📐 Geometry Formula Reference
This page serves as a quick-start guide for the most common geometry formulas used in middle school.
🟦 2D Shapes: Area and Perimeter
| Shape | Perimeter/Circumference | Area |
|---|---|---|
| Square | $P = 4s$ | $A = s^2$ |
| Rectangle | $P = 2(l + w)$ | $A = l \times w$ |
| Triangle | $P = a + b + c$ | $A = \frac{1}{2}bh$ |
| Circle | $C = 2\pi r$ | $A = \pi r^2$ |
| Trapezoid | $P = a + b_1 + c + b_2$ | $A = \frac{a+b}{2}h$ |
🧊 3D Shapes: Volume and Surface Area
1. Rectangular Prism
- Volume: $V = lwh$
- Surface Area: $SA = 2(lw + lh + wh)$
2. Cylinder
- Volume: $V = \pi r^2 h$
- Surface Area: $SA = 2\pi rh + 2\pi r^2$
3. Sphere
- Volume: $V = \frac{4}{3}\pi r^3$
- Surface Area: $SA = 4\pi r^2$
📐 Angles and Triangles
Pythagorean Theorem
For any right-angled triangle, the square of the hypotenuse ($c$) is equal to the sum of the squares of the other two sides ($a$ and $b$). $$a^2 + b^2 = c^2$$
Angle Sums
- Triangle: The interior angles always sum to $180^\circ$.
- Quadrilateral: The interior angles always sum to $360^\circ$.
💡 Geometry and Proportion Connection
Remember:
- Direct Proportion: If you double the radius of a circle, the circumference doubles (Linear).
- Square-Cube Law: If you double the side of a cube, the surface area increases by 4x ($2^2$), but the volume increases by 8x ($2^3$)!