Audited 29 Jul 2026·Last updated 31 Jul 2026·4 citations·Tier 2·0 uses

30-60-90 Triangle Calculator

Enter any one measurement of a 30-60-90 triangle — a side, the area or the perimeter — and get every other side plus the area, perimeter and altitude.

30-60-90 Triangle Calculator

Which measurement do you already know?
Any unit you like — millimetres, inches, feet, metres. Lengths come back in the same unit and the area comes back in that unit squared. Must be greater than zero.
Long leg (opposite 60°)
8.6603
The medium-length side, equal to the short leg × √3. Because √3 is irrational, this figure is an approximation unless it is the number you typed — exact to the 10 decimal places shown and no further.
Short leg (opposite 30°)
5
Hypotenuse (opposite 90°)
10
Area
21.6506sq units
Perimeter
23.6603
Altitude to the hypotenuse
4.3301

Background.

The Quanta 30-60-90 triangle calculator takes any single measurement of a 30-60-90 triangle and returns everything else it has: all three sides, the area, the perimeter, and the altitude dropped from the right angle onto the hypotenuse. Enter the short leg, the long leg, the hypotenuse, the area or the perimeter — five modes, one number each. With the default short leg of 5, the long leg comes back as 8.6602540378, the hypotenuse as 10, the area as 21.6506350946, the perimeter as 23.6602540378 and the altitude as 4.3301270189.

A 30-60-90 triangle has exactly one degree of freedom. An ordinary right triangle needs two numbers to pin it down, because the two legs are independent. Here the angles are fixed, so the side ratios are fixed too — 1 : √3 : 2 — and a single measurement determines the whole figure. That is what separates this page from a general Pythagorean-theorem solver, which needs two sides before it can give you a third and can do nothing at all with one.

The ratio comes from cutting an equilateral triangle in half. Euclid's Elements Book I, Proposition 1 shows how to construct an equilateral triangle; drop a perpendicular from one vertex to the opposite side and it bisects both that side and the 60° angle above it, leaving two identical right triangles with angles of 30°, 60° and 90°. If the equilateral triangle had side 2s, each half has a short leg of s (half the base), a hypotenuse of 2s (an original side), and a long leg found by Euclid Book I, Proposition 47 — the Pythagorean theorem — as √(4s² − s²) = √(3s²) = s√3. So short leg, long leg and hypotenuse stand in the ratio 1 : √3 : 2, forever, whatever the size.

Two consequences are worth committing to memory because they let you do most of these problems in your head. The hypotenuse is always exactly double the short leg — no square roots involved. And the long leg is always the short leg times about 1.732. Everything else on this page is bookkeeping on top of those two facts. The area, for instance, is half the product of the legs, ½ × s × s√3, which simplifies to (√3 ÷ 2)s². Watch that coefficient: it is √3 ÷ 2, not the √3 ÷ 4 that belongs to an equilateral triangle. The two get mixed up constantly, and the reason is visible in the derivation above — this triangle is half of an equilateral of side 2s, so its area is half of (√3 ÷ 4)(2s)², and the 2s squared brings a factor of 4 that cancels down to √3 ÷ 2.

One caveat belongs beside the answer rather than three screens below it. √3 is irrational, so no 30-60-90 triangle has all three sides as whole numbers, or even as exact decimals. Fix the short leg at 5 and the long leg is 8.660254037844386… going on forever. Every figure this page returns other than the one you typed is exact to the ten decimal places shown and no further. That also means a 30-60-90 triangle can never be a Pythagorean triple, which is the exact complement of the triples page: one family has tidy angles and irrational sides, the other has whole-number sides and awkward angles.

And a scope limit: this page assumes the triangle really is 30-60-90. If your angles are 31, 59 and 90, the answers here will be close but wrong, and the triangle solver is the right tool. A fixed-shape calculator is only ever as true as the assumption behind it.

What is 30-60-90 triangle calculator?

A 30-60-90 triangle is a right triangle whose other two angles are 30° and 60°. Because all three angles are fixed, all similar 30-60-90 triangles have the same side ratios, and those ratios are 1 : √3 : 2 — short leg (opposite the 30° angle) to long leg (opposite the 60° angle) to hypotenuse (opposite the right angle). Wolfram MathWorld's entry states it trigonometrically: for a hypotenuse of length a, the two legs are a·sin(60°) = ½a√3 and a·sin(30°) = ½a, which divided through by ½a is 1 : √3 : 2 again. The classical derivation uses no trigonometry at all. Take an equilateral triangle of side 2s (Euclid, Elements Book I, Proposition 1) and drop a perpendicular from any vertex to the opposite side. By symmetry the perpendicular bisects that side into two halves of length s and bisects the 60° apex angle into two 30° angles, producing two congruent right triangles. Each has a short leg of s, a hypotenuse of 2s and a long leg equal to the perpendicular, which the Pythagorean theorem (Elements Book I, Proposition 47) fixes at √((2s)² − s²) = √(3s²) = s√3. Together with the 45-45-90 triangle, the 30-60-90 is one of the two 'special right triangles' taught in every geometry course, and the pair between them supply the exact values of every trigonometric function at 30°, 45° and 60° — sin 30° = 1/2, sin 60° = √3/2, tan 30° = 1/√3, tan 60° = √3, and so on. That is why they are memorised rather than looked up: they are the only angles at which the trigonometric functions have simple closed forms in elementary terms.

How to use this calculator.

  1. Choose which measurement you already have from the dropdown: short leg, long leg, hypotenuse, area or perimeter. Any one of them fixes the entire triangle.
  2. Identify the sides correctly before you type. The short leg is opposite the 30° angle and is the shortest side; the long leg is opposite the 60° angle; the hypotenuse is opposite the right angle and is always the longest. Getting the two legs the wrong way round is the commonest input error, and the giveaway is that the hypotenuse should come out at exactly twice the short leg.
  3. Enter the value in any unit. Lengths come back in the same unit; the area comes back in that unit squared.
  4. Read the long leg at the top — in the default short-leg mode it is the number most people came for, because it is the one you cannot do in your head.
  5. Read the hypotenuse and confirm it is exactly double the short leg. That is a free sanity check on whether you entered the right side.
  6. Use the area and perimeter for material estimates, and the altitude to the hypotenuse when you need the height measured from the long side rather than from a leg — for a roof brace, a gusset plate, or a cut list.
  7. If your triangle is not exactly 30-60-90, do not use this page. Enter the three measurements you actually have into the triangle solver instead.

The formula.

short : long : hyp = 1 : √3 : 2 , A = (√3 ⁄ 2)s² , P = s(3 + √3)
s · short leg, opposite the 30° angles√3 · long leg, opposite the 60° angle2s · hypotenuse, opposite the right angleA · area, P · perimeteraltitude to the hypotenuse = (s√3) ⁄ 2, half the long leg

Every mode on this page solves for the short leg s first, then rebuilds the rest of the triangle from it.

WHERE THE RATIO COMES FROM. Construct an equilateral triangle of side 2s (Euclid, Elements Book I, Proposition 1). Drop a perpendicular from one vertex to the opposite side. By symmetry it bisects that side, so each half is s, and it bisects the 60° angle at the top into two 30° angles. Each of the two right triangles produced therefore has angles 30°, 60° and 90°, a short leg of s, and a hypotenuse of 2s. The Pythagorean theorem (Elements Book I, Proposition 47) gives the third side: long leg = √((2s)² − s²) = √(4s² − s²) = √(3s²) = s√3. Hence 1 : √3 : 2.

THE FIVE INVERSIONS. Short-leg mode is trivial: s is what you typed. Long-leg mode divides by √3, since long = s√3. Hypotenuse mode halves, since hyp = 2s. Area mode uses A = ½ × s × s√3 = (√3 ÷ 2)s², so s = √(2A ÷ √3). Perimeter mode uses P = s + s√3 + 2s = s(3 + √3), so s = P ÷ (3 + √3), where 3 + √3 ≈ 4.7320508076.

WORKING THE DEFAULT ALL THE WAY THROUGH. With s = 5: the long leg is 5 × 1.732050807568877293 = 8.660254037844386467, displayed as 8.6602540378. The hypotenuse is 2 × 5 = 10 exactly. The area is ½ × 5 × 8.660254037844386467 = 21.650635094610966, displayed as 21.6506350946. The perimeter is 5 + 8.660254037844386467 + 10 = 23.660254037844386, displayed as 23.6602540378. The altitude to the hypotenuse is (5 × 8.660254037844386467) ÷ 10 = 4.330127018922193, displayed as 4.3301270189 — which is exactly half the long leg, as it must be, because the altitude of a right triangle onto its hypotenuse is (leg₁ × leg₂) ÷ hypotenuse and the hypotenuse here is 2s.

A CROSS-CHECK ON THE AREA THAT USES A DIFFERENT FORMULA. The parent equilateral triangle has side 10, so its area is (√3 ÷ 4) × 100 = 43.30127018922193, and half of that is 21.650635094610966 — the same figure the page reports, derived without ever using ½ × base × height. That check is in the test suite, not just in this paragraph.

WATCH THE AREA COEFFICIENT. It is √3 ÷ 2 for the 30-60-90 triangle and √3 ÷ 4 for an equilateral triangle, and mixing them up doubles or halves your answer. The reason both numbers appear in the same derivation is that this triangle is half of an equilateral whose side is 2s rather than s: half of (√3 ÷ 4)(2s)² = half of √3 s² = (√3 ÷ 2)s².

ROUNDING STAGE. All arithmetic runs at 40 significant digits in Decimal, and √3 is computed from scratch rather than entered as the literal 1.7320508076 — a literal would inject error into the ninth decimal place of the area, which is quadratic in s. The short leg is never rounded before the other five outputs are derived from it. Rounding happens once, at the return boundary, to 10 decimal places. Nothing on this page is sorted into bands or graded, so no threshold depends on display rounding.

A worked example.

Example

A cabinetmaker is cutting a triangular brace for a shelf and wants it at 30 degrees, with the short edge — the one that sits against the upright — measuring 5 inches. Leaving the mode on short leg and entering 5 returns a long leg of 8.6602540378 inches, a hypotenuse of 10 inches, an area of 21.6506350946 square inches, a perimeter of 23.6602540378 inches and an altitude to the hypotenuse of 4.3301270189 inches. In workshop terms: the piece runs 5 inches up the upright, 8⅝ inches (8.660 is a hair over 8 and 21/32) along the shelf, and the long diagonal edge is exactly 10 inches. That the hypotenuse comes out as a round 10 is not luck — it is always exactly twice the short leg, with no square root in sight, which is the single most useful thing to remember about this triangle. The long leg is the awkward one: 5 × 1.7320508076 = 8.6602540378, and no choice of units will ever make it a round number, because √3 is irrational. The area of 21.6506350946 square inches is half the product of the two legs, ½ × 5 × 8.6602540378. Cross-checking it a different way: this brace is exactly half of an equilateral triangle with 10-inch sides, whose area is (√3 ÷ 4) × 100 = 43.3012701892 square inches — and half of that is 21.6506350946, the same number. The altitude to the hypotenuse, 4.3301270189 inches, is what you would measure from the middle of the long diagonal edge straight back to the right-angled corner; it is exactly half the long leg, and it is the figure to use if you are laying the brace out from its diagonal rather than from its square corner. Switching to hypotenuse mode and entering 10 reproduces every one of these numbers, which is the check to run if the only measurement you can take on an existing part is the long edge.

value5
known QuantityshortLeg

Frequently asked questions.

What are the rules for a 30-60-90 triangle?
Three of them, and the first two do almost all the work. The hypotenuse is exactly twice the short leg. The long leg is the short leg times √3, about 1.7320508076. And the short leg is always opposite the 30° angle, the long leg opposite the 60° angle, and the hypotenuse opposite the right angle. Written as a ratio, short : long : hypotenuse = 1 : √3 : 2. Everything else follows: the area is (√3 ÷ 2) × short leg², the perimeter is short leg × (3 + √3), and the altitude onto the hypotenuse is half the long leg.
Why is the ratio 1 : √3 : 2?
Because a 30-60-90 triangle is exactly half an equilateral triangle. Take an equilateral triangle of side 2s and drop a perpendicular from one vertex to the opposite side. By symmetry it cuts that side into two halves of s and cuts the 60° apex angle into two 30° angles, so each half is a 30-60-90 triangle with short leg s and hypotenuse 2s — the hypotenuse is an original side of the equilateral triangle, which is why it is double. The Pythagorean theorem then gives the perpendicular itself: √((2s)² − s²) = √(3s²) = s√3. No trigonometry is needed anywhere in that argument, which is why the result was known long before trigonometry existed.
How do I find the sides if I only know the long leg?
Divide by √3 to get the short leg, then double that for the hypotenuse. If the long leg is 12, the short leg is 12 ÷ 1.7320508076 = 6.9282032303 and the hypotenuse is 13.8564064606. Many textbooks write the division as multiplying by √3 ÷ 3, which is the same thing rationalised — 12 × 0.5773502692 lands on the same 6.928203230 to nine decimal places. Select 'Long leg' from the dropdown and this calculator does it for you, along with the area, perimeter and altitude.
What is the area of a 30-60-90 triangle?
Half the product of the two legs: ½ × s × s√3, which simplifies to (√3 ÷ 2) × s², roughly 0.8660254038 × s², where s is the short leg. For a short leg of 5 that is 21.6506350946 square units. Be careful not to reach for the equilateral triangle's formula: that one is (√3 ÷ 4) × side², with a 4 in the denominator rather than a 2, and using it here halves your answer. The reason both coefficients turn up in the same derivation is that this triangle is half of an equilateral whose side is 2s, not s — and squaring the 2 supplies the factor that turns √3 ÷ 4 into √3 ÷ 2.
Can a 30-60-90 triangle have whole-number sides?
No. √3 is irrational, so if the short leg is a whole number the long leg cannot be — it is that number times an unending, non-repeating decimal — and if the long leg is whole then the short leg and hypotenuse are not. Any 30-60-90 triangle you actually cut is an approximation, and the useful question is how close you need to get. That also means a 30-60-90 triangle can never be a Pythagorean triple, which is the exact complement of the triples page on this site: triples have whole-number sides and untidy angles (3-4-5 has angles of about 36.87° and 53.13°), while this triangle has tidy angles and irrational sides. You cannot have both.
How is this different from the triangle solver?
Number of inputs. Quanta's triangle solver takes three measurements — three sides, or two sides and the angle between them, or two angles and the side between them — and returns all six elements of any triangle. You can in fact push a 30-60-90 problem through its ASA mode by entering 30, 60 and the hypotenuse, but you have to already know that the side between those two vertices is the hypotenuse, and you have to match the calculator's vertex labelling. It cannot start from the short leg or the long leg on its own, and it has no way to work backwards from an area or a perimeter. This page takes one number and no angles, because the angles are the thing it already knows.
Where does the 30-60-90 triangle actually come up?
Wherever a hexagon or an equilateral triangle does, because it is the building block of both. A regular hexagon is six equilateral triangles, so each one splits into two 30-60-90s — which is why the distance across the flats of a hex nut is its side length times √3. Roof framing at a 30° pitch, isometric drawing (which uses 30° axes), standard drafting triangles, truss bracing, and the standard set-square in every geometry kit are all the same triangle. In trigonometry it is one of the two triangles that supply exact values: sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3 and tan 60° = √3 all come straight off its side ratios.

References& sources.

  1. [1]Wolfram Research, MathWorld — '30-60-90 Triangle' (live revision retrieved 2026-07-29). States that for a hypotenuse of length a the two legs are a·sin(60°) = ½a√3 and a·sin(30°) = ½a, which is the 1 : √3 : 2 ratio this page uses. Open access.
  2. [2]Euclid, Elements, Book I, Proposition 1 — 'To construct an equilateral triangle on a given finite straight line.' The parent figure of the bisection derivation. David E. Joyce web edition of the Heath translation, Clark University; statement confirmed verbatim on retrieval 2026-07-29. Open access.
  3. [3]Euclid, Elements, Book I, Proposition 47 — 'In right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle.' Applied to the bisected equilateral triangle it gives the long leg as √(4s² − s²) = s√3, an entirely geometric derivation of the ratio that uses no trigonometry and serves as the independent check on this page's constant. Joyce web edition; confirmed verbatim on retrieval 2026-07-29. Open access.
  4. [4]NIST Digital Library of Mathematical Functions, §4.14 — definitions and periodicity of the circular functions, the basis of the trigonometric statement of the side ratios (sin 30° = 1/2, sin 60° = √3/2). Open access.
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