Volume & Surface Area

For shapes with hollow spaces, holes, or cutouts, we subtract the volumes:

💡 Stickers for the Top, Front, and Right faces hovering outside the box.

• The top face loses an area of $w_{\text{inner}} \times l_{\text{outer}}$, but this is exactly replaced by the bottom of the cut channel.
• The front and back faces each lose a notch of size $w_{\text{inner}} \times h_{\text{inner}}$, reducing the surface area by $2 \times (w_{\text{inner}} \times h_{\text{inner}})$.
• Inside the channel, two new side wall faces are exposed, each of area $l_{\text{outer}} \times h_{\text{inner}}$, increasing the surface area by $2 \times (l_{\text{outer}} \times h_{\text{inner}})$.

💡 Circular stickers for the Top and Bottom bases, and a flat rectangular sticker for the curved side hovering outside.

• Two sector bases: each sector has area $\frac{\theta}{360^\circ} \pi r^2$, giving a total base area of $2 \times \frac{\theta}{360^\circ} \pi r^2 = \frac{\theta}{180^\circ} \pi r^2$.
• Two flat rectangular side faces: each rectangle has width $r$ and height $h$, giving a total area of $2 \times r h$.
• One curved lateral face: a rectangular sticker of height $h$ and length equal to the sector's arc length $\frac{\theta}{180^\circ} \pi r$, giving an area of $\frac{\theta}{180^\circ} \pi r h$.

💡 Two sector stickers, two rectangle stickers for the flat sides, and one curved sticker of length $\pi r \theta / 180$ hovering outside.

• Base Area ($B$): For a square base of side $s$, $B = s^2$.
• Lateral Area ($L$): Composed of four identical triangles with base $s$ and slant height $l$. The total lateral area is $L = 4 \times \left(\frac{1}{2} s l\right) = 2 s l$.
• Slant Height ($l$): If the perpendicular height $h$ is given instead of the slant height, calculate the slant height using the Pythagorean theorem on half the base side: $l = \sqrt{h^2 + \left(\frac{s}{2}\right)^2}$.

• Base Area ($B$): For a circle of radius $r$, $B = \pi r^2$.
• Lateral Area ($L$): The unwrapped curved side forms a circular sector with area $L = \pi r l$, where $l$ is the slant height.
• Slant Height ($l$): If height $h$ is given, find the slant height using the Pythagorean theorem: $l = \sqrt{h^2 + r^2}$.

• A line or boundary is 1-dimensional (1D) — it has only length.
• A flat surface is 2-dimensional (2D) — it has width and height.
• A solid object is 3-dimensional (3D) — it has length, width, and height.

• Metric Volume Units: $\text{mm}^3$ (cubic millimeter), $\text{cm}^3$ (cubic centimeter), $\text{dm}^3$ (cubic decimeter), $\text{m}^3$ (cubic meter).
• US Customary Volume Units: $\text{in}^3$ (cubic inch), $\text{ft}^3$ (cubic foot), $\text{yd}^3$ (cubic yard).

• 1D Dimensions (radius, height, perimeter, side length) scale linearly by $\mathbf{k}$. The ratio of the new dimension to the old is $k:1$.
• 2D Areas (surface area, base area, lateral area) scale quadratically by $\mathbf{k^2}$. The ratio of the new area to the old is $k^2:1$.
• 3D Volumes scale cubically by $\mathbf{k^3}$. The ratio of the new volume to the old is $k^3:1$.
  
  Example: If you double all linear dimensions of a box ($k = 2$):
• The height/width/length doubles (ratio $2:1$).
• The surface area increases by $2^2 = 4$ times (ratio $4:1$).
• The volume increases by $2^3 = 8$ times (ratio $8:1$).

• $2.2 \times 10^4$ (or $2.2\text{e+}4$ / $2.2\text{e}4$): represents $2.2 \times 10{,}000 = 22{,}000$. The exponent $+4$ tells us to move the decimal point 4 places to the right.
• $2.2 \times 10^{-4}$ (or $2.2\text{e-}4$): represents $2.2 \times 0.0001 = 0.00022$. The exponent $-4$ tells us to move the decimal point 4 places to the left.

Mastering SealMath: Ratio & Unit Answers