30-60-90 Triangle Calculator
Created by: Daniel Hayes
Last updated:
This 30-60-90 Triangle Calculator determines all sides and properties of a special right triangle based on any known side. Features trigonometric ratios, geometric relationships, and comprehensive triangle analysis.
30-60-90 Triangle Calculator
MathCalculate all properties of a 30-60-90 triangle from any known side
Related Calculators
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
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)
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)
30-60-90 Triangle Calculation Examples
Example 1: Given Short Side
Calculate all sides when the short side (opposite 30°) is 6 units.
- Given: Short side = 6 units
- Apply ratios (1 : √3 : 2)
- Long side = 6 × √3 = 6 × 1.732 = 10.39 units
- Hypotenuse = 6 × 2 = 12 units
- Area = (1/2) × 6 × 10.39 = 31.17 square units
Example 2: Given Hypotenuse
Calculate all sides when the hypotenuse is 20 units.
- Given: Hypotenuse = 20 units
- Short side = 20 ÷ 2 = 10 units
- Long side = 20 × (√3 ÷ 2) = 10√3 = 17.32 units
- Area = (1/2) × 10 × 17.32 = 86.6 square units
Common Applications
- Architecture and Construction: Calculate roof angles, ramp inclinations, structural support angles with precise 30°, 60°, or 90° angles
- Engineering Design: Design mechanical components, structural frameworks requiring specific angular relationships and proportional measurements
- Navigation and Surveying: Calculate distances and heights using triangulation methods, compass bearings, and elevation measurements
- Mathematics Education: Teach trigonometry concepts, special triangle relationships, and geometric principles to students
- Manufacturing and Machining: Calculate cutting angles, tool positioning, and part dimensions for precision manufacturing
- Physics and Science: Analyze force vectors, wave propagation, optics calculations involving angular measurements
Frequently Asked Questions
What is a 30-60-90 triangle and how do I calculate its sides?
A 30-60-90 triangle is a special right triangle with angles of 30°, 60°, and 90°. The sides are in the ratio 1 : √3 : 2. If the short side (opposite 30°) is s, then the long side is s√3 and the hypotenuse is 2s.
Why are the side ratios of a 30-60-90 triangle always 1 : √3 : 2?
These ratios come from trigonometry. In a 30-60-90 triangle, sin(30°) = 1/2, cos(30°) = √3/2, and these relationships create the fixed proportional ratios between the sides regardless of the triangle's size.
How do I find all sides if I only know one side of a 30-60-90 triangle?
Use the fixed ratios: if you know the short side, multiply by √3 for the long side and by 2 for the hypotenuse. If you know the hypotenuse, divide by 2 for the short side and multiply by √3/2 for the long side.
What are the practical applications of 30-60-90 triangles?
Common in construction (roof angles), engineering (structural supports), navigation (compass bearings), and manufacturing (cutting angles). The predictable ratios make calculations simple for precise angular work.
How accurate are the calculations for 30-60-90 triangles?
Calculations are mathematically exact using the precise ratios. Real-world applications depend on measurement accuracy. The calculator uses high-precision values for √3 to ensure reliable results for both educational and professional use.
30-60-90 Triangle Calculation Tips
- Remember the ratios: The side ratios 1 : √3 : 2 are always consistent for any 30-60-90 triangle
- Identify the angle: Determine which side you know (opposite 30°, 60°, or 90°) to apply the correct formula
- Use precise values: √3 = 1.732050808... Use sufficient decimal places for your precision needs
- Check your work: Verify using the Pythagorean theorem: a² + b² = c²
- Convert units: Ensure all measurements are in the same unit before calculating
Sources and References
- Larson, R., & Edwards, B. H. (2020). Trigonometry. Cengage Learning.
- Jacobs, H. R. (2003). Geometry: Seeing, Doing, Understanding. W. H. Freeman and Company.
- Aufmann, R. N., & Barker, V. C. (2018). Trigonometry. Cengage Learning.