30-60-90 Triangle Calculator

One triangle, three fixed angles

A 30-60-90 triangle is a right triangle whose two sharp corners always measure 30 degrees and 60 degrees. Fix the angles and you fix the shape, so every 30-60-90 triangle is a scaled copy of every other one. The sides come in one ratio that never budges: the short leg is x, the long leg is x × 1.732 (the square root of 3), and the hypotenuse is 2x. That one ratio is the whole tool.

You can see where it comes from with a single cut. Take an equilateral triangle with sides of 2x and slice straight down from the top corner to the middle of the base. Each half is a right triangle: the half base is x, the original side becomes the hypotenuse of 2x, and the height works out to x × 1.732. Half of an equilateral triangle is always a 30-60-90 triangle, which is one reason it keeps showing up in shops, on roofs and in 3D printed parts.

The short trigonometry course at Clark University covers the right triangle conventions behind this, and NASA's beginner trigonometry pages work the same ground for wing and ramp problems. Enter the one side you know above, and the calculator returns the other two sides, the area and the perimeter.

The ratio rules, in both directions

Three rules run the show, and each one runs backward:

Two of the multipliers come straight from the square root of 3: half of it is 0.866, and 2 ÷ 1.732 is 1.155. Check the bottom row against a short leg calculation: a hypotenuse of 18 gives a short leg of 18 ÷ 2 = 9 and a long leg of 18 × 0.866 = 15.59, which is the same triangle you get from 9 × 1.732 = 15.59.

The easiest anchor to remember is the hypotenuse: it is exactly twice the short leg, with no rounding involved. Find the short leg and you have everything. The same NASA pages explain why: once an angle is fixed, the side ratios are fixed too, for a triangle of any size.

Worked example: bracing a picture frame from the short leg

A cross brace for a large picture frame has to run at 30 degrees off the frame edge, and the short side of the triangle measures 9 inches. Here is the full calculation, one step at a time:

Step 1: long leg = short leg × 1.732 = 9 × 1.732 = 15.588, which rounds to 15.59 inches.
Step 2: hypotenuse = short leg × 2 = 9 × 2 = 18 inches.
Step 3: area = half × short leg × long leg = 0.5 × 9 × 15.588 = 70.15 square inches.
Step 4: perimeter = 9 + 15.59 + 18 = 42.59 inches.

So the brace reaches 15.59 inches along the frame, and the diagonal piece is 18 inches long. Cut from a full 18 inches and there is no room for the saw kerf, so most woodworkers add a little extra. The calculator returns the same four numbers from the single input of 9, with nothing to retype. The ratio carries no units, so the same numbers work in centimeters or feet: 9 becomes 9 of whatever you measured.

Worked example: a phone stand built from the hypotenuse

A desk phone stand leans back at 30 degrees, and the back panel, which is the hypotenuse, measures 8 inches. Working from the hypotenuse means dividing instead of multiplying:

Step 1: short leg = hypotenuse ÷ 2 = 8 ÷ 2 = 4 inches.
Step 2: long leg = short leg × 1.732 = 4 × 1.732 = 6.928, which rounds to 6.93 inches.
Step 3: area = 0.5 × 4 × 6.928 = 13.86 square inches.
Step 4: perimeter = 4 + 6.93 + 8 = 18.93 inches.

The panel touches the desk 4 inches from the back of the base, and the phone rests on a 6.93 inch face. If a taller version calls for a 10 inch back panel, every length scales by 10 ÷ 8 = 1.25, because the shape is locked. That is the payoff of a fixed ratio: one measured side resizes the whole triangle, and no angle has to be rechecked.

Worked example: a shelf brace from the long leg

Long leg inputs are the ones people get wrong, because the instinct is to halve the number, and halving is wrong here. Say a shelf brace spans 12 inches along the wall, and that wall span is the long leg:

Step 1: short leg = long leg ÷ 1.732 = 12 ÷ 1.732 = 6.928, which rounds to 6.93 inches.
Step 2: hypotenuse = short leg × 2 = 6.928 × 2 = 13.856, which rounds to 13.86 inches.
Step 3: area = 0.5 × 6.928 × 12 = 41.57 square inches.
Step 4: perimeter = 6.93 + 12 + 13.86 = 32.79 inches, or 32.78 carrying the unrounded values.

Halving 12 would give 6, and a 6 inch short leg would demand a hypotenuse of 12, which leaves no diagonal at all. The long leg is the middle sized side, so the number you divide by is 1.732, not 2. The calculator takes the long leg straight from the menu and does the division for you.

Common sizes at a glance

Because the ratio never changes, one small table covers most shop math. Scale any row up or down and it still holds:

These sizes cover the places where a 30 or 60 degree angle is the natural choice. Hexagon layouts, gazebos and any six sided layout are equilateral triangles in disguise, so every cut across them is a 30-60-90. Some shed roofs and dormer walls land at 30 degrees. 3D printed gussets often use a 30 degree brace because it stiffens a corner without much material. And a door wedge, a keyboard riser or a phone stand set at 30 degrees is a 30-60-90 triangle whether or not anyone names it.

One caution on the table: these are the math lengths, with no allowance for the thickness of the material or the kerf of a blade. The ratios hold at any size, so measure your real short leg first, find its row, and cut to the numbers that follow from it.

How it differs from the 45-45-90 triangle

There are two special right triangles, and it pays to keep them straight:

The quickest tell is the legs. Equal legs mean 45 degree corners and a hypotenuse of a leg times 1.414. Different legs, with the hypotenuse exactly double the shorter one, mean you are holding a 30-60-90. One angle reading settles any doubt: 45 degrees points to the square family, while 30 or 60 points to the equilateral family.

Three mistakes that throw the answer off

First, the fake 1, 2, 3 ratio. The sides are not 1, 2 and 3; they are 1, 1.732 and 2. A 3-4-5 triangle fails the test on sight, because the hypotenuse would have to be twice the short leg, and 5 is not 6. Its real angles are about 36.87 and 53.13 degrees. And no triangle with sides of 1, 2 and 3 exists anywhere, since 1 + 2 = 3 collapses into a straight line.

Second, rounding 1.732 too soon. On a 12 inch short leg, using 1.73 instead of 1.732 puts the long leg off by about 0.02 inch, which is enough to show up in joinery. Carry four digits, 1.732, or let the calculator work from the full value of 1.7320508.

Third, swapping the legs. The long leg sits across from the 60 degree angle and equals the short leg times 1.732. It is not half the hypotenuse; the short leg is. If a plan calls out a hypotenuse of 20, the short leg is 20 ÷ 2 = 10 and the long leg is 10 × 1.732 = 17.32.