Geometry Reference Formulas

A quickreference guide for the most common plane and solid geometry formulas. It is organized by shape, with sections for perimeter, area, surface area, and volume where applicable.

1. Plane Figures

1.1 Triangles

Property	Formula	Notes
Perimeter	P = a + b + c	a, b, c are the three sides
Area (baseheight)	A = \frac{1}{2} bh	b is the base, h is the altitude to that base
Area (Herons formula)	A = \sqrt{s(s-a)(s-b)(s-c)}	s = \frac{a+b+c}{2}
Area (using two sides & included angle)	A = \frac{1}{2}ab\sin C	C is the angle between sides a and b

1.2 Quadrilaterals

Shape	Perimeter	Area
Square	P = 4s	A = s^2
Rectangle	P = 2(l + w)	A = lw
Parallelogram	P = 2(a + b)	A = bh
Rhombus	P = 4a	A = \frac{d_1 d_2}{2}
Trapezoid (US) / Trapezium (UK)	P = a + b + c + d	A = \frac{1}{2}(a+b)h

1.3 Circles

Property	Formula
Circumference	C = 2\pi r = \pi d
Area	A = \pi r^2
Sector area	A_{sector}= \frac{\theta}{360^\circ}\pi r^2
Segment area	A_{segment}= \frac{r^2}{2}\left(\theta - \sin\theta\right)	in radians

2. Solid Figures

2.1 Prisms

Shape	Surface Area	Volume
Cylinder	SA = 2\pi r(h+r)	V = \pi r^2 h
Rectangular prism	SA = 2(lw+lh+wh)	V = lwh
Triangular prism	SA = bh + 2\left(\frac{1}{2}ab\right)	V = \frac{1}{2}abh

2.2 Pyramids

Base Shape	Surface Area	Volume
Square pyramid	SA = b^2 + 2b\sqrt{\left(\frac{b}{2}\right)^2+h^2}	V = \frac{1}{3}b^2 h
Triangular pyramid (tetrahedron)	SA = \sqrt{3}a^2	V = \frac{a^3}{6\sqrt{2}}	a = edge length

2.3 Spheres and Spherical Segments

Property	Formula
Surface area	SA = 4\pi r^2
Volume	V = \frac{4}{3}\pi r^3
Greatcircle area (hemisphere)	A = 2\pi r^2
Spherical segment (cap) volume	V = \frac{\pi h^2}{3}(3r-h)

3. Useful Constants & Conversions

3.14159
1 radian 57.2958
Area of a regular ngon with side s: A = \frac{n s^2}{4\tan(\pi/n)}
Volume of a regular pyramid with regular polygon base (area B): V = \frac{1}{3}Bh

4. Quick Tips for Using the Formulas

Always label the quantities you know before selecting a formula. Identify which variables are given and which you need to find.
Pay attention to units. Convert all lengths to the same unit before performing calculations.
When a problem involves a circle or sphere, check whether the radius (r) or the diameter (d) is provided. Remember that d = 2r.
For triangles, the law of sines and cosines are handy when you have nonright configurations:
- 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
Check whether a shape is regular (all sides and angles equal). Regular shapes often have simplified formulas, as shown for regular polygons and regular pyramids.