Where does the 30-60-90 triangle come from?

Cut an equilateral triangle in half by dropping an altitude from one vertex to the opposite side. Each half is a 30-60-90 triangle. The altitude bisects the top angle (60° → 30°) and the base, creating the 1 : √3 : 2 ratio naturally.

How do you find the long leg of a 30-60-90 triangle?

Multiply the short leg by √3. If you know the hypotenuse instead, first divide it by 2 to get the short leg, then multiply that result by √3. For example, if the hypotenuse is 14, the short leg is 7 and the long leg is 7√3.

Can you use the Pythagorean theorem instead of the 30-60-90 ratio?

Yes, the Pythagorean theorem always works for right triangles. However, in a 30-60-90 triangle you only need one side to find the other two using the ratio, which is faster. The ratio is actually derived from the Pythagorean theorem, so both methods give the same answer.

30-60-90 Triangle — Definition, Formula & Examples

A 30-60-90 triangle is a right triangle whose three interior angles measure 30°, 60°, and 90°. Its sides are always in the fixed ratio 1 : √3 : 2, which lets you find any side length when you know just one.

A 30-60-90 triangle is a special right triangle formed by bisecting an equilateral triangle along an altitude. Because the angle measures are fixed at 30°, 60°, and 90°, the side opposite each angle maintains the constant ratio $1 : \sqrt{3} : 2$ . Specifically, if the shortest side (opposite 30°) has length $x$ , then the side opposite 60° has length $x\sqrt{3}$ and the hypotenuse (opposite 90°) has length $2x$ .

Key Formula

\text{Sides} = x \;:\; x\sqrt{3} \;:\; 2x

Where:

$x$ = Length of the shortest side (opposite the 30° angle)
$x\sqrt{3}$ = Length of the medium side (opposite the 60° angle)
$2x$ = Length of the hypotenuse (opposite the 90° angle)

How It Works

Start by identifying which side you know. The shortest side is always opposite the 30° angle, the medium side is opposite the 60° angle, and the longest side (the hypotenuse) is opposite the 90° angle. If you know the shortest side

x

, multiply by

\sqrt{3}

to get the medium side and by 2 to get the hypotenuse. If you know the hypotenuse, divide by 2 to get the shortest side, then multiply that result by

\sqrt{3}

. If you know the medium side, divide by

\sqrt{3}

to recover the shortest side, then double it for the hypotenuse. This ratio works for every 30-60-90 triangle regardless of size, so memorizing it saves significant calculation time.

Worked Example

Problem: In a 30-60-90 triangle, the side opposite the 30° angle is 5 cm. Find the other two sides.

Step 1: Identify the known side. The side opposite 30° is the shortest side, so x = 5.

x = 5

Step 2: Find the side opposite 60° by multiplying x by √3.

x\sqrt{3} = 5\sqrt{3} \approx 8.66 \text{ cm}

Step 3: Find the hypotenuse by multiplying x by 2.

2x = 2(5) = 10 \text{ cm}

Step 4: Verify with the Pythagorean theorem: the sum of the squares of the two legs should equal the square of the hypotenuse.

5^2 + (5\sqrt{3})^2 = 25 + 75 = 100 = 10^2 \;\checkmark

Answer: The side opposite 60° is

5\sqrt{3} \approx 8.66

cm, and the hypotenuse is 10 cm.

Another Example

This example starts from the hypotenuse instead of the shortest side, showing how to work backward through the ratio.

Problem: The hypotenuse of a 30-60-90 triangle is 12 in. Find the lengths of the two legs.

Step 1: The hypotenuse equals 2x, so set 2x = 12 and solve for x.

2x = 12 \implies x = 6

Step 2: The short leg (opposite 30°) is x = 6 in.

\text{Short leg} = 6 \text{ in}

Step 3: The longer leg (opposite 60°) is x√3.

6\sqrt{3} \approx 10.39 \text{ in}

Step 4: Check: 6² + (6√3)² = 36 + 108 = 144 = 12².

36 + 108 = 144 = 12^2 \;\checkmark

Answer: The short leg is 6 in and the longer leg is

6\sqrt{3} \approx 10.39

in.

Why It Matters

You will use the 30-60-90 ratio repeatedly in geometry, trigonometry, and precalculus—especially when evaluating sin, cos, and tan of 30° and 60° without a calculator. Standardized tests like the SAT and ACT regularly feature problems that become straightforward once you recognize this triangle. Engineers and architects rely on it when calculating roof pitches, ramp grades, and hexagonal structures.

Common Mistakes

Mistake: Putting √3 on the hypotenuse instead of the longer leg.

Correction: The hypotenuse is always the cleanest multiple: 2x. The √3 factor belongs to the medium side (opposite 60°). Remember: the longest side in any right triangle is the hypotenuse, and 2 > √3 ≈ 1.73.

Mistake: Mixing up the 30-60-90 ratio with the 45-45-90 ratio.

Correction: The 45-45-90 ratio is 1 : 1 : √2 (equal legs). The 30-60-90 ratio is 1 : √3 : 2 (unequal legs). A quick check: count how many different angle measures exist—two distinct angles (besides 90°) means 30-60-90.

Mistake: Forgetting to divide by √3 (or rationalize) when the longer leg is given.

Correction: If the side opposite 60° is known, divide by √3 to find x. For instance, if the longer leg is 6, then x = 6/√3 = 2√3, not 6. Rationalize if your teacher requires it: 6/√3 = 6√3/3 = 2√3.

Check Your Understanding

The longer leg of a 30-60-90 triangle is

9\sqrt{3}

. What are the short leg and hypotenuse?

Hint: The longer leg equals x√3, so set x√3 = 9√3 and solve for x.

Answer: Short leg = 9, hypotenuse = 18.

A 30-60-90 triangle has a hypotenuse of 20. What is the area of the triangle?

Hint: Find both legs first (10 and 10√3), then use A = ½ × base × height.

Answer: Area =

50\sqrt{3} \approx 86.6

square units.

True or false: In a 30-60-90 triangle, the hypotenuse is always exactly twice the shortest side.

Answer: True. The ratio 1 : √3 : 2 guarantees hypotenuse = 2 × (short leg) for every 30-60-90 triangle.

Related Terms

Angle Sum Property of a Triangle — Guarantees the three angles total 180°
Altitude of a Triangle — Dropping an altitude creates 30-60-90 triangles
Acute Triangle — Equilateral triangles are acute and split into 30-60-90s
AA Similarity — All 30-60-90 triangles are similar by AA
ASA Congruence — Proves two 30-60-90 triangles congruent if a side matches
AAS Congruence — Another way to prove 30-60-90 triangles congruent
Centroid — Located using coordinates from known side lengths
Base of a Triangle — Either leg can serve as the base for area calculations