Triangle & Angle Calculator | Complete Geometry Solver
The Triangle & Angle Calculator determines complete triangle properties from partial inputs (sides and/or angles). It delivers precise results for area, perimeter, centers, radii, and more, while offering unit conversions and step‑by‑step solutions. Designed to serve both students learning geometry and engineers or designers needing accurate calculations, it eliminates guesswork and ensures clarity in every workflow.
🔺 Triangle & Angle Calculator
A comprehensive tool to solve triangles, calculate angles, and analyze geometric properties. Input any 3 known values to solve for all unknowns.
Triangle Configuration
Enter Known Values
Triangle Visualization
Triangle Classification
Enter values to classify triangle.
Calculation Results
Basic Properties
Advanced Geometric Properties
Formulas Used
After calculation, formulas used will appear here.
About This Calculator
This comprehensive triangle calculator solves for all unknown sides, angles, area, perimeter, and geometric properties of any triangle. It handles multiple input methods (SSS, SAS, ASA, AAS, HL) and provides step-by-step solutions.
Accuracy Note
Calculations are performed with double-precision floating-point arithmetic. Results are accurate to at least 6 decimal places. For engineering applications, always verify critical calculations with alternative methods.
Common Mistakes to Avoid
- Ensure angles sum to 180° (for non-degenerate triangles)
- Check triangle inequality: sum of any two sides > third side
- Right triangles require one 90° angle
- SSA (side-side-angle) may have 0, 1, or 2 solutions (ambiguous case)
Key Features
- Solve triangles using SSS, SAS, ASA, AAS, and SSA methods
- Support for right, isosceles, equilateral, and general triangles
- Calculate area, perimeter, angles, and sides
- Advanced properties: altitudes, medians, centroids, and circle radii
- Visual triangle representation with labeled dimensions
- Multiple unit systems (metric and imperial)
- Angle units: degrees, radians, and gradians
- Step-by-step formula display
- Mobile-responsive design
- Print and export functionality
Common Use Cases
- Education: Students learning geometry and trigonometry
- Engineering: Structural calculations and design verification
- Architecture: Roof pitch calculations and spatial planning
- Construction: Site measurements and material estimation
- Surveying: Land measurements and boundary calculations
- Navigation: Distance and bearing calculations
- Physics: Vector analysis and force resolution
Tips for Best Results
- Always provide at least 3 values, including at least 1 side length
- Ensure inputs satisfy the triangle inequality theorem
- For SSA cases, be aware of potential multiple solutions
- Use consistent units for all side measurements
- Double-check angle measurements are in the correct unit
- For right triangles, use the dedicated right triangle mode
Triangle Properties Reference
| Property | Formula | Description |
|---|---|---|
| Area (Heron's) | \( \sqrt{s(s-a)(s-b)(s-c)} \) | When all three sides are known |
| Area (SAS) | \( \frac{1}{2}ab \sin C \) | When two sides and included angle are known |
| Law of Sines | \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \) | Relates sides to their opposite angles |
| Law of Cosines | \( a^2 = b^2 + c^2 - 2bc \cos A \) | Generalization of Pythagorean theorem |
| Inradius | \( r = \frac{Area}{s} \) | Radius of inscribed circle |
| Circumradius | \( R = \frac{abc}{4 \cdot Area} \) | Radius of circumscribed circle |
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📐 Triangle & Angle Calculator: Complete User Guide
This comprehensive guide explains how to use the calculator, all formulas employed, and tips for accurate results.
🎯 Quick Start Guide
Step 1: Select Triangle Type
Choose from four triangle types:
- Any Triangle: Most flexible option
- Right Triangle: One 90° angle automatically set
- Isosceles Triangle: Two equal sides and angles
- Equilateral Triangle: All sides and angles equal (60° each)
Step 2: Choose Input Method
Select how you want to input your known values:
SSS (Side-Side-Side)
When you know all three side lengths.
Example: a = 5cm, b = 6cm, c = 7cm
SAS (Side-Angle-Side)
Two sides and the included angle.
Example: b = 8cm, A = 60°, c = 10cm
ASA (Angle-Side-Angle)
Two angles and the included side.
Example: A = 45°, c = 12cm, B = 60°
AAS (Angle-Angle-Side)
Two angles and any non-included side.
Example: A = 30°, B = 45°, a = 5cm
Step 3: Enter Your Values
Input your known measurements in the appropriate fields:
- Sides: Use positive numbers only (e.g., 5.5, 12, 7.25)
- Angles: Enter in degrees (e.g., 45, 90, 60.5)
- Units: Select appropriate units from dropdowns
Step 4: Click Calculate
Press the "Calculate Triangle" button to compute all unknown values.
Step 5: Review Results
Examine the comprehensive results including:
- All three sides and angles
- Area and perimeter
- Advanced properties (inradius, circumradius, altitudes, medians)
- Visual triangle diagram
- Formulas used in calculations
📊 Visual Reference: Triangle Notation
| Symbol | Meaning | Standard Position |
|---|---|---|
| a, b, c | Side lengths | Opposite angles A, B, C respectively |
| A, B, C | Angle measures | At vertices A, B, C respectively |
| α, β, γ | Angle measures (Greek) | Alternative notation for A, B, C |
| P | Perimeter | P = a + b + c |
| s | Semi-perimeter | s = P/2 |
| A△ | Area | Various formulas depending on known values |
🧮 Complete Formula Reference
Accuracy Note
All calculations use double-precision floating-point arithmetic with precision up to 15 decimal places. Results are accurate for practical applications in engineering, architecture, and education.
1. Fundamental Triangle Properties
Angle Sum Theorem
For any triangle, the sum of interior angles is always 180°:
Triangle Inequality Theorem
For any valid triangle, the sum of any two sides must be greater than the third side:
2. Solving Methods by Input Type
| Input Method | Formulas Used | Minimum Required Inputs |
|---|---|---|
| SSS (Side-Side-Side) |
Law of Cosines: $$ A = \cos^{-1}\left(\frac{b^2 + c^2 - a^2}{2bc}\right) $$
Then Law of Sines: $$ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} $$
|
All three sides: a, b, c |
| SAS (Side-Angle-Side) |
Law of Cosines: $$ a = \sqrt{b^2 + c^2 - 2bc\cos A} $$
Then Law of Sines for remaining angles
|
Two sides and included angle (e.g., b, A, c) |
| ASA (Angle-Side-Angle) |
Angle sum: $$ C = 180^\circ - A - B $$
Law of Sines: $$ a = \frac{c \sin A}{\sin C}, \quad b = \frac{c \sin B}{\sin C} $$
|
Two angles and included side (e.g., A, c, B) |
| AAS (Angle-Angle-Side) |
Angle sum: $$ C = 180^\circ - A - B $$
Law of Sines: $$ b = \frac{a \sin B}{\sin A}, \quad c = \frac{a \sin C}{\sin A} $$
|
Two angles and any side (e.g., A, B, a) |
| Right Triangle |
Pythagorean Theorem: $$ a^2 + b^2 = c^2 $$
Trigonometric ratios: $$ \sin A = \frac{a}{c}, \quad \cos A = \frac{b}{c}, \quad \tan A = \frac{a}{b} $$
|
Any two sides, or one side and one acute angle |
3. Area Calculation Formulas
Base-Height Formula
When to use: When you know base and corresponding altitude
Heron's Formula
When to use: When you know all three sides (SSS)
SAS Formula
When to use: When you know two sides and included angle
AAS/ASA Formula
When to use: When you know two angles and a side
4. Advanced Geometric Properties
| Property | Formula | Description |
|---|---|---|
| Perimeter | $$ P = a + b + c $$ | Total length around the triangle |
| Semi-perimeter | $$ s = \frac{P}{2} = \frac{a + b + c}{2} $$ | Half the perimeter, used in many formulas |
| Inradius (r) | $$ r = \frac{A_{\triangle}}{s} $$ | Radius of inscribed circle |
| Circumradius (R) | $$ R = \frac{abc}{4A_{\triangle}} $$ | Radius of circumscribed circle |
| Altitude to side a | $$ h_a = \frac{2A_{\triangle}}{a} $$ | Height perpendicular to side a |
| Median to side a | $$ m_a = \frac{1}{2}\sqrt{2b^2 + 2c^2 - a^2} $$ | Line from vertex A to midpoint of BC |
⚠️ Common Mistakes & Troubleshooting
Input Validation Errors
This means your side lengths cannot form a triangle. Check that:
• a + b > c
• b + c > a
• c + a > b
Example: Sides 3, 4, 8 cannot form a triangle because 3 + 4 < 8
The sum of all three angles must equal exactly 180°. If you entered two angles, the third is automatically calculated. If you entered three angles, they must sum to 180°.
Without at least one side, the triangle size cannot be determined. Similar triangles have the same angles but different sizes.
Unit Conversion Issues
Always use consistent units. Don't mix centimeters with inches. Convert all measurements to the same system before inputting.
The calculator accepts degrees, radians, and gradians. Ensure you've selected the correct unit for your inputs. Most users work with degrees.
Ambiguous Case (SSA)
When given two sides and a non-included angle (SSA), there may be 0, 1, or 2 possible triangles. The calculator handles this ambiguity and will show both solutions when they exist.
📋 Example Calculations
Example 1: Right Triangle
Given: Right triangle with legs a = 3 cm, b = 4 cm
Solution:
Example 2: SSS Triangle
Given: Sides a = 7 cm, b = 8 cm, c = 9 cm
Solution:
🔧 Technical Specifications
| Specification | Details |
|---|---|
| Calculation Precision | Double-precision floating point (≈15 decimal digits) |
| Supported Units |
Length: mm, cm, m, km, inches, feet, yards, miles Angles: degrees, radians, gradians Area: Auto-converted based on length unit |
| Browser Compatibility | Chrome, Firefox, Safari, Edge, Opera (modern versions) |
| Mobile Support | iOS Safari, Android Chrome, Tablet browsers |
| Data Storage | No data sent to servers - all calculations happen locally in your browser |
| Export Options | Copy to clipboard, Print-friendly format, Visual diagram |
