30-60-90 Triangle Calculator
Special right triangle with sides in ratio 1 : √3 : 2
Enter Any Side
Side Ratios
1 : √3 : 2
• Short leg (30°) = x
• Long leg (60°) = x√3
• Hypotenuse (90°) = 2x
30-60-90 Triangle
Understanding the 30-60-90 Triangle
The 30-60-90 triangle is one of the two special right triangles (the other being 45-45-90). It has angles of 30°, 60°, and 90°, with sides always in the ratio 1 : √3 : 2.
Origin: Half of an Equilateral Triangle
A 30-60-90 triangle is exactly half of an equilateral triangle. When you draw a height (altitude) in an equilateral triangle, it bisects the triangle into two congruent 30-60-90 triangles.
The Side Relationships
- Short leg (x): Opposite the 30° angle
- Long leg (x√3): Opposite the 60° angle
- Hypotenuse (2x): Opposite the 90° angle, always twice the short leg
Conversion Formulas
- If short leg = x: long leg = x√3, hypotenuse = 2x
- If long leg = y: short leg = y/√3, hypotenuse = 2y/√3
- If hypotenuse = z: short leg = z/2, long leg = z√3/2
Common Applications
- Calculating heights in equilateral triangles
- Trigonometry problems involving 30° or 60°
- Construction and architectural design
- Hexagonal patterns and honeycomb structures
Frequently Asked Questions
Why is the ratio 1 : √3 : 2?
This ratio comes from the relationship between an equilateral triangle and its altitude. When you bisect an equilateral triangle, the altitude has length (side × √3)/2, creating this specific ratio.
Why is the hypotenuse exactly twice the short leg?
In a 30-60-90 triangle, sin(30°) = opposite/hypotenuse = short leg/hypotenuse = 1/2. This means hypotenuse = 2 × short leg.
How is this related to an equilateral triangle?
A 30-60-90 triangle is exactly half of an equilateral triangle. The altitude of an equilateral triangle creates two congruent 30-60-90 triangles.
What is √3 approximately equal to?
√3 ≈ 1.732. So if the short leg is 1, the long leg is about 1.732, and the hypotenuse is 2.
When should I use this instead of trig functions?
Use this special triangle ratio whenever you have a 30° or 60° angle in a right triangle. It's often faster than calculating with sine, cosine, or tangent.
How do I rationalize √3 in the denominator?
Multiply by √3/√3. For example, x/√3 = (x√3)/3. This gives an equivalent form without a radical in the denominator.
What are the exact trig values for 30° and 60°?
sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = 1/√3. For 60°: sin(60°) = √3/2, cos(60°) = 1/2, tan(60°) = √3.
What's the area formula for a 30-60-90 triangle?
Area = (1/2) × short leg × long leg = (1/2) × x × x√3 = (x²√3)/2, where x is the short leg.