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30-60-90 Triangle Calculator

A 30-60-90 triangle is a special right triangle with angles of 30°, 60°, and 90°. Its sides follow a predictable ratio of 1 : √3 : 2, making calculations straightforward once you know a single measurement. Engineers, architects, and students rely on these properties to solve geometric problems without complex trigonometry. Enter any known value—a leg, hypotenuse, area, or perimeter—and instantly find all remaining dimensions.

Last updated: May 12, 2026

Creators Jasmine J. Mah, BSc

Reviewers Mateusz Mucha and Anna Szczepanek

4,843 people find this calculator helpful

Understanding the 30-60-90 Triangle

The 30-60-90 triangle is one of two special right triangles in geometry, prized for its predictable side ratios. Named for its three interior angles, this triangle appears frequently in construction, design, and mathematics because its measurements scale proportionally.

The defining characteristic is the relationship between sides. If the shortest leg (opposite the 30° angle) has length a, then the longer leg (opposite the 60° angle) is a√3, and the hypotenuse is 2a. This consistency means you need only one measurement to determine everything else.

The angles themselves form an arithmetic sequence—30°, 60°, 90°—making this the only right triangle with this property. This mathematical elegance is why it appears so often in real-world applications, from roof pitch calculations to navigation problems.

30-60-90 Triangle Formulas

Given the shortest leg a, use these formulas to find all other measurements:

b = a × √3

c = 2 × a

Area = (a² × √3) ÷ 2

Perimeter = a × (3 + √3)

a — length of the shorter leg (opposite the 30° angle)
b — length of the longer leg (opposite the 60° angle)
c — length of the hypotenuse (opposite the 90° angle)

Deriving Sides from Different Known Values

When the longer leg b is your starting point, divide by √3 to recover the shorter leg:

a = b ÷ √3 = b × √3 ÷ 3
c = b × 2 ÷ √3 = b × 2√3 ÷ 3

If you know the hypotenuse c, it's even simpler. Since the hypotenuse equals 2a, the shorter leg is half the hypotenuse:

a = c ÷ 2
b = c × √3 ÷ 2

These inverse relationships let you work backward from any measurement to reconstruct the entire triangle.

Practical Example: Working with Hypotenuse

Suppose you have a 30-60-90 triangle with a hypotenuse of 10 cm. Find both legs and the area:

Shorter leg: 10 ÷ 2 = 5 cm
Longer leg: 5 × √3 ≈ 8.66 cm
Area: (5² × √3) ÷ 2 = (25 × √3) ÷ 2 ≈ 21.65 cm²

Notice how the area formula simplifies when you substitute a = 5. The hypotenuse-to-leg relationships make mental arithmetic feasible, which is why builders and surveyors favour this triangle type for field calculations.

Common Pitfalls and Tips

Avoid these frequent mistakes when solving 30-60-90 triangles:

Don't confuse which leg is which — The shorter leg is always opposite the 30° angle, and the longer leg opposite the 60° angle. Mixing these up inverts your calculations. Always label before computing.
Remember √3 ≈ 1.732, not 1.7 — Rounding too early—to 1.7 or 2—introduces cumulative error. If precision matters (construction, engineering), keep at least 3 decimal places or use exact form (leave √3 in your answer).
Area uses the shorter leg squared, not the longer one — The formula (a² × √3) ÷ 2 assumes a is the shortest side. If you mistakenly square the longer leg, you'll overestimate area by a factor of 3.
Verify the 1 : √3 : 2 ratio before trusting results — After computing all sides, check that b ÷ a ≈ 1.732 and c ÷ a = 2. This sanity check catches arithmetic errors quickly.

Frequently Asked Questions

What makes the 30-60-90 triangle special in geometry?

This triangle's angles form an arithmetic sequence (30°, 60°, 90°), a property unique among right triangles. More importantly, its side lengths follow a fixed ratio of 1 : √3 : 2, which means you need only one measurement to find everything else. This predictability makes it invaluable in construction, surveying, and design work where quick calculations are essential. Unlike triangles with arbitrary angles, there's no need for trigonometric tables or calculators—just multiplication by simple constants.

How do I find all sides if I only know the longer leg?

If the longer leg (opposite 60°) has length b, divide by √3 to get the shorter leg: a = b ÷ √3. Then double the shorter leg to find the hypotenuse: c = 2a = 2b ÷ √3. For example, if the longer leg is 10 cm, the shorter leg is 10 ÷ 1.732 ≈ 5.77 cm, and the hypotenuse is 11.55 cm. This approach works whenever the problem gives you the middle-length side first.

Can I use the 30-60-90 triangle ratios for larger or smaller triangles?

Yes, absolutely. The 1 : √3 : 2 ratio holds for any 30-60-90 triangle, regardless of scale. A 30-60-90 triangle with a shorter leg of 1 inch and one with a shorter leg of 1 metre both follow identical proportions. This is why the ratio is so powerful: it lets you scale solutions up or down without recalculating. Contractors often use this property to enlarge blueprints or shrink designs while preserving geometric accuracy.

What's the relationship between hypotenuse and area in a 30-60-90 triangle?

Area grows with the square of the hypotenuse. Since the shorter leg is half the hypotenuse, area = ((c ÷ 2)² × √3) ÷ 2 = (c² × √3) ÷ 8. For a hypotenuse of 10, the area is 100 × 1.732 ÷ 8 ≈ 21.65 square units. For a hypotenuse of 20, area becomes 86.6 square units—exactly four times larger, not twice, because area scales with the square of linear dimensions.

How do I verify I've solved a 30-60-90 triangle correctly?

Check two quick relationships. First, confirm that longer leg ÷ shorter leg ≈ 1.732 (the ratio of √3). Second, verify hypotenuse ÷ shorter leg = 2 exactly. If both ratios check out, your sides are correct. For area, multiply the two legs and divide by 2, then verify this equals (a² × √3) ÷ 2. These spot checks catch arithmetic errors before they propagate further.

Why would I encounter a 30-60-90 triangle in real life?

Engineers use them for roof designs (sloped at 30° or 60° to horizontal), structural bracing, and landscape grading. Surveyors employ 30-60-90 triangles for sight line calculations and distance measurements. In manufacturing, they appear in angled cuts, joints, and assembly fixtures. Because the ratio is so clean, 30-60-90 triangles let professionals avoid complex trigonometry on site and reduce mental calculation time, making them practical for field work where speed and accuracy matter equally.

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