Step 2: Polyhedral Nets, Cuboids, Cubes & Right Cylinders
Calculating boundary areas and volumetric capacities begins with unfolding three-dimensional skins into two-dimensional nets.
1. Cuboid and Cube Mensuration
Consider a cuboid with length $l$, breadth $b$, and height $h$:
- Lateral Surface Area (LSA / Area of 4 Walls):
$$\mathbf{\text{LSA} = 2h(l + b)}$$
- Total Surface Area (TSA):
$$\mathbf{\text{TSA} = 2(lb + bh + hl)}$$
- Volume ($V$):
$$\mathbf{V = l \cdot b \cdot h}$$
- Longest Space Diagonal ($d$):
$$\mathbf{d = \sqrt{l^2 + b^2 + h^2}}$$
- Cube (Side length $a$):
- $\text{LSA} = 4a^2$
- $\text{TSA} = 6a^2$
- $V = a^3$
- Diagonal $d = a\sqrt{3}$
2. Right Circular Cylinder
A cylinder of base radius $r$ and perpendicular height $h$:
- Curved Surface Area (CSA):
$$\mathbf{\text{CSA} = 2\pi rh}$$
- Total Surface Area (TSA):
$$\mathbf{\text{TSA} = 2\pi rh + 2\pi r^2 = 2\pi r(r + h)}$$
- Volume ($V$):
$$\mathbf{V = \pi r^2 h}$$
3. Hollow Cylinder (Pipes)
For outer radius $R$, inner radius $r$, and height $h$:
- $\text{CSA} = 2\pi h(R + r)$
- $\text{TSA} = 2\pi h(R + r) + 2\pi(R^2 - r^2)$
- $\text{Volume of Material} = \pi h(R^2 - r^2)$