The 30-60-90 Triangle

A 30-60-90 triangle is a right triangle whose acute angles are 30° and 60°. Its sides are always in the ratio 1 : √3 : 2: if the short leg (opposite 30°) is x, the long leg (opposite 60°) is x√3 and the hypotenuse is 2x. Know one side and you know all three.

The ratio is not a fact to memorize blindly. It falls out of the Pythagorean theorem in three lines once you notice that a 30-60-90 triangle is exactly half of an equilateral triangle.

Where the 1 : √3 : 2 ratio comes from

Start with an equilateral triangle: all three sides equal and all three angles 60°. Call each side 2x (using 2x instead of x avoids fractions later). Now drop the altitude, the perpendicular line from the top vertex to the base.

By symmetry the altitude does three things at once. It meets the base at a right angle, it cuts the base into two equal halves of length x, and it splits the 60° angle at the top into two 30° angles. Each half is a right triangle with angles 30°, 60° and 90°. Its hypotenuse is an original side, 2x, and its short leg is half the base, x.

The only unknown is the long leg, the altitude itself. Call it h and apply the theorem:

So the three sides are x, x√3 and 2x, in the ratio 1 : √3 : 2. Because √3 is irrational, the three sides can never all be whole numbers, which is why this shape never shows up among the Pythagorean triples.

The same construction explains the height of an equilateral triangle with side s: it is the long leg of a half with hypotenuse s, so h = s over 2√3 = √3 over 2s. The page on the height of a triangle takes that idea further.

30-60-90 triangle rules

Before you calculate, label the sides by the angle they face. The smallest side is opposite the smallest angle, so the order of the angles (30, 60, 90) matches the order of the sides (1, √3, 2). Not sure which side is the hypotenuse? It is always the side across from the right angle; see parts of a right triangle.

Then follow the rule for the side you were given. Every route goes through the short leg, so if you don't start with it, find it first.

Decimals are rounded to the nearest hundredth. Two more formulas follow from the short leg: the area is 1 over 2 · x · x√3 = √3 over 2x², and the perimeter is x + x√3 + 2x = x(3 + √3).

Given the short leg

This is the easy case: multiply by √3 for the long leg and by 2 for the hypotenuse.

Given the hypotenuse

Halve it to get the short leg, then multiply the short leg by √3 for the long leg. No radicals appear in the denominator, so there is nothing to tidy up.

Given the long leg

Divide by √3 to get the short leg, then double it. Dividing by a root leaves a radical in the denominator, so rationalize by multiplying top and bottom by √3:

If the long leg already contains √3, the division is instant: a long leg of 7√3 means a short leg of 7 and a hypotenuse of 14. If radicals are rusty, review simplifying square roots.

Worked examples

The special right triangle calculator repeats any of these with your own numbers, showing the exact and decimal answers.

Exact trig values for 30° and 60°

Shrink the triangle so that its hypotenuse is 1. The short leg becomes 1 over 2 and the long leg √3 over 2. Placed inside a unit circle, the legs are the cosine and the sine of the angle at the center.

Reading the sides with SOH CAH TOA gives values you can write down without a calculator:

The 45° row comes from the 45-45-90 triangle. Notice that sin30° = cos60° and cos30° = sin60°: the side opposite one acute angle is adjacent to the other. Each row also passes the Pythagorean identity test, for example (1 over 2)² + (√3 over 2)² = 1 over 4 + 3 over 4 = 1.

When a problem gives an angle that is not 30°, 45° or 60°, these shortcuts no longer apply and you need a calculator's sine, cosine or tangent. The guide to choosing between the theorem and trigonometry explains which tool fits which problem.

Memory tips

• The 2 belongs to the hypotenuse. The hypotenuse is the longest side, and 2 is bigger than √3 ≈ 1.73. Putting √3 on the hypotenuse is the most common slip.
• Small to large. 1 < √3 < 2, and 30° < 60° < 90°. Match them in order: each side faces the angle in the same position.
• The squares are 1, 3 and 4. Square the sides of the 1 : √3 : 2 triangle and you get 1 + 3 = 4, the Pythagorean theorem in its simplest form. If your three sides don't pass this test, something is wrong.
• Half the hypotenuse. If you remember only one thing, remember that the side opposite 30° is half the hypotenuse (that is just sin30° = 1 over 2). The long leg then follows from the theorem.
• Draw the equilateral triangle. In a test, sketching the full equilateral triangle rebuilds the whole rule in seconds.

Where 30-60-90 triangles show up

• Equilateral triangles. Every equilateral triangle hides two of them, which gives its height √3 over 2s and area √3 over 4s².
• Hexagons. A regular hexagon is six equilateral triangles, so it splits into twelve 30-60-90 triangles. That is how you find the distance across the flats of a hex bolt (s√3) compared with the distance across its corners (2s), or the area of a hexagonal tile.
• Slopes at 30°. A ramp or roof inclined at 30° rises exactly half its sloping length: a 10 m slope at 30° climbs 5 m.
• Forces and velocities. A force or velocity at 30° to the horizontal splits into components √3 over 2 and 1 over 2 of its size. See the theorem in physics.
• Drawing and tests. One of the two standard set squares is a 30-60-90 triangle, and the SAT reference sheet prints the ratio; the ACT does not, so know it cold. More on the theorem on the SAT and ACT.

Common mistakes

• Hypotenuse = x√3. No: the hypotenuse is 2x. The long leg carries the √3.
• Mixing up the legs. The short leg is opposite the 30° angle, not next to it. Find the angle, then look straight across the triangle.
• Halving the long leg. To get the short leg from the long leg, divide by √3, not by 2. Only the hypotenuse gets halved.
• Assuming the shape. A triangle that merely looks like a 30-60-90 triangle might not be one. You need a 30° or 60° angle, or a leg that is exactly half the hypotenuse, before the shortcut applies; otherwise use a² + b² = c² or trigonometry.

The common mistakes page covers errors that apply to every right triangle.