30-60-90 Triangle Calculator - Calculate Triangle Properties & Ratios
Free 30-60-90 triangle calculator. Calculate sides, angles, area, and perimeter of 30-60-90 triangles with step-by-step solutions and ratios. Our calculator uses the special right triangle properties to determine all triangle measurements from any given side length.
Last updated: October 19, 2025
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5
opposite 30°
8.660254
opposite 60°
10
opposite 90°
21.650635 square units
23.660254 units
Short : Hypotenuse
1 : 2
Actual: 0.5 : 1
Long : Hypotenuse
√3 : 2
Actual: 0.866025 : 1
Short : Long
1 : √3
Actual: 0.57735 : 1
Key Relationships:
- • Hypotenuse = 2 × Short Side
- • Long Side = √3 × Short Side
- • Long Side = √3/2 × Hypotenuse
30-60-90 Triangle Calculator Types & Features
Side relationships
1 : √3 : 2
Calculates short side, long side, and hypotenuse from any input
Area formula
A = (√3/2) × s²
Uses the special 30-60-90 triangle area formula
Perimeter formula
P = s(3 + √3)
Calculates perimeter from short side length
Triangle type
30°-60°-90°
Special right triangle with unique angle relationships
Key ratios
Short:Long:Hyp = 1:√3:2
Shows fundamental ratios of 30-60-90 triangles
Angles
30°, 60°, 90°
Fixed angle measurements for 30-60-90 triangles
Quick Example Result
For a 30-60-90 triangle with short side = 5:
Short Side
5
Long Side
8.66
Hypotenuse
10
How Our 30-60-90 Triangle Calculator Works
Our 30-60-90 triangle calculator uses the fundamental properties of special right triangles to calculate all measurements from any given side length. The calculation applies geometric relationships and trigonometric principles to determine sides, angles, area, and perimeter.
The 30-60-90 Triangle Ratios
Short Side : Long Side : Hypotenuse = 1 : √3 : 2Long Side = Short Side × √3Hypotenuse = Short Side × 2Area = (√3/2) × Short Side²These ratios are fundamental to 30-60-90 triangles and allow calculation of all properties from any single side length. The relationships are based on the geometric properties of equilateral triangles.
Shows the side relationships and angle measurements
Mathematical Foundation
The 30-60-90 triangle is derived from an equilateral triangle by drawing an altitude, which creates two congruent 30-60-90 triangles. The side ratios come from the geometric properties of equilateral triangles and the Pythagorean theorem.
- 30-60-90 triangles are half of equilateral triangles
- The short side is opposite the 30° angle
- The long side is opposite the 60° angle
- The hypotenuse is opposite the 90° angle
- All angles are fixed: 30°, 60°, 90°
- Side ratios are always 1:√3:2
Sources & References
- Geometry: A Comprehensive Course - Dan PedoeStandard reference for geometric properties and special triangles
- Trigonometry - Michael SullivanComprehensive coverage of special right triangles and trigonometric ratios
- Khan Academy - Special Right TrianglesEducational resources for understanding 30-60-90 triangle properties
Need help with other triangle calculations? Check out our triangle area calculator and Pythagorean theorem calculator.
Get Custom Calculator for Your Platform30-60-90 Triangle Calculator Examples
Given Information:
- Short side: 6 units
- Angle opposite: 30°
- Triangle type: 30-60-90
Calculation Steps:
- Long side = 6 × √3 = 10.39
- Hypotenuse = 6 × 2 = 12
- Area = (6 × 10.39) ÷ 2 = 31.18
- Perimeter = 6 + 10.39 + 12 = 28.39
Result: Short side = 6, Long side = 10.39, Hypotenuse = 12
Area = 31.18 square units, Perimeter = 28.39 units
Given Long Side Example
Long side = 9√3
Short side = 9, Hypotenuse = 18
Given Hypotenuse Example
Hypotenuse = 14
Short side = 7, Long side = 7√3
Frequently Asked Questions
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