30-60-90 Triangle Calculator
Enter any one side, the area, or the perimeter to solve every dimension of this special right triangle.
Known measurement
Choose the one measurement you already know.
Use a positive decimal with a period, such as 11 or 6.35.
Changing units converts the current known value, including square units for area.
Using b = a√3, so a = b/√3.
Solved triangle
Hypotenuse (c)
12.7017 in
Shorter leg (a)
6.3509 in
Opposite 30°
Longer leg (b)
11 in
Opposite 60°
Area
34.9297 in²
One-half of leg product
Perimeter
30.0526 in
Sum of all three sides
Enter one complete positive measurement to solve the triangle.
Dimension breakdown
| Quantity | Relationship | Opposite angle | Solved value |
|---|---|---|---|
| Shorter leg (a) | a | 30° | 6.3509 in |
| Longer leg (b) | a√3 | 60° | 11 in |
| Hypotenuse (c) | 2a | 90° | 12.7017 in |
| Area | a²√3 / 2 | – | 34.9297 in² |
| Perimeter | a(3 + √3) | – | 30.0526 in |
All values come from one canonical side ratio. Display values are rounded to four decimal places; the Excel workbook retains full numeric precision.
How to use this 30-60-90 triangle calculator
What this calculator does
This calculator solves a 30°-60°-90° right triangle from any one known side, its area, or its perimeter. It returns the shorter leg, longer leg, hypotenuse, area, and perimeter in one consistent unit system. The result is an exact geometric identity apart from displayed rounding; it does not determine whether a real object was measured accurately or whether a physical corner is truly 30°, 60°, or 90°.
When to use it
Use it to check special-triangle homework, size a drafting or woodworking layout, convert a known diagonal into perpendicular dimensions, or verify a trigonometry calculation. The fixed ratio is especially useful when only one measurement is available. OpenStax explains why the sides of this special right triangle always follow the ratio 1 : √3 : 2 in right-triangle trigonometry.
How to calculate
- The calculator opens with a ready-to-use example: a longer leg of 11 inches. Its results and a validated example Excel workbook are immediately available.
- Under Known measurement, choose the quantity you have: shorter leg, longer leg, hypotenuse, area, or perimeter.
- Replace Measurement value with a positive number. Use a period as the decimal separator. Grouping commas, scientific notation, mixed fractions, and unit text inside the field are intentionally rejected.
- Select the Unit. A unit change converts the current value rather than merely relabeling it. Area values are converted as square units.
- Read the solved cards and the Dimension breakdown table. Select Download Excel to export the current validated model. Reset clears the demonstration data, results, validation residue, and workbook state; Excel remains unavailable until a complete valid value is entered again.
Input guide
Known measurement is required and identifies the meaning of your single numeric input. Choose Shorter leg (a) for the side opposite 30°, Longer leg (b) for the side opposite 60°, or Hypotenuse (c) for the side opposite 90°. Choose Area for a square-unit measurement or Perimeter for the sum of the three sides. Selecting the wrong side name is the most common interpretation error because a long leg and a hypotenuse are not interchangeable.
Measurement value is a required positive decimal no greater than 10150 in the selected unit. Examples include 11, 6.35, or 21.65. Higher side or perimeter values scale every length proportionally and area quadratically. Higher area values scale each side with the square root of area. Zero, negatives, ambiguous decimal-comma input such as 1,5, scientific notation, and pasted labels such as “11 in” are rejected visibly.
Unit is required and supports millimeters, centimeters, meters, inches, and feet. A side or perimeter is expressed in the selected linear unit; area is expressed in the corresponding square unit. For example, changing 11 inches to centimeters displays 27.94 cm. NIST's guidance on SI units of length and derived square units is useful when documenting metric measurements.
Output guide
Hypotenuse (c) is the longest side and the primary result card. Shorter leg (a) is opposite 30°; Longer leg (b) is opposite 60°. Each is a calculated length driven by the selected known measurement. Area measures enclosed surface in square units and can never be negative for a valid triangle. Perimeter is the exact sum a + b + c. The header's Known pill restates the active input, while Pythagorean check confirms that a² + b² agrees with c² within floating-point tolerance.
The Dimension breakdown table repeats every result with its relationship to the shorter leg and, for sides, its opposite angle. A zero or missing table means the input is incomplete or invalid. Very large values are accepted only while all derived results remain finite. These outputs are exact identities of the special triangle, not statistical estimates or recommendations.
Worked example
The startup example sets the longer leg b to 11 in. Since b = a√3, the shorter leg is a = 11/√3 = 6.3509 in. The hypotenuse is c = 2a = 12.7017 in. The area is ab/2 = 34.9297 in², and the perimeter is a + b + c = 30.0526 in. These are the same first-open values used in the results, table, and downloadable workbook.
Learn more
For a compact reference that also derives area from the hypotenuse, see Wolfram MathWorld's 30-60-90 triangle formulas.
Core formulas
Let a be the shorter leg. Every other measurement follows from one scale factor:
b = a√3
c = 2a
Area = a²√3 / 2
Perimeter = a(3 + √3)
If a different measurement is known, the calculator reverses the appropriate formula first, then derives the complete model from a.
Why the ratio is fixed
Bisecting an equilateral triangle creates two congruent right triangles. The original side becomes the hypotenuse, half that side becomes the shorter leg, and the altitude becomes the longer leg. The Pythagorean theorem gives the altitude as √3 times the shorter leg, producing 1 : √3 : 2.
Common mistakes
- Calling the longer leg the hypotenuse.
- Using linear conversion factors for area.
- Rounding √3 too early and compounding error.
- Entering a general right triangle that does not have 30° and 60° acute angles.