30-60-90 Triangle
The 30-60-90 triangle has fixed side ratio 1:√3:2. It is exactly half of an equilateral triangle. Shortest opposite 30°, middle opposite 60°, hypotenuse opposite 90°. One side determines all.
Why This Mathematical Concept Matters
Why: 30-60-90 triangles appear in hexagonal structures, roof trusses, and give exact trig values: sin 30°=½, sin 60°=√3/2. Honeycomb cells use this geometry.
How: Given shortest: middle = shortest×√3, hypotenuse = 2×shortest. Given middle: shortest = middle/√3, hypotenuse = 2×shortest. Given hypotenuse: shortest = hyp/2, middle = shortest×√3.
- ●A 30-60-90 triangle is half of an equilateral triangle—bisect with an altitude.
- ●The ratio gives exact trig values without a calculator.
- ●Hexagons divide into six 30-60-90 triangles radiating from the center.
30-60-90 — The 1:√3:2 Ratio
Half of an equilateral triangle. One side gives you all sides, area, and perimeter. Essential for trigonometry and hexagonal geometry.
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Enter Triangle Values
Triangle Visualization
Side Length Ratios
Angle Proportions (30° : 60° : 90°)
Properties Radar
Step-by-Step Breakdown
⚠️For educational and informational purposes only. Verify with a qualified professional.
🧮 Fascinating Math Facts
30-60-90 is half of an equilateral triangle—cut with an altitude.
— Wolfram MathWorld
sin(30°)=½, cos(30°)=√3/2, tan(30°)=1/√3—exact values from the ratio.
— Khan Academy
Key Takeaways
- The 30-60-90 triangle has a fixed side ratio of 1 : √3 : 2 (shortest : middle : hypotenuse)
- It is exactly half of an equilateral triangle — bisect any equilateral triangle to get two 30-60-90 triangles
- Angles are always 30°, 60°, and 90° — the shortest side is opposite 30°, middle opposite 60°, hypotenuse opposite 90°
- Knowing one side is enough to find all others — no trigonometry needed
- Essential for hexagonal geometry, roof trusses, and trigonometric exact values (sin 30° = ½, sin 60° = √3/2)
Did You Know?
How 30-60-90 Triangles Work
The 30-60-90 triangle derives from an equilateral triangle. When you draw an altitude from one vertex to the opposite side, you split the equilateral triangle into two congruent right triangles.
Derivation from Equilateral Triangle
Start with an equilateral triangle of side length 2. The altitude has length √3 (by Pythagorean theorem: √(2² − 1²) = √3). This creates a 30-60-90 triangle with sides 1, √3, and 2.
Scaling from One Known Side
If shortest = s, then middle = s√3 and hypotenuse = 2s. If middle = m, then shortest = m/√3 and hypotenuse = 2m/√3. If hypotenuse = h, then shortest = h/2 and middle = h√3/2.
Applications
Used in trigonometry for exact values, in architecture for hexagonal and triangular structures, in navigation for 30° and 60° bearings, and in graphics for mesh subdivision.
Expert Tips
Memorize the Ratio 1:√3:2
The shortest side is always 1, middle is √3, hypotenuse is 2. Scale by your known side to get actual lengths.
Don't Confuse with 45-45-90
45-45-90 has ratio 1:1:√2. The 30-60-90 is not isosceles — its legs have different lengths.
Keep √3 Exact Until the End
Use √3 ≈ 1.732 only when rounding final answers. Preserve exact form (e.g., 10√3) when possible.
Area = ½ × Shortest × Middle
The right angle is between the shortest and middle sides, so area = ½ab using those two legs.
30-60-90 vs 45-45-90
| Property | 30-60-90 | 45-45-90 |
|---|---|---|
| Angles | 30°, 60°, 90° | 45°, 45°, 90° |
| Side ratio | 1 : √3 : 2 | 1 : 1 : √2 |
| Isosceles? | No | Yes |
| Derived from | Half of equilateral | Half of square |
| Hypotenuse formula | 2 × shortest | √2 × leg |
Frequently Asked Questions
Why is it called a 30-60-90 triangle?
The three angles are always 30°, 60°, and 90°. The name tells you the angle measures. The shortest side is opposite the 30° angle, the middle side opposite 60°, and the hypotenuse opposite the 90° right angle.
What is the 1:√3:2 ratio?
It means if the shortest side has length 1, the middle side has length √3 (≈1.732), and the hypotenuse has length 2. Scale this ratio by any factor to get any 30-60-90 triangle.
How is a 30-60-90 triangle related to an equilateral triangle?
Bisect an equilateral triangle with an altitude. You get two congruent 30-60-90 triangles. The altitude of an equilateral triangle of side 2 is √3, giving the classic 1:√3:2 triangle.
Can I use this for any right triangle?
No. Only right triangles with angles 30°, 60°, and 90° have this ratio. A 3-4-5 triangle is right but not 30-60-90.
What are exact trig values for 30° and 60°?
sin(30°)=½, cos(30°)=√3/2, tan(30°)=1/√3. sin(60°)=√3/2, cos(60°)=½, tan(60°)=√3. These come directly from the 30-60-90 ratio.
Where are 30-60-90 triangles used in real life?
Roof trusses, hexagonal structures (honeycombs, bolts), ramps with 30° incline, surveying with 30° bearings, and 3D graphics mesh subdivision.
How do I find the area?
Area = ½ × shortest × middle. The right angle is between these two legs, so they serve as base and height.
What if I know the hypotenuse?
Shortest = hypotenuse ÷ 2, middle = shortest × √3. The hypotenuse is always exactly twice the shortest side.
30-60-90 by the Numbers
Official & Trusted Sources
Disclaimer: This calculator provides mathematically precise results based on the 30-60-90 triangle ratio 1:√3:2. Results use standard floating-point precision. For critical engineering or surveying applications, verify with domain-specific tools. Not a substitute for professional analysis.