Geometry Formulas

Circle

Formula	Expression
Area	$A = \pi r^2$
Circumference	$C = 2\pi r = \pi d$
Diameter	$d = 2r$
Arc Length	$s = r\theta$ ($\theta$ in radians)
Sector Area	$A = \frac{1}{2}r^2\theta$ ($\theta$ in radians)

Triangle

Formula	Expression
Area	$A = \frac{1}{2}bh$
Area (3 sides)	$A = \sqrt{s(s-a)(s-b)(s-c)}$ (Heron's)
Semi-perimeter	$s = \frac{a + b + c}{2}$
Perimeter	$P = a + b + c$

Special Triangles

Equilateral Triangle (side = $s$):

• Area: $A = \frac{\sqrt{3}}{4}s^2$
• Height: $h = \frac{\sqrt{3}}{2}s$

Right Triangle:

• Area: $A = \frac{1}{2} \times \text{leg}_1 \times \text{leg}_2$
• Pythagorean: $a^2 + b^2 = c^2$

Rectangle

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

Square

Formula	Expression
Area	$A = s^2$
Perimeter	$P = 4s$
Diagonal	$d = s\sqrt{2}$

Parallelogram

Formula	Expression
Area	$A = bh$
Perimeter	$P = 2(a + b)$

Trapezoid

Formula	Expression
Area	$A = \frac{1}{2}(b_1 + b_2)h$
Perimeter	$P = a + b_1 + c + b_2$

Rhombus

Formula	Expression
Area	$A = \frac{1}{2}d_1 d_2$
Area	$A = s^2\sin(\theta)$
Perimeter	$P = 4s$

Regular Polygons

General Formula ($n$-sided polygon with side $s$)

Formula	Expression
Interior angle	$\frac{(n-2) \times 180°}{n}$
Sum of interior angles	$(n-2) \times 180°$
Exterior angle	$\frac{360°}{n}$
Area	$A = \frac{1}{4}ns^2\cot\left(\frac{\pi}{n}\right)$
Perimeter	$P = ns$

Common Regular Polygons

Polygon	Sides	Interior Angle	Area (side = $s$)
Pentagon	5	$108°$	$\frac{1}{4}s^2\sqrt{25 + 10\sqrt{5}}$
Hexagon	6	$120°$	$\frac{3\sqrt{3}}{2}s^2$
Heptagon	7	$\approx 128.57°$	$\frac{7}{4}s^2\cot\left(\frac{\pi}{7}\right)$
Octagon	8	$135°$	$2(1 + \sqrt{2})s^2$
Nonagon	9	$140°$	$\frac{9}{4}s^2\cot\left(\frac{\pi}{9}\right)$
Decagon	10	$144°$	$\frac{5}{2}s^2\sqrt{5 + 2\sqrt{5}}$

3D Shapes

Rectangular Prism (Box)

Formula	Expression
Volume	$V = lwh$
Surface Area	$SA = 2(lw + lh + wh)$
Space Diagonal	$d = \sqrt{l^2 + w^2 + h^2}$

Cube (side = $s$)

Formula	Expression
Volume	$V = s^3$
Surface Area	$SA = 6s^2$
Space Diagonal	$d = s\sqrt{3}$

Cylinder

Formula	Expression
Volume	$V = \pi r^2 h$
Lateral Surface	$LSA = 2\pi rh$
Total Surface	$TSA = 2\pi r^2 + 2\pi rh$

Cone

Formula	Expression
Volume	$V = \frac{1}{3}\pi r^2 h$
Slant Height	$l = \sqrt{r^2 + h^2}$
Lateral Surface	$LSA = \pi rl$
Total Surface	$TSA = \pi r^2 + \pi rl$

Sphere

Formula	Expression
Volume	$V = \frac{4}{3}\pi r^3$
Surface Area	$SA = 4\pi r^2$