30 60 90 Calculator: Solve Any Triangle Fast
A 30-60-90 triangle locks its sides into the ratio 1: √3: 2, so a single known value fixes the whole triangle. This calculator accepts any one starting point (the short leg, the long leg, the hypotenuse, the area, the altitude to the hypotenuse, or the perimeter) and returns every side, the area, the perimeter, the altitude, the exact radical form, and a six-step working you can copy into homework.
Triangle solved
Results are shown to four decimal places. The exact form line keeps √3 unevaluated, so a homework answer such as 5√3 is never rounded away.
Quick Answer
With a short leg of 5, the long leg is 5√3 = 8.6603, the hypotenuse is 10.0000, the area is 21.6506, the perimeter is 23.6603 and the altitude to the hypotenuse is 4.3301. The same triangle comes back if you start from its long leg 8.6603, its hypotenuse 10, its area 21.6506, its altitude 4.3301, or its perimeter 23.6603: every route recovers the short leg 5 first, then applies the 1: √3: 2 ratio.
Key takeaways
- One value is enough. A side, the area, the altitude or the perimeter each determine the short leg, and the ratio does the rest.
- The short leg sits opposite 30°, the long leg opposite 60°, and the hypotenuse is exactly twice the short leg.
- The exact form keeps √3 symbolic, so 5√3 stays exact instead of becoming a rounded 8.6603.
- The unit selector is a label, not a converter. Pick cm and every result is labelled cm, with the area in cm².
- Quick Answer
- How to use the 30-60-90 calculator
- Where the 1: √3: 2 ratio comes from, and every inverse formula
- Worked example: short leg 5, checked from three other directions
- 30-60-90 vs 45-45-90: the two special right triangles side by side
- What this calculator covers, and why each part matters
- Frequently asked questions
- Add This Calculator to Your Website
- References
How to use the 30-60-90 calculator
Pick what you know, type its value, and click Calculate. The results stay hidden until you calculate, and a Show steps button under them reveals the full working.
- Say what the value is: the short leg, long leg or hypotenuse, or one of the three derived measures: area, altitude to the hypotenuse, or perimeter. Problems that hand you the altitude or the perimeter are the ones most single-side tools refuse.
- Enter the number: it must be greater than zero. Blank, text and negative entries stop with a message rather than a wrong triangle.
- Add a unit label if you need one: mm through mi, or unitless for pure numbers. The label is carried through every result and squared on the area.
- Read the solved triangle: all three sides, area, perimeter, altitude, the exact radical form, and a ratio check line that divides your three sides back down to 1: 1.7321: 2.0000 so you can see the answer is consistent.
Where the 1: √3: 2 ratio comes from, and every inverse formula
Halve an equilateral triangle and the ratio falls out. Split a side-2 equilateral triangle along an altitude and each half is a right triangle with hypotenuse 2, short leg 1, and long leg √(2² − 1²) = √3 by the Pythagorean theorem, a construction set out in reference works such as Wolfram MathWorld’s 30-60-90 triangle entry. Scale that triangle by a and the sides become a, a√3 and 2a.
Each starting point inverts one of those formulas to recover the short leg a, which is all the tool ever needs:
| You know | Short leg a = | Why |
|---|---|---|
| Short leg | the value itself | Given directly |
| Long leg b | b ÷ √3 | Inverts b = a√3 |
| Hypotenuse c | c ÷ 2 | The hypotenuse is always double |
| Area A | √(2A ÷ √3) | Inverts A = (√3/2)a² |
| Altitude h | 2h ÷ √3 | Inverts h = a√3/2 |
| Perimeter P | P ÷ (3 + √3) | P = a + a√3 + 2a = a(3 + √3), and 3 + √3 ≈ 4.7321 |
The ratio also fixes the standard trig values for 30° and 60°: sin 30° = 1/2, cos 30° = √3/2, tan 60° = √3, matching the tabulated values in the NIST Digital Library of Mathematical Functions. If a problem gives you an angle and a side instead of naming the triangle, the sin calculator covers the ratio side of the same work.
Worked example: short leg 5, checked from three other directions
A short leg of 5 gives a long leg of 5√3 = 8.6603 and a hypotenuse of 10.0000. The calculator reaches that answer in six visible steps:
The stronger test is starting from a derived measure and landing back on the same triangle. Every row below is a real run of this calculator:
| Starting value | Short leg | Long leg | Hypotenuse | Area | Altitude |
|---|---|---|---|---|---|
| Long leg 10 cm | 5.7735 cm | 10.0000 cm | 11.5470 cm | 28.8675 cm² | 5.0000 cm |
| Hypotenuse 12 | 6.0000 | 10.3923 | 12.0000 | 31.1769 | 5.1962 |
| Altitude 6 | 6.9282 | 12.0000 | 13.8564 | 41.5692 | 6.0000 |
| Perimeter 32.7846 | 6.9282 | 12.0000 | 13.8564 | 41.5692 | 6.0000 |
| Area 100 m² | 10.7457 m | 18.6121 m | 21.4914 m | 100.0000 m² | 9.3060 m |
Notice the altitude and perimeter rows describe the same triangle from two different givens, which is the quickest self-check the tool offers: solve from the altitude your problem states, then confirm the perimeter line matches the perimeter your problem states. For perimeter work on shapes in general, the perimeter formula calculator covers the wider family.
30-60-90 vs 45-45-90: the two special right triangles side by side
Both special right triangles trade trigonometry for a fixed ratio, but the ratios differ and so do the traps. The 45-45-90 has two equal legs, which makes it easy to mislabel; the 30-60-90 has no equal sides, so the whole job is keeping the short and long legs straight.
30-60-90 (this page)
- Ratio 1: √3: 2, no equal sides
- Hypotenuse is exactly twice the short leg
- Comes from halving an equilateral triangle
- Solvable here from the area, altitude or perimeter too
45-45-90
- Ratio 1: 1: √2, the two legs are equal
- Hypotenuse is a leg times √2
- Comes from halving a square along its diagonal
- Covered by the 45-45-90 triangle calculator
What this calculator covers, and why each part matters
This calculator takes all six starting points directly, and returns every side, the area, the perimeter, the altitude, the exact radical form, and a six-step working you can copy into homework.
- All six starting points. The short leg, the long leg, the hypotenuse, the area, the altitude to the hypotenuse, or the perimeter: a problem that states any one of them can be entered directly, with no hand inversion first.
- One measure solves the whole triangle. Every route recovers the short leg first, then applies the 1: √3: 2 ratio, so the same triangle comes back whichever starting point you enter.
- Exact form kept exact. The exact form line keeps √3 unevaluated, so a homework answer such as 5√3 is never rounded away into decimals.
- What 30-60-90 problems ask for, and nothing padded. The result box stays with the sides, area, perimeter, altitude, exact form and steps: the measures these problems ask for.
Frequently asked questions
Halve an equilateral triangle of side 2 along an altitude. You get a right triangle with hypotenuse 2, one leg 1, and the other leg √(2² − 1²) = √3.
The short leg. The side opposite 60° is the long leg, and the hypotenuse sits opposite the right angle.
(√3/2) × a², which is half the product of the legs.
Yes. From area A the short leg is the square root of (2A / √3), and everything else follows.
Values written with √3 left unevaluated, like 5√3, which are exact rather than rounded decimals.
Use a = 2h / √3. The altitude from the right angle to the hypotenuse is h = a√3/2, so inverting gives the short leg, and the ratio 1: √3: 2 gives the rest. Try h = 6 in the calculator above.
Divide by (3 + √3), about 4.7321. Since P = a + a√3 + 2a = a(3 + √3), the short leg is a = P / 4.7321, and the long leg and hypotenuse follow from the ratio.
Yes. Press Show steps under the results to see each substitution, from the known value through every side, the area, the perimeter, and the altitude.
Add This Calculator to Your Website
Embed this calculator on your own site with the code below. Pick a size to preview it live, then copy the code.
Powered by Calculator Zilo
References
- Weisstein, Eric W., Wolfram MathWorld. 30-60-90 Triangle. Accessed October 24, 2026.
- Weisstein, Eric W., Wolfram MathWorld. Right Triangle. Accessed October 24, 2026.
- National Institute of Standards and Technology. NIST Digital Library of Mathematical Functions, Section 4.14: Trigonometric Functions, Values. Accessed October 24, 2026.
- Encyclopaedia Britannica. Trigonometry. Accessed October 24, 2026.
- OpenStax, Rice University. Precalculus 2e. Accessed October 24, 2026.
- Weisstein, Eric W., Wolfram MathWorld. Pythagorean Theorem. Accessed October 24, 2026.