🧊 Solid Geometry Formulas
🎯 Volume Formulas
Cube
$$V = s^3$$
Usage: Find volume of cube with side length $s$ Micro-example: Cube with side 4 has volume $4^3 = 64$
Rectangular Prism
$$V = lwh$$
Usage: Find volume of rectangular box Micro-example: Box with dimensions $3 \times 4 \times 5$ has volume $3 \cdot 4 \cdot 5 = 60$
Cylinder
$$V = \pi r^2 h$$
Usage: Find volume of circular cylinder Micro-example: Cylinder with radius 3 and height 7 has volume $\pi \cdot 3^2 \cdot 7 = 63\pi$
Sphere
$$V = \frac{4}{3}\pi r^3$$
Usage: Find volume of sphere Micro-example: Sphere with radius 2 has volume $\frac{4}{3}\pi \cdot 2^3 = \frac{32\pi}{3}$
Cone
$$V = \frac{1}{3}\pi r^2 h$$
Usage: Find volume of circular cone Micro-example: Cone with radius 4 and height 6 has volume $\frac{1}{3}\pi \cdot 4^2 \cdot 6 = 32\pi$
Pyramid
$$V = \frac{1}{3}Bh$$
Usage: Find volume of pyramid with base area $B$ and height $h$ Micro-example: Square pyramid with base side 5 and height 8 has volume $\frac{1}{3} \cdot 5^2 \cdot 8 = \frac{200}{3}$
📐 Surface Area Formulas
Cube
$$SA = 6s^2$$
Usage: Find total surface area of cube Micro-example: Cube with side 3 has surface area $6 \cdot 3^2 = 54$
Rectangular Prism
$$SA = 2(lw + lh + wh)$$
Usage: Find total surface area of rectangular box Micro-example: Box $2 \times 3 \times 4$ has surface area $2(2 \cdot 3 + 2 \cdot 4 + 3 \cdot 4) = 2(6 + 8 + 12) = 52$
Cylinder
$$SA = 2\pi r^2 + 2\pi rh$$
Usage: Find total surface area of cylinder Micro-example: Cylinder with radius 2 and height 5 has surface area $2\pi \cdot 2^2 + 2\pi \cdot 2 \cdot 5 = 8\pi + 20\pi = 28\pi$
Sphere
$$SA = 4\pi r^2$$
Usage: Find surface area of sphere Micro-example: Sphere with radius 3 has surface area $4\pi \cdot 3^2 = 36\pi$
Cone
$$SA = \pi r^2 + \pi rl$$
Usage: Find total surface area of cone where $l$ is slant height Micro-example: Cone with radius 3 and slant height 5 has surface area $\pi \cdot 3^2 + \pi \cdot 3 \cdot 5 = 9\pi + 15\pi = 24\pi$
🔗 Special Relationships
Scaling Volume
$$V_{new} = V_{old} \times k^3$$
Usage: When all dimensions are scaled by factor $k$ Micro-example: Doubling all dimensions increases volume by $2^3 = 8$ times
Scaling Surface Area
$$SA_{new} = SA_{old} \times k^2$$
Usage: When all dimensions are scaled by factor $k$ Micro-example: Doubling all dimensions increases surface area by $2^2 = 4$ times
Cylinder Lateral Surface
$$SA_{lateral} = 2\pi rh$$
Usage: Just the curved surface, not including bases Micro-example: Cylinder with radius 4 and height 6 has lateral surface area $2\pi \cdot 4 \cdot 6 = 48\pi$
💡 Quick Reference
| Solid | Volume | Surface Area |
|---|---|---|
| Cube | $s^3$ | $6s^2$ |
| Rectangular Prism | $lwh$ | $2(lw + lh + wh)$ |
| Cylinder | $\pi r^2 h$ | $2\pi r^2 + 2\pi rh$ |
| Sphere | $\frac{4}{3}\pi r^3$ | $4\pi r^2$ |
| Cone | $\frac{1}{3}\pi r^2 h$ | $\pi r^2 + \pi rl$ |
| Pyramid | $\frac{1}{3}Bh$ | Base + lateral faces |
⚠️ Common Pitfalls
- Missing $\frac{1}{3}$ factor in cone and pyramid volume
- Confusing radius and diameter in circular solids
- Forgetting to include all faces in surface area
- Wrong slant height in cone surface area
- Unit conversion errors in calculations