid is a three-dimensional shape with six rectangular faces. The dimensions given are:

• Length (l) = 4 cm
• Breadth (b) = 6 cm
• Height (h) = 10 cm

Formula for Total Surface Area of a Cuboid

The total surface area of a cuboid is the sum of the areas of all six faces. Since opposite faces are identical, the formula is derived by adding the areas of the three unique pairs of faces:

Area of front and back faces = $2 \times (length \times height) = 2lh$

Area of top and bottom faces = $2 \times (length \times breadth) = 2lb$

Area of left and right faces = $2 \times (breadth \times height) = 2bh$

Total Surface Area (TSA) = Area of front/back + Area of top/bottom + Area of left/right

The formula for the total surface area (TSA) of a cuboid is:

TSA = $2(lb + bh + hl)$

Step-by-Step Calculation of Total Surface Area

Now, let's substitute the given dimensions into the formula:

• Length (l) = 4 cm
• Breadth (b) = 6 cm
• Height (h) = 10 cm

TSA = $2 \times ((4 \text{ cm} \times 6 \text{ cm}) + (6 \text{ cm} \times 10 \text{ cm}) + (10 \text{ cm} \times 4 \text{ cm}))$

First, calculate the area of each pair of faces:

• $l \times b = 4 \text{ cm} \times 6 \text{ cm} = 24 \text{ cm}^2$
• $b \times h = 6 \text{ cm} \times 10 \text{ cm} = 60 \text{ cm}^2$
• $h \times l = 10 \text{ cm} \times 4 \text{ cm} = 40 \text{ cm}^2$

Next, sum these areas:

$lb + bh + hl = 24 \text{ cm}^2 + 60 \text{ cm}^2 + 40 \text{ cm}^2 = 124 \text{ cm}^2$

Finally, multiply the sum by 2 to get the total surface area:

TSA = $2 \times 124 \text{ cm}^2 = 248 \text{ cm}^2$

Final Answer Confirmation

The total surface area of the cuboid with length 4 cm, breadth 6 cm, and height 10 cm is 248 cm².

Revision Table: Cuboid Properties

Additional Information: Understanding Surface Area

Surface area is a measure of the total area that the surface of a three-dimensional object occupies. For a cuboid, it's like unfolding the box and finding the total area of the resulting 2D shape (net). Calculating surface area is important in various real-world applications, such as determining the amount of paint needed to cover a wall, the material required to wrap a gift, or the amount of heat that can be radiated or absorbed by an object.

The units for surface area are always square units (e.g., cm², m², in²) because it represents an area.