Comprehensive Geometry Formulas (2D & 3D)

How to use: This sheet uses native MathML for clean, LaTeX‑style math rendering in modern browsers (Chrome, Edge, Firefox, Safari). Click Print / Export to PDF to save an A4 PDF for study. Variables are consistently named: a, b, c (sides), l, w, h (length, width, height), r, d (radius, diameter), P (perimeter), A (area), SA (surface area), V (volume), s (semi‑perimeter or slant height by context).

Downloadable Formula Sheets

Keystone Geometry Formula Sheet — concise reference aligned with Keystone exam topics.
M-G2 Geometric Formulas — collection of 2D and 3D formulas with illustrative diagrams.
Math Resources Geometry Formulas — general resource covering common geometry formulas.
Comprehensive Geometric Formulas — compact sheet summarizing fundamental equations.

2D Shapes

Triangle

Perimeter P Area A Right triangle hypotenuse c Semi‑perimeter s

Perimeter: $P = a + b + c$
Area (with base b, height h): $A = \frac{1}{2} b h$
Heron's formula:
Let $s = \frac{a + b + c}{2}$ . Then
$A = \sqrt{s (s - a) (s - b) (s - c)}$
Pythagorean theorem (right triangle): $c^{2} = a^{2} + b^{2}$
Law of cosines: $c^{2} = a^{2} + b^{2} - 2 a b \cos γ$

Square

side adiagonal d

Perimeter: $P = 4 a$
Area: $A = a^{2}$
Diagonal: $d = a \sqrt{2}$

Rectangle

length lwidth wdiagonal d

Perimeter: $P = 2 (l + w)$
Area: $A = l w$
Diagonal: $d = \sqrt{l^{2} + w^{2}}$

Parallelogram

adjacent sides a,bheight h

Perimeter: $P = 2 (a + b)$
Area: $A = b h$

Rhombus

diagonals d1, d2

Perimeter: $P = 4 a$ (all sides equal to a)
Area: $A = \frac{d_{1} \cdot d_{2}}{2}$

Trapezoid (US) / Trapezium (UK)

bases b1, b2height h

Perimeter: Sum of all side lengths.
Area: $A = \frac{b 1 + b 2}{2} h$
Median length: $m = \frac{b 1 + b 2}{2}$

Circle

radius rdiameter d=2r

Circumference: $C = 2 π r = π d$
Area: $A = π r^{2}$
Sector (angle θ in radians):
Arc length $L = θ r$ , Area $A = \frac{θ}{2} r^{2}$

Ellipse

semi‑major asemi‑minor b

Area: $A = π a b$
Circumference (Ramanujan approx.): $C \approx π [3 (a + b) - \sqrt{(3 a + b) (a + 3 b)}]$

Regular n-gon

side length sapothem an sides

Perimeter: $P = n s$
Area (two forms): $A = \frac{P a}{2} = \frac{n s^{2}}{4 \tan (\frac{π}{n})}$

3D Shapes

Cube

edge a

Surface area: $SA = 6 a^{2}$
Volume: $V = a^{3}$
Space diagonal: $d = a \sqrt{3}$

Rectangular Prism (Cuboid)

length lwidth wheight h

Surface area: $SA = 2 (l w + l h + w h)$
Volume: $V = l w h$
Space diagonal: $d = \sqrt{l^{2} + w^{2} + h^{2}}$

Right Prism (General base)

base area A_basebase perimeter P_baseheight h

Lateral area: $LA = P_{base} h$
Surface area: $SA = LA + 2 A_{base}$
Volume: $V = A_{base} h$

Right Circular Cylinder

radius rheight h

Lateral area: $LA = 2 π r h$
Surface area: $SA = 2 π r h + 2 π r^{2}$
Volume: $V = π r^{2} h$

Right Circular Cone

radius rheight hslant s

Slant height: $s = \sqrt{r^{2} + h^{2}}$
Surface area: $SA = π r s + π r^{2}$
Volume: $V = \frac{1}{3} π r^{2} h$

Right Square Pyramid

base side bheight hslant (face) s

Slant height (face center): $s = \sqrt{{\frac{b}{2}}^{2} + h^{2}}$
Surface area: $SA = b^{2} + 2 b s$
Volume: $V = \frac{1}{3} b^{2} h$

Sphere

radius r

Surface area: $SA = 4 π r^{2}$
Volume: $V = \frac{4}{3} π r^{3}$

Hemisphere

radius r

Curved surface area (no base): $CSA = 2 π r^{2}$
Total surface area (with base): $SA = 3 π r^{2}$
Volume: $V = \frac{2}{3} π r^{3}$

Frustum of a Right Circular Cone

top radius r₁bottom radius r₂height hslant s

Slant height: $s = \sqrt{h^{2} + {(r_2 - r_1)}^{2}}$
Lateral area: $LA = π (r_{1} + r_{2}) s$
Surface area: $SA = LA + π {r_{1}}^{2} + π {r_{2}}^{2}$
Volume: $V = \frac{1}{3} π h ({r_{1}}^{2} + r_{1} r_{2} + {r_{2}}^{2})$

Frustum of a Right Regular Pyramid

top base area A₁, perimeter P₁bottom base area A₂, perimeter P₂height hslant s

Lateral area (regular): $LA = \frac{1}{2} (P_{1} + P_{2}) s$
Surface area: $SA = LA + A_{1} + A_{2}$
Volume (general): $V = \frac{1}{3} h (A_{1} + A_{2} + \sqrt{A_{1} A_{2}})$

Torus & Ellipsoid (quick refs)

Torus: major R, minor rEllipsoid semiaxes a,b,c

Torus:
Surface area and volume (donut with tube radius r around circle radius R)
$SA = 4 π^{2} R r$ $V = 2 π^{2} R r^{2}$
Ellipsoid: $V = \frac{4}{3} π a b c$
Surface area requires elliptic integrals; a common approximation (Knud Thomsen) uses exponent p ≈ 1.6075:
$SA \approx 4 π {\frac{a^{p} + b^{p} + c^{p}}{3}}^{1 / p}$

Comprehensive Geometry Formulas — 2D & 3D (Student Edition)

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