30-60-90 Triangle Calculator

Calculate all sides of a 30-60-90 special right triangle

Enter the length of any side to calculate all sides of a 30-60-90 triangle. This special right triangle has fixed ratios: 1 : √3 : 2.

Must be a positive number

Examples

Click on any example to load it into the calculator

Short Side Known

short

Calculate from the shortest side (opposite 30°)

Side Type: short

Length: 5

Long Side Known

long

Calculate from the longer side (opposite 60°)

Side Type: long

Length: 8.660

Hypotenuse Known

hypotenuse

Calculate from the hypotenuse (opposite 90°)

Side Type: hypotenuse

Length: 10

Unit Triangle

short

Standard unit 30-60-90 triangle

Side Type: short

Length: 1

Other Titles
Understanding 30-60-90 Triangle Calculator: A Comprehensive Guide
Explore the properties of 30-60-90 special right triangles, their applications in geometry, trigonometry, and real-world problem solving

Understanding 30-60-90 Triangle Calculator: A Comprehensive Guide

  • 30-60-90 triangles are special right triangles with unique side ratios
  • They appear frequently in geometry, trigonometry, and engineering
  • Understanding these triangles simplifies many mathematical calculations
A 30-60-90 triangle is a special right triangle where the angles measure 30°, 60°, and 90°. This triangle is called 'special' because its sides have a consistent, predictable ratio.
The sides of a 30-60-90 triangle are always in the ratio 1 : √3 : 2, where 1 corresponds to the side opposite the 30° angle, √3 to the side opposite the 60° angle, and 2 to the hypotenuse.
This consistent ratio makes 30-60-90 triangles extremely useful in geometry and trigonometry, as knowing one side immediately allows calculation of the other two sides.
These triangles commonly appear in construction, engineering, and design applications where precise angle measurements are required.

Basic Side Calculations

  • If short side = 4, then long side = 4√3 ≈ 6.928 and hypotenuse = 8
  • If hypotenuse = 12, then short side = 6 and long side = 6√3 ≈ 10.392
  • If long side = 9, then short side = 9/√3 ≈ 5.196 and hypotenuse = 18/√3 ≈ 10.392

Step-by-Step Guide to Using the 30-60-90 Triangle Calculator

  • Learn how to identify which side you know
  • Understand the calculator's input requirements
  • Master interpreting and verifying results
Our 30-60-90 triangle calculator simplifies the process of finding all sides when you know just one side length.
Step 1: Identify the Known Side
  • Short Side: The side opposite the 30° angle (shortest side)
  • Long Side: The side opposite the 60° angle (medium length)
  • Hypotenuse: The side opposite the 90° angle (longest side)
Step 2: Enter the Value
Input the length of your known side. The calculator accepts any positive number and will compute the other sides using the exact ratios.
Step 3: Verify Results
Check that the calculated sides maintain the 1 : √3 : 2 ratio. The short side should be half the hypotenuse, and the long side should be √3 times the short side.

Calculator Usage Examples

  • Known: Short side = 7 → Results: Long side = 7√3 ≈ 12.124, Hypotenuse = 14
  • Known: Hypotenuse = 20 → Results: Short side = 10, Long side = 10√3 ≈ 17.321
  • Known: Long side = 15 → Results: Short side = 15/√3 ≈ 8.660, Hypotenuse = 30/√3 ≈ 17.321

Real-World Applications of 30-60-90 Triangle Calculations

  • Architecture and Construction: Roof angles and structural design
  • Engineering: Component design and stress analysis
  • Navigation: Bearing calculations and route planning
  • Art and Design: Perspective drawing and geometric patterns
30-60-90 triangles appear frequently in practical applications across many fields:
Architecture and Construction:
  • Roof Design: Many roof pitches use 30° or 60° angles for optimal water drainage and structural integrity.
  • Staircase Construction: 30-60-90 triangles help determine optimal rise and run ratios for comfortable stair climbing.
Engineering Applications:
  • Mechanical Design: Components often incorporate 30° and 60° angles for strength and aesthetic appeal.
  • Bridge Construction: Truss designs frequently use these angles for optimal load distribution.
Navigation and Surveying:
  • Triangulation: Surveyors use 30-60-90 triangles for precise distance and angle measurements.
  • GPS Calculations: Satellite positioning systems utilize these triangles in coordinate transformations.

Practical Applications

  • Roof with 30° pitch: If rise = 6 feet, then run = 6√3 ≈ 10.39 feet, rafter = 12 feet
  • Hexagonal bolt head: Each triangle has 30-60-90 angles with predictable side ratios
  • Triangular road sign: 30-60-90 proportions ensure stability and visibility
  • Solar panel angle: 30° tilt optimizes sun exposure in many latitudes

Common Misconceptions and Correct Methods in 30-60-90 Triangles

  • Addressing frequent errors in triangle identification
  • Clarifying the difference between 30-60-90 and other special triangles
  • Explaining why the ratios are exact, not approximations
Understanding common misconceptions about 30-60-90 triangles helps avoid calculation errors and builds confidence in using these special triangles.
Misconception 1: All Right Triangles Have Special Ratios
Wrong: Assuming any right triangle follows the 1 : √3 : 2 ratio.
Correct: Only right triangles with 30° and 60° angles have this specific ratio. Other right triangles require different calculation methods.
Misconception 2: The Ratios Are Approximations
Wrong: Treating 1 : √3 : 2 as rough estimates.
Correct: These ratios are mathematically exact. √3 is an irrational number, so decimal approximations are rounded, but the ratio itself is precise.
Misconception 3: Confusing 30-60-90 with 45-45-90 Triangles
Wrong: Using the 1 : 1 : √2 ratio (from 45-45-90 triangles) for 30-60-90 calculations.
Correct: 30-60-90 triangles use 1 : √3 : 2, while 45-45-90 triangles use 1 : 1 : √2. These are completely different ratios.

Common Error Corrections

  • Wrong: A right triangle with sides 3, 4, 5 follows 30-60-90 ratios
  • Correct: A right triangle with sides 3, 4, 5 is a general right triangle
  • Wrong: In a 30-60-90 triangle, if short side = 5, then long side ≈ 5.2
  • Correct: If short side = 5, then long side = 5√3 ≈ 8.660

Mathematical Derivation and Examples

  • Understanding why the 1 : √3 : 2 ratio exists
  • Deriving the ratios using trigonometry
  • Connecting to unit circle and trigonometric functions
The 1 : √3 : 2 ratio in 30-60-90 triangles can be derived from fundamental trigonometric principles.
Trigonometric Derivation:
Starting with a 30-60-90 triangle where the hypotenuse = 2 and the short side = 1:
  • sin(30°) = opposite/hypotenuse = short side/hypotenuse = 1/2
  • cos(30°) = adjacent/hypotenuse = long side/hypotenuse = √3/2
  • Therefore: long side = √3, confirming the 1 : √3 : 2 ratio
Unit Circle Connection:
On the unit circle, the point at 30° is (√3/2, 1/2), and at 60° is (1/2, √3/2). These coordinates directly reflect the 30-60-90 triangle ratios.
Pythagorean Verification:
For sides in ratio 1 : √3 : 2, the Pythagorean theorem confirms: 1² + (√3)² = 1 + 3 = 4 = 2², proving these form a valid right triangle.

Mathematical Proofs

  • Verification: If sides are 6, 6√3, 12, then 6² + (6√3)² = 36 + 108 = 144 = 12²
  • Trigonometry: sin(60°) = √3/2, confirming the ratio relationships
  • Unit circle: Point (√3/2, 1/2) represents cos(30°) and sin(30°)
  • Scaling: Any 30-60-90 triangle is just a scaled version of the 1:√3:2 fundamental triangle