📐 Geometry Formulas

Master geometry with formulas for shapes, theorems, and more.

📌 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:
\( A = \frac{1}{2} \times P \times a \)
Circumference of a Circle:
\( C = 2\pi r \)
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:
\( A = \frac{\pi d^2}{4} \)

📌 3D Shapes

Volume of a Cube:
\( V = s^3 \)
Surface Area of a Cube:
\( SA = 6s^2 \)
Volume of a Rectangular Prism:
\( V = l \times w \times h \)
Surface Area of a Rectangular Prism:
\( SA = 2(lw + lh + wh) \)
Volume of a Cylinder:
\( V = \pi r^2 h \)
Surface Area of a Cylinder:
\( SA = 2\pi r h + 2\pi r^2 \)
Volume of a Cone:
\( V = \frac{1}{3} \pi r^2 h \)
Surface Area of a Cone:
\( SA = \pi r l + \pi r^2 \), where \( l = \sqrt{r^2 + h^2} \)
Volume of a Sphere:
\( V = \frac{4}{3} \pi r^3 \)
Surface Area of a Sphere:
\( SA = 4\pi r^2 \)
Volume of a Pyramid (Square Base):
\( V = \frac{1}{3} b^2 h \)
Surface Area of a Pyramid (Square Base):
\( SA = b^2 + 2b l \), where \( l = \sqrt{\left(\frac{b}{2}\right)^2 + h^2} \)
Volume of a Prism:
\( V = B \times h \), where \( B \) is base area
Surface Area of a Prism:
\( SA = 2B + P h \), where \( P \) is base perimeter

📌 Coordinate Geometry

Distance Formula:
\( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
Midpoint Formula:
\( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)
Slope of a Line:
\( m = \frac{y_2 - y_1}{x_2 - x_1} \)
Slope-Intercept Form:
\( y = mx + b \)
Point-Slope Form:
\( y - y_1 = m (x - x_1) \)
General Form of a Line:
\( Ax + By + C = 0 \)
Distance from Point to Line:
\( d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \)
Angle Between Two Lines:
\( \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \)
Parallel Lines:
\( m_1 = m_2 \)
Perpendicular Lines:
\( m_1 \cdot m_2 = -1 \)

📌 Trigonometry

Sine:
\( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)
Cosine:
\( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \)
Tangent:
\( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
Pythagorean Identity:
\( \sin^2 \theta + \cos^2 \theta = 1 \)
Angle Sum Identity (Sine):
\( \sin(A + B) = \sin A \cos B + \cos A \sin B \)
Angle Sum Identity (Cosine):
\( \cos(A + B) = \cos A \cos B - \sin A \sin B \)
Double Angle Identity (Sine):
\( \sin 2\theta = 2 \sin \theta \cos \theta \)
Double Angle Identity (Cosine):
\( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \)
Law of Sines:
\( \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} \)
Law of Cosines:
\( c^2 = a^2 + b^2 - 2ab \cos C \)

📌 Advanced Theorems

Euclid’s Theorem:
For a right triangle, \( a^2 + b^2 = c^2 \)
Ceva’s Theorem:
In a triangle, cevians are concurrent if \( \frac{BD}{DC} \cdot \frac{CE}{EA} \cdot \frac{AF}{FB} = 1 \)
Menelaus’ Theorem:
In a triangle, for a transversal, \( \frac{BD}{DC} \cdot \frac{CE}{EA} \cdot \frac{AF}{FB} = -1 \)
Ptolemy’s Theorem:
In a cyclic quadrilateral, \( AC \cdot BD = AB \cdot CD + AD \cdot BC \)
Stewart’s Theorem:
For a cevian, \( a^2 m + b^2 n = c (d^2 + mn) \), where \( m+n=c \)
Brahmagupta’s Formula:
Area of cyclic quadrilateral: \( A = \sqrt{(s-a)(s-b)(s-c)(s-d)} \), where \( s = \frac{a+b+c+d}{2} \)
Heron’s Formula:
Area of triangle: \( A = \sqrt{s(s-a)(s-b)(s-c)} \), where \( s = \frac{a+b+c}{2} \)
Thales’ Theorem:
If a triangle is inscribed in a circle with one side as diameter, it is a right triangle
Angle Bisector Theorem:
\( \frac{BD}{DC} = \frac{AB}{AC} \)
Inscribed Angle Theorem:
Inscribed angle is half the measure of its intercepted arc