Geometry Formulas - Complete Reference

Triangle Formulas

Angle Sum

\angle A + \angle B + \angle C = 180°

Area (Base × Height)

S = \frac{1}{2}bh

Heron's Formula

S = \sqrt{s(s-a)(s-b)(s-c)}

where $s = \frac{a+b+c}{2}$

Area (SAS)

S = \frac{1}{2}ab\sin C

Pythagorean Theorem

a^2 + b^2 = c^2

(right triangle, c = hypotenuse)

Equilateral Area

S = \frac{\sqrt{3}}{4}s^2

Law of Sines

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

Law of Cosines

c^2 = a^2 + b^2 - 2ab\cos C

Perimeter

P = a + b + c

Quadrilateral Formulas

Angle Sum

\sum \text{angles} = 360°

Rectangle Area

A = l \times w

Rectangle Perimeter

P = 2(l + w)

Square Area

A = s^2

Square Perimeter

P = 4s

Parallelogram Area

A = b \times h

Rhombus Area

A = \frac{1}{2}d_1 \times d_2

Trapezoid Area

A = \frac{1}{2}(a + b) \times h

Rectangle Diagonal

d = \sqrt{l^2 + w^2}

Circle Formulas

Circumference

C = 2\pi r = \pi d

Area

A = \pi r^2

Diameter

d = 2r

Arc Length (radians)

s = r\theta

Sector Area (radians)

A = \frac{1}{2}r^2\theta

Arc Length (degrees)

s = \frac{\theta°}{360°} \times 2\pi r

Sector Area (degrees)

A = \frac{\theta°}{360°} \times \pi r^2

Semicircle Area

A = \frac{1}{2}\pi r^2

Pi (π)

\pi \approx 3.14159

Vector Formulas

Magnitude

|\vec{a}| = \sqrt{a_1^2 + a_2^2}

Addition

\vec{a} + \vec{b} = (a_1+b_1, a_2+b_2)

Scalar Multiplication

k\vec{a} = (ka_1, ka_2)

Dot Product

\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2

Angle Formula

\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}

Unit Vector

\hat{a} = \frac{\vec{a}}{|\vec{a}|}

Perpendicular Test

\vec{a} \perp \vec{b} \iff \vec{a} \cdot \vec{b} = 0

Position Vector

\vec{AB} = (x_B - x_A, y_B - y_A)

Dot Product Property

\vec{a} \cdot \vec{a} = |\vec{a}|^2

Conic Sections Formula Sheet

Ellipse (horizontal)

\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\quad(a>b>0)

Foci: $(\pm c,0)$ , $c^2=a^2-b^2$

Ellipse (vertical)

\frac{x^2}{b^2}+\frac{y^2}{a^2}=1\quad(a>b>0)

Foci: $(0,\pm c)$ , $c^2=a^2-b^2$

Hyperbola (horizontal)

\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\quad(a>0,b>0)

Asymptotes: $y=\pm\frac{b}{a}x$ , $c^2=a^2+b^2$

Hyperbola (vertical)

\frac{y^2}{a^2}-\frac{x^2}{b^2}=1\quad(a>0,b>0)

Asymptotes: $y=\pm\frac{a}{b}x$ , $c^2=a^2+b^2$

Parabola (opens right/left)

y^2=2px\quad\text{ or }\quad y^2=-2px\quad(p>0)

Focus: $(\pm\frac{p}{2},0)$ , Directrix: $x=\mp\frac{p}{2}$

Parabola (opens up/down)

x^2=2py\quad\text{ or }\quad x^2=-2py\quad(p>0)

Focus: $(0,\pm\frac{p}{2})$ , Directrix: $y=\mp\frac{p}{2}$

Parameter Relations

\text{Ellipse: }c^2=a^2-b^2,\quad e=\frac{c}{a}\ (0<e<1)

\text{Hyperbola: }c^2=a^2+b^2,\quad e=\frac{c}{a}\ (e>1)

Latus Rectum

\text{Ellipse/Hyperbola: length }=\frac{2b^2}{a}

\text{Parabola: length }=2p

Focal Radii (quick forms)

For standard forms with foci $F_1(-c,0),F_2(c,0)$ (or $(0,\pm c)$ ) and $e=\frac{c}{a}$ .

\text{Ellipse } \frac{x^2}{a^2}+\frac{y^2}{b^2}=1:\quad PF_1=a+ex,\ \ PF_2=a-ex

\text{Hyperbola } \frac{x^2}{a^2}-\frac{y^2}{b^2}=1\ (x>0):\quad PF_1=ex+a,\ \ PF_2=ex-a

Vertical types: replace $x$ by $y$ (use the branch with $y>0$ for $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$ ).

Parabola Focal Chord

For $y^2=2px$ , a chord through the focus intersects at $(x_1,y_1),(x_2,y_2)$ .

x_1x_2=\frac{p^2}{4},\qquad y_1y_2=-p^2

Vieta Toolkit (line intersections)

Substitute the line into the conic to get a quadratic $Ax^2+Bx+C=0$ (or in $y$ ). If the intersections have x-coordinates $x_1,x_2$ , then:

x_1+x_2=-\frac{B}{A},\qquad x_1x_2=\frac{C}{A}

Discriminant rule: $\Delta=B^2-4AC>0$ (two intersections), $\Delta=0$ (tangent), $\Delta<0$ (no real intersection).

Solid Geometry Formula Sheet

Prism (right)

V=A_{\text{base}}\,h

L=P_{\text{base}}\,h

S=2A_{\text{base}}+L

Pyramid (regular/right)

V=\frac13A_{\text{base}}\,h

L=\frac12P_{\text{base}}\,l

S=A_{\text{base}}+L

Frustum (regular)

V=\frac{h}{3}\left(A_1+A_2+\sqrt{A_1A_2}\right)

L=\frac12(P_1+P_2)\,l

S=A_1+A_2+L

Cylinder

V=\pi r^2h

L=2\pi rh

S=2\pi r^2+2\pi rh

Cone

V=\frac13\pi r^2h

l=\sqrt{r^2+h^2}

S=\pi r^2+\pi rl

Sphere

S=4\pi r^2

V=\frac{4}{3}\pi r^3

Euler's Formula (convex polyhedra)

V - E + F = 2

V: vertices, E: edges, F: faces

Angle Between Vectors

\cos\theta=\frac{\vec u\cdot \vec v}{|\vec u||\vec v|}

Line–Plane Angle

Direction $\vec u$ , plane normal $\vec n$ .

\sin\theta=\frac{|\vec u\cdot \vec n|}{|\vec u||\vec n|}

Plane–Plane (Dihedral) Angle

Normals $\vec n_1,\vec n_2$ .

\cos\theta=\frac{|\vec n_1\cdot \vec n_2|}{|\vec n_1||\vec n_2|}

Point-to-Plane Distance

\text{Plane }Ax+By+Cz+D=0,\quad d=\frac{|Ax_0+By_0+Cz_0+D|}{\sqrt{A^2+B^2+C^2}}

Angle Relationships

Complementary

\angle A + \angle B = 90°

Supplementary

\angle A + \angle B = 180°

Vertical Angles

\angle 1 = \angle 3

Polygon Angle Sum

(n-2) \times 180°

Regular Polygon Interior

\frac{(n-2) \times 180°}{n}

Exterior Angle (Triangle)

\angle_{ext} = \angle_1 + \angle_2