📌 Basic Shapes

Perimeter of a square:

\( P = 4s \)

Area of a square:

\( A = s^2 \)

Perimeter of a rectangle:

\( P = 2(l + w) \)

Area of a rectangle:

\( A = l \times w \)

Perimeter of a parallelogram:

\( P = 2(a + b) \)

Area of a parallelogram:

\( A = b \times h \)

Perimeter of a rhombus:

\( P = 4a \)

Area of a rhombus:

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

Perimeter of a trapezoid:

\( P = a + b + c + d \)

Area of a trapezoid:

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

Perimeter of a regular polygon:

\( P = n \times s \)

Area of a regular polygon:

\( A = \frac{1}{2} \times P \times a \)

Perimeter of a circle (Circumference):

\( C = 2\pi r \)

Area of a circle:

\( A = \pi r^2 \)

Arc length:

\( L = r \theta \)

📌 Triangles

Area:

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

Pythagorean Theorem:

\( a^2 + b^2 = c^2 \)

Sum of interior angles:

\( 180^\circ \)

Heron's Formula:

\( A = \sqrt{s(s-a)(s-b)(s-c)} \), where \( s = \frac{a+b+c}{2} \)

Law of Sines:

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

Law of Cosines:

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

Area using SAS:

\( A = \frac{1}{2} ab \sin C \)

Median length:

\( m_a = \frac{1}{2} \sqrt{2b^2 + 2c^2 - a^2} \)

Altitude length:

\( h_a = \frac{2A}{a} \)

Inradius:

\( r = \frac{A}{s} \)

Circumradius:

\( R = \frac{abc}{4A} \)

Centroid coordinates:

\( \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \)

Orthocenter:

Intersection of altitudes

Circumcenter:

Intersection of perpendicular bisectors

Incenter:

Intersection of angle bisectors

Exradius:

\( r_a = \frac{A}{s-a} \)

Right Triangle Area:

\( A = \frac{1}{2} ab \)

Equilateral Triangle Area:

\( A = \frac{\sqrt{3}}{4} s^2 \)

Isosceles Triangle Area:

\( A = \frac{1}{2} b \sqrt{a^2 - \frac{b^2}{4}} \)

Triangle Inequality:

\( a + b > c \)

📌 Quadrilaterals

Parallelogram Area:

\( A = b \times h \)

Rhombus Area:

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

Trapezoid Area:

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

Rectangle Area:

\( A = l \times w \)

Square Area:

\( A = s^2 \)

Kite Area:

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

Cyclic Quadrilateral Area:

\( A = \sqrt{(s-a)(s-b)(s-c)(s-d)} \), where \( s = \frac{a+b+c+d}{2} \)

Parallelogram Perimeter:

\( P = 2(a + b) \)

Rhombus Perimeter:

\( P = 4a \)

Trapezoid Perimeter:

\( P = a + b + c + d \)

Rectangle Perimeter:

\( P = 2(l + w) \)

Square Perimeter:

\( P = 4s \)

Kite Perimeter:

\( P = 2(a + b) \)

Diagonal of Rectangle:

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

Diagonal of Square:

\( d = s\sqrt{2} \)

📌 Circles

Circumference:

\( C = 2\pi r \)

Area:

\( A = \pi r^2 \)

Arc length:

\( L = r \theta \)

Sector Area:

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

Chord length:

\( L = 2r \sin\left(\frac{\theta}{2}\right) \)

Segment Area:

\( A = \frac{1}{2} r^2 (\theta - \sin \theta) \)

Tangent length:

\( L = \sqrt{d^2 - r^2} \)

Equation of Circle:

\( (x-h)^2 + (y-k)^2 = r^2 \)

Central Angle:

\( \theta = \frac{L}{r} \)

Inscribed Angle:

\( \theta = \frac{1}{2} \times \text{Arc} \)

Power of a Point:

\( PA \times PB = PC \times PD \)

Intersecting Chords:

\( (AE)(EB) = (CE)(ED) \)

Intersecting Secants:

\( (PA)(PB) = (PC)(PD) \)

Tangent-Secant:

\( (PA)^2 = (PB)(PC) \)

Circle Area in Terms of Diameter:

\( A = \frac{\pi d^2}{4} \)