Here are essential geometry formulas (2D and 3D), organized for quick reference.
2D — Triangles & Polygons
- Triangle area (base & height): $A=\tfrac{1}{2}bh$
- Triangle (Heron): $s=\tfrac{a+b+c}{2}$, $$A=\sqrt{s(s-a)(s-b)(s-c)}$$
- Triangle (trig): $A=\tfrac{1}{2}ab\sin C$
- Pythagorean theorem: $a^2+b^2=c^2$ (right triangle)
- Law of sines: $\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}$
- Law of cosines: $c^2=a^2+b^2-2ab\cos C$
- Regular n-gon area (apothem $a$, perimeter $p$): $A=\tfrac{1}{2}ap$
- Regular n-gon area (side $s$): $A=\tfrac{1}{4}ns^2\cot\!\left(\tfrac{\pi}{n}\right)$
- Sum of interior angles of n-gon: $(n-2)\times180^\circ$
- Number of diagonals: $\dfrac{n(n-3)}{2}$
2D — Quadrilaterals & Other
- Rectangle area: $A=lw$; perimeter: $P=2(l+w)$
- Square area: $A=s^2$; perimeter: $P=4s$
- Parallelogram area: $A=bh$
- Trapezoid area: $A=\tfrac{1}{2}(b_1+b_2)h$
- Rhombus area: $A=\tfrac{1}{2}d_1d_2$ (using diagonals $d_1,d_2$)
Circles & Ellipses
- Circumference: $C=2\pi r=\pi d$
- Area: $A=\pi r^2$
- Sector area (radians): $A=\tfrac{1}{2}r^2\theta$
- Arc length (radians): $s=r\theta$
- Ellipse area (semi-axes $a,b$): $A=\pi ab$
Coordinate Geometry
- Distance between points: $$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$
- Midpoint: $\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)$
- Slope: $m=\dfrac{y_2-y_1}{x_2-x_1}$
- Line: point-slope $y-y_1=m(x-x_1)$; slope-intercept $y=mx+b$
- Shoelace formula (polygon vertices $(x_i,y_i)$ cyclic): $$A=\tfrac{1}{2}\left|\sum_{i=1}^{n}(x_i y_{i+1}-x_{i+1}y_i)\right|$$
3D — Prisms, Cylinders, Pyramids, Cones, Spheres
- Prism/Cylinder volume: $V=Bh$ (base area $B$)
- Rectangular prism: $V=lwh$; surface area $SA=2(lw+lh+wh)$
- Right circular cylinder: $V=\pi r^2 h$; $SA=2\pi r(h+r)$
- Pyramid/Cone volume: $V=\tfrac{1}{3}Bh$
- Right circular cone: $V=\tfrac{1}{3}\pi r^2 h$; lateral area $=\pi r l$; total $=\pi r l+\pi r^2$
- Frustum of cone (radii $r_1,r_2$, height $h$): $V=\tfrac{1}{3}\pi h(r_1^2+r_1r_2+r_2^2)$
- Sphere: $V=\tfrac{4}{3}\pi r^3$; $SA=4\pi r^2$
- Hemisphere: $V=\tfrac{2}{3}\pi r^3$; total surface area (including base) $=3\pi r^2$
Trigonometry (right triangles)
- $\sin\theta=\dfrac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\dfrac{\text{adjacent}}{\text{hypotenuse}}$, $\tan\theta=\dfrac{\text{opposite}}{\text{adjacent}}$
Note: Use radians for sector/arc formulas with $\theta$ in radians. For many solids, $B$ denotes base area and $P$ denotes base perimeter (used in lateral area formulas).
If you want, I can:
- Create a printable one-page formula sheet,
- Show worked examples using any of these formulas,
- Or provide formulas grouped for a specific exam/topic. Which would help most?