30-60-90 Triangle Calculator

What is a 30-60-90 Triangle Calculator?

A 30-60-90 Triangle Calculator is a specialized geometric tool that calculates all sides and properties of a 30-60-90 degree triangle based on any one known side. This type of triangle is a special right triangle with unique proportional relationships that make calculations straightforward and predictable.

The 30-60-90 triangle is one of two special right triangles (along with the 45-45-90 triangle) that frequently appears in geometry, trigonometry, and engineering applications. Its sides are always in the ratio 1 : √3 : 2, making it possible to determine all measurements from just one known side length.

Beyond basic calculation, this tool helps students and professionals understand trigonometric relationships, apply geometric principles, and solve real-world problems involving angles and measurements in construction, navigation, and design applications.

30-60-90 Triangle Formulas

The 30-60-90 triangle has fixed side ratios that allow calculation of all sides from any known side:

Side Ratio Relationship

Opposite 30° : Opposite 60° : Hypotenuse = 1 : √3 : 2 Short Side : Long Side : Hypotenuse = 1 : √3 : 2

Formulas Based on Known Side

If the short side (opposite 30°) = s:

Short Side = s Long Side = s × √3 Hypotenuse = s × 2

If the long side (opposite 60°) = l:

Short Side = l ÷ √3 Long Side = l Hypotenuse = l × (2 ÷ √3) = l × (2√3 ÷ 3)

If the hypotenuse = h:

Short Side = h ÷ 2 Long Side = h × (√3 ÷ 2) Hypotenuse = h

Additional Properties

Area = (1/2) × Short Side × Long Side Perimeter = Short Side + Long Side + Hypotenuse

Where:

• √3 ≈ 1.732 (square root of 3)
• Short Side = Side opposite the 30° angle
• Long Side = Side opposite the 60° angle
• Hypotenuse = Side opposite the 90° angle (longest side)

How to Calculate 30-60-90 Triangle: Example

Example Scenario:

Calculate all sides of a 30-60-90 triangle when the short side (opposite 30°) is 6 units.

1. Identify the given information:
   • Triangle type: 30-60-90
   • Known side: Short side = 6 units
   • Need to find: Long side and hypotenuse
2. Apply the 30-60-90 triangle ratios:Short Side : Long Side : Hypotenuse = 1 : √3 : 2
3. Calculate the long side:
   • Long Side = Short Side × √3
   • Long Side = 6 × √3
   • Long Side = 6 × 1.732
   • Long Side = 10.39 units
4. Calculate the hypotenuse:
   • Hypotenuse = Short Side × 2
   • Hypotenuse = 6 × 2
   • Hypotenuse = 12 units
5. Calculate additional properties:
   • Area = (1/2) × 6 × 10.39 = 31.17 square units
   • Perimeter = 6 + 10.39 + 12 = 28.39 units
6. Final Results:
   • Short Side (30°): 6 units
   • Long Side (60°): 10.39 units
   • Hypotenuse (90°): 12 units
   • Area: 31.17 square units

Common Applications

• Architecture and Construction: Calculate roof angles, ramp inclinations, structural support angles, and building design elements with precise 30°, 60°, or 90° angles
• Engineering Design: Design mechanical components, structural frameworks, and systems requiring specific angular relationships and proportional measurements
• Navigation and Surveying: Calculate distances and heights using triangulation methods, compass bearings, and elevation measurements in field work
• Mathematics Education: Teach trigonometry concepts, special triangle relationships, and geometric principles to students at various levels
• Computer Graphics and Animation: Calculate coordinates, rotations, and transformations in 2D and 3D graphics programming and design applications
• Physics and Science: Analyze force vectors, wave propagation, optics calculations, and experimental setups involving angular measurements
• Art and Design: Create proportional designs, geometric patterns, and artistic compositions using mathematical relationships and angular precision
• Manufacturing and Machining: Calculate cutting angles, tool positioning, and part dimensions for precision manufacturing and quality control