2D, 3D, trigonometry, coordinate geometry — area, volume, surface area formulas.
2D Shapes — Area & Perimeter
| Shape | Area | Perimeter / Circumference |
|---|---|---|
| Square (side a) | a² | 4a |
| Rectangle (w×h) | w·h | 2(w+h) |
| Triangle (base b, height h) | ½·b·h | a+b+c (sides) |
| Equilateral triangle (side a) | (√3/4)·a² | 3a |
| Circle (radius r) | πr² | 2πr |
| Ellipse (a,b semi-axes) | πab | ≈ π·√(2(a²+b²)) (Ramanujan approx) |
| Trapezoid (parallel sides a,b; height h) | ½(a+b)·h | a+b+c+d |
| Parallelogram (base b, height h) | b·h | 2(a+b) |
| Regular hexagon (side a) | (3√3/2)·a² | 6a |
| Sector (radius r, angle θ rad) | ½r²θ | rθ + 2r |
3D Solids — Volume & Surface Area
| Solid | Volume | Surface Area |
|---|---|---|
| Cube (side a) | a³ | 6a² |
| Cuboid (l×w×h) | lwh | 2(lw+lh+wh) |
| Sphere (radius r) | (4/3)πr³ | 4πr² |
| Cylinder (r, h) | πr²h | 2πr(r+h) |
| Cone (r, h, slant l) | (1/3)πr²h | πr(r+l) |
| Pyramid (base B, height h) | (1/3)Bh | B + ½·perimeter·slant |
| Torus (R, r) | 2π²Rr² | 4π²Rr |
| Ellipsoid (a,b,c) | (4/3)πabc | ≈ 4π·((a^1.6b^1.6+…)/3)^(1/1.6) (Knud Thomsen) |
Trigonometry
| Identity | Description |
|---|---|
| sin²θ + cos²θ = 1 | Pythagorean identity |
| tan θ = sin θ / cos θ | Tangent definition |
| sin(A±B) = sinA·cosB ± cosA·sinB | Sum/difference |
| cos(A±B) = cosA·cosB ∓ sinA·sinB | Sum/difference |
| sin(2θ) = 2·sinθ·cosθ | Double angle |
| cos(2θ) = cos²θ − sin²θ = 1 − 2sin²θ | Double angle |
| a/sinA = b/sinB = c/sinC | Law of Sines |
| c² = a² + b² − 2ab·cosC | Law of Cosines |
| Area = ½ab·sinC | Triangle area from two sides |
Coordinate Geometry
| Formula | Description |
|---|---|
d = √((x₂−x₁)² + (y₂−y₁)²) | Distance between two points |
M = ((x₁+x₂)/2, (y₁+y₂)/2) | Midpoint |
slope m = (y₂−y₁)/(x₂−x₁) | Slope of line |
y = mx + b | Slope-intercept form |
(x−h)² + (y−k)² = r² | Circle with center (h,k) |
x²/a² + y²/b² = 1 | Ellipse |
y = ax² + bx + c | Parabola |
