Triangle Calculator – Solve Any Triangle with Sides, Angles, Area, Perimeter & Properties

  • Adam
  • November 4, 2025

Free online triangle calculator to solve any triangle by entering three values. Calculates all sides, angles, area, perimeter, medians, inradius, circumradius with step-by-step solutions and complete geometric analysis.

Triangle Calculator

Comprehensive Tool for Solving Triangles with Complete Analysis and Geometry Calculations

Quick Navigation

Solve Any Triangle

Enter 3 values including at least 1 side. Leave other fields blank.

Triangle Properties Reference

Triangle Classification

By Sides
Equilateral: All sides equal
Isosceles: Two sides equal
Scalene: No sides equal
By Angles
Acute: All angles < 90°
Right: One angle = 90°
Obtuse: One angle > 90°
Key Facts
Sum of angles = 180°
Sum of any two sides > third
Longest side opposite largest angle

Triangle Inequalities

  • Triangle Inequality: Sum of any two sides must be greater than the third side
  • Angle Sum: All three interior angles must sum to 180 degrees
  • No Angle Zero: Each angle must be greater than 0 degrees
  • Maximum Angle: No angle can be greater than or equal to 180 degrees

Special Triangles

Triangle Type Angles Side Ratios Properties
30-60-90 30°, 60°, 90° 1 : √3 : 2 Right triangle with specific angles
45-45-90 45°, 45°, 90° 1 : 1 : √2 Isosceles right triangle
Equilateral 60°, 60°, 60° 1 : 1 : 1 All sides and angles equal

Essential Triangle Formulas

Pythagorean Theorem (Right Triangles)

a² + b² = c²

Used for right triangles where c is hypotenuse (longest side opposite 90° angle)

Law of Cosines

a² = b² + c² - 2bc·cos(A)
b² = a² + c² - 2ac·cos(B)
c² = a² + b² - 2ab·cos(C)

Used when you know two sides and included angle, or all three sides

Law of Sines

a/sin(A) = b/sin(B) = c/sin(C) = 2R

Used when you know angle-side pairs. R is circumradius.

Area Formulas

Area = (1/2) × base × height
Area = (1/2) × a × b × sin(C)
Area = √(s(s-a)(s-b)(s-c)) [Heron's Formula]
where s = (a+b+c)/2 (semiperimeter)

Inradius and Circumradius

Inradius: r = Area / s
Circumradius: R = a / (2×sin(A))

Medians

mₐ = √(2b² + 2c² - a²) / 2
m_b = √(2a² + 2c² - b²) / 2
m_c = √(2a² + 2b² - c²) / 2

Medians connect vertices to opposite side midpoints

Understanding Triangles

Triangles are fundamental geometric shapes consisting of three straight sides, three vertices (corners), and three interior angles. Named after their vertices, a triangle with vertices A, B, and C is denoted as △ABC. Triangles form the basis of trigonometry and appear throughout mathematics, engineering, architecture, and natural phenomena.

Triangle geometry combines multiple disciplines: plane geometry (angles and sides), trigonometry (sine, cosine, tangent), and algebra (solving for unknowns). Understanding triangles is essential for surveying, navigation, architecture, physics, and countless practical applications. Whether calculating building roof angles, determining distances in surveying, or analyzing forces in physics, triangle calculations are indispensable.

This calculator solves any triangle given sufficient information—typically three values including at least one side. It employs the law of sines, law of cosines, and Pythagorean theorem to compute all remaining values. Beyond basic side and angle calculations, it provides area, perimeter, medians, inradius, and circumradius for complete geometric analysis.

Key Insight: Any three values uniquely determine a triangle (with rare ambiguous cases). This triangle solver systematically uses mathematical laws to find all remaining measurements.

Key Features & Capabilities

This comprehensive calculator provides complete triangle analysis:

📐 Complete Solver Solves any valid triangle configuration
📐 All Six Values Computes sides a, b, c and angles A, B, C
📐 Area Calculation Multiple area computation methods
📐 Perimeter & Semiperimeter Automatic calculation
📐 Inradius & Circumradius Circle geometry calculations
📐 All Three Medians Centroid analysis
📐 Triangle Classification Identifies triangle type automatically
📐 Flexible Input Degrees or radians for angles
📐 Validity Checking Verifies triangle satisfies all constraints
📐 Step-by-Step Shows calculation methods used
📐 Property Reference Complete formulas and theorems
📱 Fully Responsive Works perfectly on all devices

How to Use This Calculator

Step-by-Step Guide

  1. Gather Information: Identify what you know about your triangle. You need exactly three values, including at least one side.
  2. Identify Valid Combinations: Valid input combinations include:
    • Three sides (SSS)
    • Two sides + included angle (SAS)
    • Two angles + included side (ASA)
    • Two angles + non-included side (AAS)
    • Two sides + non-included angle (SSA - may have ambiguous cases)
  3. Select Angle Unit: Choose degrees (°) or radians (rad). Degrees recommended unless working with trigonometric functions.
  4. Enter Known Values: Input the three known measurements in corresponding fields. Leave unknown fields blank—do not enter zeros for unknown values.
  5. Click Calculate: Press "Calculate Triangle" to solve.
  6. Review Complete Solution: See all sides, angles, area, perimeter, and properties instantly calculated.
  7. Verify Triangle Type: Note the automatically identified triangle type (e.g., isosceles acute).
  8. Copy or Continue: Use results or perform new calculations.

What Information You Need

The calculator requires exactly three values with at least one side. No configuration with only angles (no sides) can be solved, as angles alone determine shape but not size. Here's what works:

  • Three Sides (SSS): Most straightforward. Law of Cosines computes all angles.
  • Two Sides + Included Angle (SAS): Law of Cosines finds third side; Law of Sines finds remaining angles.
  • Two Angles + Any Side (AAS/ASA): Third angle from sum = 180°. Law of Sines finds remaining sides.
  • Two Sides + Non-included Angle (SSA): May have 0, 1, or 2 solutions (ambiguous case). Calculator handles all possibilities.

Complete Formulas Reference

The Law of Cosines

Finding Sides or Angles
To find side c: c² = a² + b² - 2ab·cos(C)
To find angle C: C = arccos((a² + b² - c²) / (2ab))

Used when: You know two sides and included angle, or all three sides

The Law of Sines

Finding Sides or Angles
a / sin(A) = b / sin(B) = c / sin(C)

Rearranged:
a = b × sin(A) / sin(B)
sin(A) = a × sin(B) / b

Used when: You know one angle-side pair and need to find others

Pythagorean Theorem (Right Triangles Only)

For Right Triangles (One 90° Angle)
a² + b² = c²

Where c is hypotenuse (side opposite 90° angle)
Special right triangles: 30-60-90 and 45-45-90

Area Calculations

Multiple Area Methods
Method 1: Area = (1/2) × base × height
Method 2: Area = (1/2) × a × b × sin(C)
Method 3: Area = √(s(s-a)(s-b)(s-c)) [Heron's]
where s = (a + b + c) / 2 (semiperimeter)

Choose based on what you know

Inradius and Circumradius

Circle Geometry
Inradius (inscribed circle): r = Area / s
Circumradius (circumscribed circle): R = a / (2×sin(A))

Where s is semiperimeter and Area computed using Heron's formula or other method

Medians (From Vertices to Opposite Side Midpoints)

Median Length
Median to side a: mₐ = (1/2)√(2b² + 2c² - a²)
Median to side b: m_b = (1/2)√(2a² + 2c² - b²)
Median to side c: m_c = (1/2)√(2a² + 2b² - c²)

All three medians intersect at centroid (center of mass)

Triangle Types and Properties

Classification by Sides

Equilateral Triangle

All three sides equal length (a = b = c). All angles equal 60°. Highest symmetry. Formula simplifies: Area = (√3/4) × a².

Isosceles Triangle

Two sides equal length (a = b, or b = c, or a = c). Two angles equal. The side that differs is the base. Angles opposite equal sides are equal.

Scalene Triangle

All three sides different lengths (a ≠ b ≠ c). All angles different. Most general case. No special symmetries.

Classification by Angles

Acute Triangle

All angles less than 90° (A < 90°, B < 90°, C < 90°). All angles are "acute" (sharp). Can be equilateral, isosceles, or scalene.

Right Triangle

Exactly one angle equals 90°. The side opposite the right angle is called the hypotenuse (longest side). Pythagorean theorem applies: a² + b² = c².

Obtuse Triangle

One angle greater than 90° (obtuse angle). Other two angles must be acute. Cannot have more than one obtuse angle (angles sum to 180°).

Important Triangle Properties

Property Description Mathematical Statement
Angle Sum All interior angles sum to 180 degrees A + B + C = 180°
Triangle Inequality Sum of any two sides exceeds third a + b > c, b + c > a, a + c > b
Exterior Angle Exterior angle equals sum of non-adjacent interior angles Exterior = B + C (if exterior to A)
Largest Angle Largest angle opposite longest side If a is longest, then A is largest
Isosceles Base Angles Base angles of isosceles triangle equal If a = b, then A = B

Key Theorems and Laws

Pythagorean Theorem

For any right triangle with legs a and b, and hypotenuse c:

a² + b² = c²

This fundamental theorem connects algebra and geometry. The converse is also true: if a² + b² = c² for a triangle's sides, the triangle is right-angled. Special right triangles with simple integer side ratios (Pythagorean triples) include 3-4-5, 5-12-13, and 8-15-17.

Law of Sines

For any triangle:

a / sin(A) = b / sin(B) = c / sin(C) = 2R

Where R is the circumradius (radius of circle passing through all vertices)

This law connects sides and opposite angles proportionally. Useful for finding unknown sides or angles when you know an angle-side pair.

Law of Cosines

For any triangle:

a² = b² + c² - 2bc·cos(A)
b² = a² + c² - 2ac·cos(B)
c² = a² + b² - 2ab·cos(C)

Generalizes the Pythagorean theorem (when any angle is 90°, the cosine term becomes zero, yielding a² + b² = c²). Useful when you know two sides and included angle, or all three sides.

Heron's Formula

To find area when all three sides are known:

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

Elegant formula requiring only side lengths—no angles or height needed. Derived from other area formulas but remarkably efficient.

Worked Examples

Example 1: Solving with Three Sides (SSS)

Problem: Given a = 5, b = 6, c = 7. Find all angles and area.

Solution - Find Angles using Law of Cosines:
cos(A) = (b² + c² - a²) / (2bc) = (36 + 49 - 25) / (2×6×7) = 60/84 ≈ 0.714
A = arccos(0.714) ≈ 44.4°

cos(B) = (a² + c² - b²) / (2ac) = (25 + 49 - 36) / (2×5×7) = 38/70 ≈ 0.543
B = arccos(0.543) ≈ 57.1°

C = 180° - 44.4° - 57.1° = 78.5°

Find Area using Heron's Formula:
s = (5 + 6 + 7) / 2 = 9
Area = √(9×4×3×2) = √216 ≈ 14.7

Example 2: Right Triangle with Pythagorean Theorem

Problem: Right triangle with legs a = 3, b = 4. Find hypotenuse c and angles.

Solution:
Using Pythagorean Theorem:
c² = a² + b² = 9 + 16 = 25
c = 5

Find angles:
sin(A) = a / c = 3 / 5 = 0.6
A = arcsin(0.6) ≈ 36.87°

sin(B) = b / c = 4 / 5 = 0.8
B = arcsin(0.8) ≈ 53.13°

C = 90° (given)

Area = (1/2) × 3 × 4 = 6

Example 3: Two Sides and Included Angle (SAS)

Problem: Given a = 7, b = 8, C = 60°. Solve triangle.

Find side c using Law of Cosines:
c² = a² + b² - 2ab·cos(C)
c² = 49 + 64 - 2(7)(8)·cos(60°)
c² = 113 - 112(0.5) = 113 - 56 = 57
c ≈ 7.55

Find remaining angles using Law of Sines:
sin(A) / a = sin(C) / c
sin(A) = 7 × sin(60°) / 7.55 = 7 × 0.866 / 7.55 ≈ 0.803
A ≈ 53.5°

B = 180° - 60° - 53.5° = 66.5°

Example 4: Finding Area Different Ways

Problem: Triangle with a = 6, b = 8, C = 45°. Find area using two methods.

Method 1 - Using SAS Formula:
Area = (1/2) × a × b × sin(C)
Area = (1/2) × 6 × 8 × sin(45°)
Area = 24 × 0.707 ≈ 17.0

Method 2 - Find third side, then use Heron's Formula:
c² = a² + b² - 2ab·cos(C)
c² = 36 + 64 - 96·cos(45°) = 100 - 67.9 = 32.1
c ≈ 5.67

s = (6 + 8 + 5.67) / 2 = 9.835
Area = √(9.835 × 3.835 × 1.835 × 4.165) ≈ 17.0

Example 5: Equilateral Triangle Properties

Problem: Equilateral triangle with side a = 10. Find area, inradius, circumradius.

Solution:
Area = (√3 / 4) × a² = (√3 / 4) × 100 ≈ 43.3

Semiperimeter s = (10 + 10 + 10) / 2 = 15

Inradius r = Area / s = 43.3 / 15 ≈ 2.89

Circumradius R = a / (2 × sin(60°)) = 10 / (2 × 0.866) ≈ 5.77

Note: R = 2r for equilateral triangles

Frequently Asked Questions

❓ What's the difference between a triangle solver and this calculator?
A basic solver finds missing sides/angles. This calculator provides complete analysis: all measurements, area, perimeter, medians, inradius, circumradius, and automatically identifies triangle type. It's a comprehensive geometry tool.
❓ Can this solve any triangle?
Any valid triangle can be solved given three appropriate values (at least one side). Invalid inputs (violating triangle inequality or angle sum) are detected and reported. Some SSA configurations have ambiguous solutions.
❓ What's the SSA ambiguous case?
When given two sides and non-included angle, sometimes 0, 1, or 2 different triangles satisfy the conditions. This depends on the specific values. The calculator identifies and handles all cases appropriately.
❓ When should I use degrees vs radians?
Use degrees for most practical problems and education. Use radians for calculus, advanced trigonometry, physics, or when working with trigonometric functions. The calculator supports both—just select your preferred unit.
❓ What's Heron's formula and when is it useful?
Heron's formula calculates area from three side lengths without needing height or angles. Extremely useful for land surveying, construction, and any situation where you measure all three sides but not angles or height.
❓ What are inradius and circumradius?
Inradius is the radius of the circle that fits inside the triangle (incircle). Circumradius is the radius of the circle passing through all three vertices (circumcircle). Both are geometric properties derived from the triangle's measurements.
❓ How are medians useful?
Medians connect each vertex to the opposite side's midpoint. They intersect at the centroid (center of mass). Used in engineering for balance calculations, in physics for center of gravity, and in structural analysis.
❓ Can negative angles or sides work?
No. Geometric triangles require positive angles (between 0° and 180°) and positive side lengths. Negative values are rejected—only positive real numbers are valid for triangle geometry.
❓ What real-world applications use triangle calculations?
Countless: surveying land, navigation, architecture (roof angles), engineering (structural analysis), astronomy (distance calculations), physics (force vectors), construction, and cartography. Any field involving spatial measurements uses triangle geometry.
❓ How accurate are the calculations?
The calculator uses standard JavaScript precision (15-17 significant digits for floating-point numbers). Results are accurate to 4 decimal places. For engineering or scientific work, always verify critical calculations independently.

Solve Your Triangle Today

Whether you're a student learning geometry, an architect designing structures, a surveyor measuring land, or an engineer analyzing forces, this comprehensive triangle calculator solves any configuration instantly with complete geometric analysis. Get all sides, angles, area, perimeter, medians, and circle properties in seconds. Fast, accurate, and completely free.

Related Calculators