Triangle Calculator - Solve All Triangle Types & Properties

Calculate triangle area, perimeter, angles, and sides using SSS, SAS, ASA methods. Comprehensive triangle solver with visual charts, detailed analysis, and educational explanations.

Triangle Input
Enter triangle measurements to calculate properties

Enter Three Sides

Common Triangles

Triangle Results
Calculated properties and measurements
Triangle Properties Guide
Understanding triangle classifications and measurements

Triangle Classifications

By Sides
Equilateral (all equal), Isosceles (two equal), Scalene (all different)
By Angles
Acute (all < 90°), Right (one = 90°), Obtuse (one > 90°)

Key Formulas

Area Formulas
Heron's: √[s(s-a)(s-b)(s-c)] | Base-Height: ½bh
Special Radii
Inradius: Area/s | Circumradius: abc/4Area
Calculation History
Your recent triangle calculations
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Geometric Foundation: Triangles are the simplest polygons and form the basis of trigonometry, engineering, and architectural design. Understanding triangle properties is essential for advanced mathematics and real-world problem solving.

Understanding Triangles

A triangle is a three-sided polygon with three vertices, three angles, and three sides. The sum of interior angles always equals 180°, making triangles fundamental to geometry and trigonometry. Understanding different triangle classifications and solving methods enables accurate calculations for area, perimeter, and unknown measurements. Our calculator supports multiple input methods and provides comprehensive analysis including advanced properties like medians, altitudes, and special radii.

📐 Basic Properties

Three sides, three angles, angle sum equals 180°, and fundamental geometric relationships.

📊 Area & Perimeter

Calculate using multiple formulas: Heron's, base-height, and trigonometric methods.

🔍 Classifications

Identify by sides (scalene, isosceles, equilateral) and angles (acute, right, obtuse).

⚖️ Triangle Inequality

Sum of any two sides must exceed the third side for a valid triangle.

Triangle Classifications

Triangles are classified by their sides and angles, each type having unique properties and calculation methods. Understanding these classifications helps choose the appropriate solving technique and identify special triangle relationships. Our calculator automatically determines triangle type and provides relevant properties for educational understanding and practical applications.

📏 Classification by Sides

Equilateral Triangle
  • All three sides equal in length
  • All angles equal to 60 degrees
  • Highest degree of symmetry
  • Regular polygon with 3 sides
Isosceles Triangle
  • Two sides of equal length
  • Base angles are equal
  • Has line of symmetry
  • Vertex angle differs from base angles
Scalene Triangle
  • All sides have different lengths
  • All angles are different
  • No lines of symmetry
  • Most general triangle type

📐 Classification by Angles

Acute Triangle
  • All angles less than 90 degrees
  • Sharp-pointed appearance
  • Circumcenter inside triangle
  • All altitudes inside triangle
Right Triangle
  • One angle exactly 90 degrees
  • Pythagorean theorem applies
  • Has hypotenuse and two legs
  • Foundation of trigonometry
Obtuse Triangle
  • One angle greater than 90 degrees
  • Longest side opposite obtuse angle
  • Circumcenter outside triangle
  • One altitude outside triangle

🎯 Triangle Classification Quick Guide

Side Comparison
All equal → Equilateral
Two equal → Isosceles
All different → Scalene
Angle Check
All < 90° → Acute
One = 90° → Right
One > 90° → Obtuse
Combined Types
Right Scalene
Right Isosceles
Obtuse Isosceles

Triangle Calculation Methods

Different combinations of known measurements require specific solving approaches. The main methods are SSS (three sides), SAS (two sides and included angle), ASA (two angles and included side), and base-height calculations. Each method uses different mathematical principles: law of cosines, law of sines, and basic trigonometry. Understanding when to apply each method ensures accurate results and helps avoid common calculation errors.

📐 SSS Method (Side-Side-Side)

When to Use:

When all three side lengths are known. This is the most straightforward method for complete triangle determination.

Process:
  1. Verify triangle inequality (a + b > c)
  2. Use law of cosines to find angles
  3. Calculate area using Heron's formula
  4. Determine triangle classification
Formula:

Area = √[s(s-a)(s-b)(s-c)]
where s = (a+b+c)/2

📏 SAS Method (Side-Angle-Side)

When to Use:

When two sides and the included angle are known. The angle must be between the two given sides.

Process:
  1. Use law of cosines to find third side
  2. Apply law of sines for remaining angles
  3. Calculate area using trigonometry
  4. Verify angle sum equals 180°
Formula:

c² = a² + b² - 2ab·cos(C)
Area = ½ab·sin(C)

🔺 ASA Method (Angle-Side-Angle)

When to Use:

When two angles and the included side are known. The side must be between the two given angles.

Key Steps:
  • Find third angle (180° - A - B = C)
  • Use law of sines to find unknown sides
  • Calculate area and classify triangle

📐 Base-Height Method

When to Use:

When base and height are known. Assumes perpendicular height to base, creating an isosceles triangle.

Simple Formula:

Area = ½ × base × height

Essential Triangle Properties

Beyond basic measurements, triangles have numerous properties including altitudes, medians, angle bisectors, and special radii. These properties are crucial for advanced geometric analysis and have practical applications in engineering, architecture, and computer graphics. Understanding these relationships helps solve complex problems and provides deeper geometric insight for educational and professional purposes.

🎯 Key Triangle Properties

Altitude
Perpendicular distance from vertex to opposite side
Median
Line from vertex to midpoint of opposite side
Bisector
Line dividing angle into two equal parts
Radii
Inradius and circumradius measurements

📏 Linear Properties

Altitudes

Perpendicular lines from each vertex to the opposite side. Used for area calculations: Area = ½ × base × altitude.

altitude = (2 × Area) / base

Medians

Lines connecting vertices to midpoints of opposite sides. They intersect at the centroid, dividing each median 2:1.

median = ½√(2b² + 2c² - a²)

Angle Bisectors

Lines dividing angles into equal halves. They intersect at the incenter, equidistant from all sides.

bisector = (2bc × cos(A/2)) / (b + c)

⭕ Special Radii

Inradius (r)

Radius of the inscribed circle (incircle) that touches all three sides. Center is the incenter.

r = Area / semiperimeter

Circumradius (R)

Radius of the circumscribed circle (circumcircle) passing through all three vertices. Center is the circumcenter.

R = (abc) / (4 × Area)

Relationship

The inradius and circumradius are related through Euler's theorem and provide insights into triangle shape.

R ≥ 2r (equality for equilateral)

Advanced Solving Techniques

Master the fundamental laws of trigonometry essential for triangle calculations. The law of sines and law of cosines form the mathematical foundation for solving triangles with partial information. Understanding when and how to apply these laws, along with recognizing special triangle cases, enables efficient problem solving and helps avoid computational errors. These techniques are fundamental for real-world applications in engineering and design.

⚖️ Law of Sines

a/sin(A) = b/sin(B) = c/sin(C)
Best for:
  • ASA (Angle-Side-Angle) problems
  • AAS (Angle-Angle-Side) problems
  • Finding unknown sides when angles are known
  • Checking triangle validity
Key Applications:
  • Navigation and surveying
  • Astronomy calculations
  • Engineering design
  • Computer graphics

📐 Law of Cosines

c² = a² + b² - 2ab·cos(C)
Best for:
  • SSS (Side-Side-Side) problems
  • SAS (Side-Angle-Side) problems
  • Finding unknown angles when sides are known
  • Determining triangle existence
Special Cases:
  • When C = 90°, becomes Pythagorean theorem
  • When C > 90°, cos(C) is negative
  • When C < 90°, cos(C) is positive
  • Generalizes Pythagorean theorem

🎯 Problem-Solving Strategy

1️⃣
Identify Given Information
Classify as SSS, SAS, ASA, AAS, or SSA to choose the appropriate method.
2️⃣
Apply Appropriate Law
Use law of sines for angle-side problems, law of cosines for side-side problems.
3️⃣
Verify Results
Check angle sum equals 180°, triangle inequality holds, and results are reasonable.

Special Triangle Types

Certain triangles have unique properties and relationships that simplify calculations and appear frequently in mathematics and engineering. These include Pythagorean triples, the famous 30-60-90 and 45-45-90 triangles, and other special right triangles. Understanding these patterns enables quick calculations and serves as a foundation for more complex geometric relationships in real-world applications.

📐 Pythagorean Triples

Common Triples:
3-4-5 (and multiples: 6-8-10, 9-12-15)
5-12-13
8-15-17
7-24-25
Applications:
  • Construction and carpentry
  • Ancient Egyptian surveying
  • Quick right angle verification
  • Integer-sided right triangles

📏 45-45-90 Triangle

Side Ratios:
1 : 1 : √2
legs : hypotenuse
Properties:
  • Isosceles right triangle
  • Half of a square diagonal
  • 45° angles at base
  • Common in coordinate geometry
Example:

If legs = 5, then hypotenuse = 5√2 ≈ 7.071

📐 30-60-90 Triangle

Side Ratios:
1 : √3 : 2
short leg : long leg : hypotenuse
Properties:
  • Half of equilateral triangle
  • Appears in hexagonal geometry
  • Common trigonometric values
  • Architectural applications
Example:

If short leg = 1, then long leg = √3, hypotenuse = 2

🎯 Special Triangle Recognition

Quick Identification Tips

  • Integer sides that satisfy a² + b² = c²: Pythagorean triple
  • Two equal sides with 90° angle: 45-45-90 triangle
  • Angles of 30°, 60°, 90°: 30-60-90 triangle
  • All sides equal: Equilateral (all angles 60°)
  • Two equal sides: Isosceles triangle

Calculation Shortcuts

  • Use special ratios instead of trigonometric functions
  • Recognize patterns in measurements for quick solutions
  • Apply symmetry properties for isosceles and equilateral
  • Use Pythagorean theorem for all right triangles
  • Remember common angles: 30°, 45°, 60°, 90°

Real-World Applications

Triangle calculations are fundamental to numerous fields including engineering, architecture, navigation, computer graphics, and surveying. From determining structural loads in construction to calculating distances in GPS systems, triangle geometry provides the mathematical foundation for solving practical problems. Understanding these applications demonstrates the relevance of geometric principles in modern technology and professional practices.

🏗️ Construction & Engineering

  • • Structural load calculations
  • • Roof truss design
  • • Bridge engineering
  • • Foundation planning
  • • Material stress analysis

🗺️ Navigation & Surveying

  • • GPS positioning systems
  • • Maritime navigation
  • • Land surveying
  • • Triangulation methods
  • • Distance measurements

💻 Technology & Graphics

  • • 3D computer graphics
  • • Game development
  • • CAD software design
  • • Animation systems
  • • Virtual reality

🔬 Science & Research

  • • Physics calculations
  • • Astronomy measurements
  • • Optics and light paths
  • • Crystallography
  • • Molecular geometry

🏗️ Construction Example

Problem: A roof truss needs to span 24 feet with a peak height of 8 feet. Calculate the length of the rafters and the angles.

Given: Base = 24 ft, Height = 8 ft
Find: Rafter length (hypotenuse)
Solution: Using Pythagorean theorem
Rafter = √(12² + 8²) = √208 ≈ 14.42 ft
Angle: tan⁻¹(8/12) ≈ 33.7°

🗺️ Navigation Example

Problem: A ship travels 50 miles east, then 30 miles northeast (45° from east). Find the direct distance to the starting point.

Given: Two sides and included angle
Method: Law of cosines (SAS)
c² = 50² + 30² - 2(50)(30)cos(45°)
c² = 2500 + 900 - 3000(0.707)
c ≈ 35.9 miles

Advanced Triangle Concepts

Beyond basic calculations, triangles exhibit sophisticated mathematical relationships including congruence, similarity, and special centers. These concepts are fundamental to advanced geometry, trigonometry, and coordinate systems. Understanding triangle congruence rules (SSS, SAS, ASA, AAS, HL) and similarity ratios enables solving complex geometric problems and forms the basis for geometric proofs and constructions.

🔄 Triangle Congruence

Congruence Rules:
  • SSS: Three sides equal → triangles congruent
  • SAS: Two sides and included angle equal
  • ASA: Two angles and included side equal
  • AAS: Two angles and non-included side equal
  • HL: Hypotenuse-leg for right triangles only
Applications:
  • Geometric proofs and constructions
  • Pattern recognition in design
  • Manufacturing quality control
  • Architectural symmetry verification

📐 Triangle Similarity

Similarity Rules:
  • AA: Two angles equal → similar triangles
  • SAS: Two sides proportional, included angle equal
  • SSS: All three sides proportional
Scale Relationships:
  • Corresponding sides in constant ratio
  • Areas relate by square of scale factor
  • Volumes relate by cube of scale factor
  • Used in map scaling and model making

🎯 Triangle Centers

Centroid:

Intersection of medians, divides each median 2:1, center of mass.

Incenter:

Intersection of angle bisectors, center of inscribed circle.

Circumcenter:

Intersection of perpendicular bisectors, center of circumscribed circle.

Orthocenter:

Intersection of altitudes, varies by triangle type.

📊 Area Relationships

Multiple Formulas:
Area = ½ × base × height
Area = √[s(s-a)(s-b)(s-c)]
Area = ½ab sin(C)
Area = rs (r = inradius)
Applications:
  • Land measurement and surveying
  • Material quantity calculations
  • Architectural planning
  • Engineering stress analysis

⚖️ Triangle Inequalities

Basic Inequality:

a + b > c
a + c > b
b + c > a

Extended Relations:
  • Largest angle opposite longest side
  • Smallest angle opposite shortest side
  • Sum of any two angles > third angle
  • Used to verify triangle validity

Common Calculation Mistakes

Understanding common errors in triangle calculations helps avoid mistakes and ensures accurate results. These mistakes often stem from incorrect formula application, unit confusion, and misunderstanding of triangle properties. Learning to recognize and avoid these pitfalls improves problem-solving accuracy and builds confidence in geometric calculations.

❌ Critical Errors

Triangle Inequality Violations:

Failing to check if a + b > c for all side combinations before calculating.

Example: Sides 2, 3, 8 are invalid because 2 + 3 = 5 < 8

Wrong Law Application:

Using law of sines for SSS problems or law of cosines for ASA problems.

Angle Unit Confusion:

Mixing degrees and radians in trigonometric calculations.

Ambiguous Case Errors:

Not recognizing SSA cases that may have 0, 1, or 2 solutions.

✅ Best Practices

Always Verify Triangle Validity:

Check triangle inequality before performing any calculations.

Match Method to Given Data:

Identify SSS, SAS, ASA, AAS patterns to choose correct approach.

Consistent Units:

Use degrees consistently or radians consistently throughout calculations.

Check Angle Sum:

Verify that calculated angles sum to 180° (or π radians).

Consider Multiple Solutions:

For SSA cases, check if both acute and obtuse angle solutions are valid.

❌ Common Formula Errors

• Incorrect Heron's formula setup: Forgetting to calculate semiperimeter first
• Law of cosines sign error: Missing the negative sign in c² = a² + b² - 2ab cos(C)
• Area formula confusion: Using wrong base-height pair or incorrect angle
• Calculator mode issues: Being in wrong degree/radian mode

✅ Verification Checklist

• Triangle inequality satisfied: Sum of any two sides > third side
• Angle sum check: All angles sum to exactly 180°
• Reasonable values: All sides positive, all angles between 0° and 180°
• Unit consistency: All measurements in same units

Triangle Calculator Features

Our comprehensive triangle calculator supports multiple input methods and provides detailed analysis beyond basic measurements. Features include SSS, SAS, and base-height calculations, automatic triangle classification, visual charts showing relationships between sides and angles, and educational explanations of geometric properties. The calculator also tracks calculation history and provides sharing capabilities for educational and professional use.

Advanced features include calculation of special properties like medians, altitudes, angle bisectors, and circumradius/inradius measurements. The interactive charts help visualize triangle properties and relationships, making it an excellent educational tool for understanding geometric concepts. Whether you're a student learning triangle properties, an engineer designing structures, or an architect planning layouts, our calculator provides the comprehensive analysis needed for accurate geometric calculations.

Key Takeaways for Triangle Mastery

Master the fundamental calculation methods (SSS, SAS, ASA) and understand when to apply each approach. The law of cosines and law of sines are essential tools for solving triangles with partial information. Always verify triangle validity using the triangle inequality theorem before performing calculations.

Recognize special triangle patterns like Pythagorean triples and 30-60-90, 45-45-90 triangles for quick calculations. Understanding triangle classifications by sides and angles helps identify properties and choose appropriate solving methods. Use our Right Triangle Calculator for Pythagorean applications.

Apply triangle geometry to real-world problems in construction, navigation, and engineering. Understanding advanced properties like altitudes, medians, and special radii provides deeper geometric insight. Avoid common calculation errors by checking triangle inequality and angle sums.

Explore connections to other geometric concepts using our Area Calculator, Angle Converter, and Distance Calculator. Triangle geometry forms the foundation for advanced topics in trigonometry, coordinate geometry, and calculus applications.

Frequently Asked Questions

The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the third side. This fundamental rule ensures a valid triangle can be formed. For example, sides of 3, 4, and 10 would be invalid because 3 + 4 = 7, which is less than 10.
SSS (Side-Side-Side) uses three known sides to find angles. SAS (Side-Angle-Side) uses two sides and the included angle. ASA (Angle-Side-Angle) uses two angles and the included side. Each method applies different formulas: law of cosines for SSS and SAS, law of sines for ASA.
Use Heron's formula: Area = √[s(s-a)(s-b)(s-c)], where s is the semiperimeter (s = (a+b+c)/2). This formula works for any triangle when all three sides are known, making it especially useful for irregular triangles where height isn't easily determined.
Right triangles have one 90° angle and follow the Pythagorean theorem: a² + b² = c² (where c is the hypotenuse). The two acute angles always sum to 90°. Right triangles are fundamental in trigonometry and have special ratios like 3-4-5, 5-12-13, and the 30-60-90 and 45-45-90 triangles.
Compare the square of the longest side to the sum of squares of the other two sides. If c² < a² + b², it's acute. If c² = a² + b², it's right. If c² > a² + b², it's obtuse. Alternatively, if all angles are less than 90°, it's acute; if one equals 90°, it's right; if one exceeds 90°, it's obtuse.
An altitude is a perpendicular line from a vertex to the opposite side, used for area calculations. A median connects a vertex to the midpoint of the opposite side and divides the triangle into two equal areas. An angle bisector divides an angle into two equal parts. Each has different lengths and geometric properties.
The inradius (radius of inscribed circle) = Area / semiperimeter. The circumradius (radius of circumscribed circle) = (abc) / (4 × Area). These radii help determine the circles that fit inside and around the triangle, respectively, and are important in geometric constructions and calculations.
Congruent triangles have identical size and shape. They can be proven congruent using SSS (three sides equal), SAS (two sides and included angle), ASA (two angles and included side), AAS (two angles and non-included side), or HL (hypotenuse-leg for right triangles). Congruent triangles have equal corresponding sides and angles.
The SSA (Side-Side-Angle) case can produce 0, 1, or 2 valid triangles. Use the law of sines to check: if sin(B) > 1, no solution exists; if sin(B) = 1, one right triangle; if sin(B) < 1, potentially two solutions. Always verify both possibilities using the triangle inequality and angle sum theorem.
Similar triangles have the same shape but different sizes, with corresponding angles equal and sides proportional. They're proven similar by AA (two angles), SAS (two sides proportional with included angle equal), or SSS (all sides proportional). The ratio of corresponding sides is the scale factor, and areas relate by the square of this ratio.

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