Geometry Formulas

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).

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