1. Cuboid
A cuboid is a 3D shape with six rectangular faces. It has:
- length \(l\)
- breadth \(b\)
- height \(h\)
Opposite faces are equal in area, and the faces come in three pairs: \(l \times b\), \(b \times h\), and \(h \times l\).
Because all faces are rectangles, surface area calculations involve adding the areas of these rectangles.
2. Lateral Surface Area of a Cuboid
The Lateral Surface Area (LSA) includes only the four vertical faces of the cuboid. The top and bottom (roof and floor) are not included.
Each pair of opposite side faces has area:
- Two faces of \(l \times h\)
- Two faces of \(b \times h\)
So, the total lateral surface area is:
\( \text{LSA} = 2lh + 2bh = 2h(l + b) \)
2.1. Example
A cuboid has dimensions \(l = 10\,\text{cm}\), \(b = 6\,\text{cm}\), and \(h = 8\,\text{cm}\). Then:
\( \text{LSA} = 2h(l + b) = 2 \times 8 \times (10 + 6) = 16 \times 16 = 256\,\text{cm}^2 \)
3. Total Surface Area of a Cuboid
The Total Surface Area (TSA) includes all six faces of the cuboid.
The areas of the face pairs are:
- Two faces of area \(lb\)
- Two faces of area \(bh\)
- Two faces of area \(hl\)
So the TSA becomes:
\( \text{TSA} = 2(lb + bh + hl) \)
3.1. Why the Formula Works
Since a cuboid has three pairs of equal faces, we simply add the areas of all three different face types and multiply by 2. This gives the full outer surface—the part you would need to paint or wrap.
3.2. Example
For a cuboid with \(l = 12\,\text{cm}\), \(b = 5\,\text{cm}\), \(h = 9\,\text{cm}\):
\( \text{TSA} = 2(lb + bh + hl) = 2(12 \times 5 + 5 \times 9 + 9 \times 12) \)
\( = 2(60 + 45 + 108) = 2 \times 213 = 426\,\text{cm}^2 \)
4. Real-Life Applications
Cuboids appear in many real-world situations, such as rooms, boxes, tanks, and furniture.
- Use LSA when painting walls (excluding floor and ceiling).
- Use TSA when wrapping a gift box or polishing all outer surfaces.
Choosing between LSA and TSA depends on whether the top and bottom faces are involved.