About this calculator
The 30-60-90 Triangle Calculator finds all three sides from any one side of this special right triangle. It is useful for exact trigonometry, geometry revision, and problems where the angles are fixed at 30, 60, and 90 degrees.
30-60-90 triangle calculator method
A 30-60-90 triangle always has side ratio 1 : sqrt(3) : 2. The short leg is opposite 30 degrees, the long leg is opposite 60 degrees, and the hypotenuse is twice the short leg.
- short leg = n
- long leg = n x sqrt(3)
- hypotenuse = 2n
How to use the 30-60-90 triangle calculator
- Choose which side is known.
- Enter the side length.
- Convert the known side into the short leg n.
- Multiply n by sqrt(3) for the long leg.
- Multiply n by 2 for the hypotenuse.
- Check the ratio 1 : sqrt(3) : 2.
Worked examples
Short leg known
Input: short leg = 5
Calculation: long = 5sqrt(3), hypotenuse = 10
Result: long about 8.66, hypotenuse 10
Hypotenuse known
Input: hypotenuse = 20
Calculation: short = 20 / 2, long = 10sqrt(3)
Result: short 10, long about 17.32
Why the ratio works
A 30-60-90 triangle is half of an equilateral triangle. That structure creates the fixed side relationship between the short leg, long leg, and hypotenuse.
Common mistakes to avoid
- Using the wrong side label
- Triangle formulas depend on matching sides with their opposite angles or with the correct right-triangle role. If a result looks impossible, recheck the labels before changing the formula.
- Forgetting triangle validity
- Not every set of side lengths can form a triangle. The longest side must be shorter than the sum of the other two sides.
- Rounding too early
- Keep extra decimal places while calculating, especially when using square roots, sine, cosine, or inverse trig. Round the final answer to the precision required.
Edge cases
- A right triangle must have one angle of exactly 90 degrees.
- The hypotenuse must be the longest side in a right triangle.
- Special triangle ratios only apply to exact 30-60-90 and 45-45-90 triangles.
- The sine rule can produce an ambiguous SSA case where two triangles are possible.
Limitations
This calculator is for educational maths support. It uses standard geometry and trigonometry formulas with decimal approximations. For exams, coursework, engineering, surveying, or construction, follow the required method, units, tolerances, and checking process.
Frequently asked questions
Which triangle calculator should I use?
Use Pythagoras or the right triangle calculator for right-angled triangles, special triangle calculators for exact 30-60-90 or 45-45-90 triangles, and the sine or cosine law calculators for non-right triangles.
What do sides a, b, and c mean?
In general triangle notation, side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. Some right-triangle pages also label c as the hypotenuse.
Why do triangle angles add to 180 degrees?
In ordinary flat Euclidean geometry, the interior angles of a triangle always add to 180 degrees. That rule is used throughout these calculators.
Can the calculator handle any units?
Yes for lengths, as long as all side inputs use the same unit. Area results are in square units based on the unit entered.
Why is my result impossible?
The inputs may not form a valid triangle, the hypotenuse may not be the longest side, or a side may have been paired with the wrong angle.
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