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Geometry Formulas

Geometry Formula Reference

Complete collection of geometry formulas with explanations, examples, and interactive tools

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Quick Formula Reference

Essential formulas for quick lookup and reference

Triangle Basics

Area (Base × Height)

A=12×b×hA = \frac{1}{2} \times b \times h

Area (Heron's Formula)

A=s(s−a)(s−b)(s−c)A = \sqrt{s(s-a)(s-b)(s-c)}

Perimeter

P=a+b+cP = a + b + c

Angle Sum

∠A+∠B+∠C=180°\angle A + \angle B + \angle C = 180°
Circle Basics

Area

A=πr2A = \pi r^2

Circumference

C=2πr=πdC = 2\pi r = \pi d

Sector Area

A=θ360°×πr2A = \frac{\theta}{360°} \times \pi r^2

Arc Length

s=θ360°×2πrs = \frac{\theta}{360°} \times 2\pi r
Quadrilateral Basics

Rectangle Area

A=l×wA = l \times w

Square Area

A=s2A = s^2

Parallelogram Area

A=b×hA = b \times h

Trapezoid Area

A=12(b1+b2)×hA = \frac{1}{2}(b_1 + b_2) \times h
Vector Basics

Vector Addition

a⃗+b⃗=(a1+b1,a2+b2)\vec{a} + \vec{b} = (a_1+b_1, a_2+b_2)

Dot Product

a⃗⋅b⃗=∣a⃗∣∣b⃗∣cos⁡θ\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta

Magnitude

∣v⃗∣=x2+y2|\vec{v}| = \sqrt{x^2 + y^2}

Unit Vector

u^=v⃗∣v⃗∣\hat{u} = \frac{\vec{v}}{|\vec{v}|}

Triangle Formulas

Essential formulas for triangle calculations, solving, and analysis

Triangle Solving
Intermediate
Complete formulas for solving triangles using sine law, cosine law, and area calculations

Key Formulas:

Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)

Law of Cosines: c² = a² + b² - 2ab·cos(C)

Area = (1/2)ab·sin(C) = √[s(s-a)(s-b)(s-c)]

Angle Sum: A + B + C = 180°

Applications:

Triangle solving
Navigation
Engineering
Architecture

Trigonometry Formulas

Comprehensive trigonometric identities, sum/difference formulas, and reductions

Trigonometry Identity Transformations
Intermediate
Fundamental trigonometric identities and transformation formulas

Key Formulas:

Pythagorean: sin²θ + cos²θ = 1

Reciprocal: csc θ = 1/sin θ, sec θ = 1/cos θ

Quotient: tan θ = sin θ/cos θ

Co-function: sin(90°-θ) = cos θ

Applications:

Identity proofs
Equation solving
Function simplification
Calculus
Trigonometry Sum-Difference
Advanced
Sum and difference formulas for trigonometric functions

Key Formulas:

sin(A ± B) = sin A cos B ± cos A sin B

cos(A ± B) = cos A cos B ∓ sin A sin B

tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)

Double Angle: sin(2θ) = 2sin θ cos θ

Applications:

Wave analysis
Oscillations
Signal processing
Physics
Trigonometry Induction
Advanced
Induction (reduction) formulas for angle transformations

Key Formulas:

sin(π ± θ) = ∓sin θ

cos(π ± θ) = -cos θ

sin(π/2 ± θ) = ±cos θ

cos(π/2 ± θ) = ∓sin θ

Applications:

Angle reduction
Period analysis
Function simplification
Proofs

📐 How to Use Geometry Formulas Effectively

Master geometry problem solving with these formula application strategies

Identify the Problem Type

Determine what you're solving for (area, perimeter, angle, side length) to choose the right formula.

Check Units & Variables

Ensure all measurements use consistent units and identify all known and unknown variables.

Verify Your Answer

Double-check calculations and ensure your answer makes sense in the context of the problem.