Surface Area and Volume of Solids

Prisms

$V = Bh \quad \text{(Base area × height)}$ $SA = 2B + Ph \quad \text{(2 bases + lateral area)}$

Rectangular prism: $V = lwh$, $SA = 2(lw + lh + wh)$

Cylinders

$V = \pi r^2 h$ $SA = 2\pi r^2 + 2\pi rh$

Pyramids

$V = \frac{1}{3}Bh$ $SA = B + \frac{1}{2}Pl \quad \text{(Base + lateral area, } l = \text{slant height)}$

Cones

$V = \frac{1}{3}\pi r^2 h$ $SA = \pi r^2 + \pi r l$

Slant height: $l = \sqrt{r^2 + h^2}$

Spheres

$V = \frac{4}{3}\pi r^3 \quad SA = 4\pi r^2$

Composite Solids

Break into simpler shapes, add (or subtract) volumes.

Example: A cylinder with a hemisphere on top: $V = \pi r^2 h + \frac{2}{3}\pi r^3$

Cavalieri's Principle

If two solids have the same height and every cross-section at the same level has the same area, then they have the same volume.

Cross-Sections

| Solid | Horizontal Cut | Vertical Cut | |-------|---------------|--------------| | Cylinder | Circle | Rectangle | | Cone | Circle | Triangle | | Sphere | Circle | Circle | | Rectangular prism | Rectangle | Rectangle |

Effect of Scaling

If a solid is scaled by factor $k$:

• Surface area scales by $k^2$
• Volume scales by $k^3$

Example: Double all dimensions ($k = 2$):

• SA is $4$ times larger
• Volume is $8$ times larger

Common mistake: Don't confuse height $h$ (perpendicular to base) with slant height $l$ (along the lateral face)!