A special right triangle is one whose angles force its sides into a fixed, exactly expressible ratio. There are two of them. The 30-60-90 triangle has sides in the ratio 1 : √3 : 2, and the 45-45-90 triangle has sides in the ratio 1 : 1 : √2. Because those ratios never change, one side length determines the whole triangle — no trigonometry required, and no rounding either. This calculator gives you both the exact radical answer and the decimal, side by side.
Arb Digital publishes free geometry tools that answer the question actually being asked. That distinction matters here more than usual: a general right-triangle solver returns 8.6603 where a homework question wants 5√3, and the two are not interchangeable in a marked answer. Our Pythagorean theorem calculator handles any right triangle numerically; this page is the one that keeps the radicals intact for the two cases where an exact form exists.
What This Special Right Triangle Calculator Does
You pick the triangle type, say which side you know, and enter its length. The tool then returns all three sides in exact radical form — reduced, with the denominator rationalised — alongside decimal values to whatever precision you set. It also gives the perimeter and the area in exact form where one exists, and draws the triangle to scale with its angles labelled so you can check that you nominated the right side.
The exact forms are built symbolically rather than by rounding a decimal back to a radical. Enter a hypotenuse of 7 on a 45-45-90 triangle and you get legs of 7√2/2, not 4.9497 dressed up. Enter a long leg of 5 on a 30-60-90 triangle and you get a short leg of 5√3/3, which is 5/√3 with the denominator rationalised in the way every textbook insists on.
How to Use It
- Choose the triangle type. A 30-60-90 has three different side lengths; a 45-45-90 is isosceles with two equal legs.
- Say which side you know. On the 30-60-90, the short leg faces the 30° angle and the long leg faces the 60° angle. Getting this backwards is the single most common error.
- Enter the length. Whole numbers produce the tidiest radicals, but any positive value works.
- Take the exact form for written work and the decimal for anything you are going to measure or cut.
- Check the diagram. The longest side is always the hypotenuse, and the largest angle always faces the longest side.
The Ratios and Where They Come From
Cut an equilateral triangle of side 2 in half through one vertex. The half has angles of 30°, 60° and 90°, a hypotenuse of 2 (an original side), and a short leg of 1 (half of the base). Pythagoras gives the third side as √(2² − 1²) = √3. That is the 1 : √3 : 2 ratio, derived rather than memorised. Now take a square of side 1 and cut along the diagonal. The half has two 45° angles, two legs of 1, and a hypotenuse of √(1² + 1²) = √2, giving the 1 : 1 : √2 ratio. Euclid's proof of the underlying theorem is Elements Book I, Proposition 47.
Work the default by hand. A 30-60-90 with a short leg of 5 has a long leg of 5√3 ≈ 8.6603 and a hypotenuse of 10. The perimeter is 15 + 5√3 ≈ 23.6603 and the area is ½ × 5 × 5√3 = 25√3/2 ≈ 21.6506. Switch to a 45-45-90 with a hypotenuse of 10 and each leg is 10 ÷ √2 = 5√2 ≈ 7.0711, with an area of exactly 25.
Why Exact Form Is Not Just Fussiness
Two reasons, one academic and one practical. Academically, √3 is irrational: its decimal expansion never terminates or repeats, so every decimal you write is wrong by some amount. Writing 5√3 is exactly right; writing 8.66 is right to three significant figures and no further. When a question says "give your answer in exact form", it is testing whether you understand that distinction, and 8.6603 will be marked down however many decimals you add.
Practically, exact forms compose without accumulating error. If a construction stacks five 30-60-90 triangles end to end, working in radicals gives 25√3 and rounding once at the end gives 43.3013. Rounding each step to two decimals first gives 43.30, and in longer chains the drift becomes visible. This is the same discipline that applies anywhere repeated rounding occurs; our significant figures calculator and rounding calculator deal with the general case.
Rationalising the Denominator
When the known side is the long leg of a 30-60-90 triangle, the short leg comes out as L ÷ √3. Almost every syllabus wants that rewritten without a radical underneath, which you do by multiplying top and bottom by √3: L ÷ √3 = L√3 ÷ 3. So a long leg of 5 gives a short leg of 5√3/3, not 5/√3, even though the two are the same number. The same move turns 1/√2 into √2/2.
The convention predates calculators and had a genuine purpose: dividing by a rational number by hand is far easier than dividing by an irrational one, so 5 × 1.732 ÷ 3 was a much less painful computation than 5 ÷ 1.732. The habit survived because it also gives everyone a single canonical way to write the same quantity, which makes marking and comparison straightforward. This tool always returns the rationalised form. Our rationalize denominator calculator handles the general expressions, including the conjugate method for binomial denominators.
Spotting a Special Triangle in a Larger Problem
These two triangles hide inside a great deal of standard geometry. The diagonal of any square creates two 45-45-90 triangles, so a screen or a photo with a 1:1 aspect ratio has a diagonal of side × √2. The altitude of any equilateral triangle splits it into two 30-60-90 triangles, which is where the formula side²√3/4 for the area of an equilateral triangle comes from. A regular hexagon is six equilateral triangles, so hexagonal layouts are full of √3 relationships. The face diagonal of a cube is s√2 and its space diagonal is s√3.
Recognising the pattern turns a trigonometry problem into arithmetic. If a ramp rises at 30° and runs 4 m horizontally, the horizontal run is the long leg, so the rise is 4√3/3 ≈ 2.31 m and the ramp itself is 8√3/3 ≈ 4.62 m — all without touching a sine button. Where the angle is not 30, 45 or 60, you do need trigonometry, and the trigonometric functions calculator takes over. The similar triangles calculator is the tool for scaling one of these triangles up or down against a matching one.
The Exact Trigonometric Values These Triangles Generate
Every exact trig value taught in a first course comes straight off these two triangles. From the 30-60-90: sin 30° = 1/2, cos 30° = √3/2, tan 30° = √3/3, sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3. From the 45-45-90: sin 45° = cos 45° = √2/2 and tan 45° = 1. There is nothing else to memorise, because each is just a ratio of two sides of a triangle you can reconstruct in ten seconds from a square or an equilateral triangle.
That reconstruction is worth practising, because it is far more reliable than a memorised table under exam pressure — and it explains why these particular values appear in every unit-circle question. Combined with reference angles it extends to every multiple of 30° and 45° around the circle, where only the signs change. Our reference angle calculator handles that step, and Euclid's Elements Book VI, Proposition 4 is the formal statement that the ratios depend on the angles alone and not on the size of the triangle.
Arb Digital designs responsive interfaces around a proportional grid, so nothing breaks when the viewport changes.
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Two quick sanity checks catch almost every error. First, the hypotenuse must be the longest side, and in a 30-60-90 triangle it must be exactly double the shortest side — so if your hypotenuse is not twice one of your legs, something has gone wrong. Second, √2 is about 1.414 and √3 is about 1.732, so the long leg of a 30-60-90 triangle should land a little over 1.7 times the short leg, and the hypotenuse of a 45-45-90 triangle should land a little over 1.4 times a leg. Those two decimals are worth knowing by heart; they turn every answer on this page into something you can verify in your head in a couple of seconds.
A third check applies to area. For the 45-45-90 triangle the area is half the square of a leg, which is a clean rational number whenever the leg is rational. For the 30-60-90 triangle the area always carries a factor of √3, so an area with no radical in it is a sign that a side has been misassigned. If the numbers still refuse to reconcile, redraw the triangle with the right angle at the bottom left and label the angles before the sides — most errors here are labelling errors rather than arithmetic ones.
Common Mistakes to Avoid
- Swapping the legs — the short leg faces the 30° angle and the long leg faces the 60° angle, never the other way round.
- Multiplying the hypotenuse by √3 to get the long leg. The hypotenuse is twice the short leg; the long leg is the short leg times √3.
- Leaving a radical in the denominator when the question expects the rationalised form.
- Giving a decimal where an exact answer was requested, which is marked wrong no matter how many places you write.
- Assuming any right triangle with a 30° angle has integer sides — the ratio is exact but irrational, so whole-number sides are the exception.
Related Free Tools From Arb Digital
Solve any right triangle numerically with the Pythagorean theorem calculator, compare two triangles with the similar triangles calculator, find an area from three sides with the triangle area calculator, tidy an expression with the rationalize denominator calculator, or work an angle back to the first quadrant with the reference angle calculator. Everything else is on the free online tools hub.
Frequently Asked Questions
1 : √3 : 2, with the short leg opposite the 30° angle, the long leg opposite the 60° angle, and the hypotenuse exactly twice the short leg.
1 : 1 : √2. The two legs are equal because the two non-right angles are equal, and the hypotenuse is a leg multiplied by the square root of two.
Both come from cutting a symmetric figure in half — an equilateral triangle and a square. That construction fixes the ratios exactly, whereas other angles produce side ratios with no simple closed form.
Divide the long leg by the square root of three, then rationalise: the short leg equals the long leg times the square root of three, divided by three.
It gives one canonical way of writing the quantity and was originally far easier to evaluate by hand. Most syllabuses still require it, even though the unrationalised form is the same number.
Only for one or two of the three sides at a time. Because the square roots of two and three are irrational, the full set of three sides can never all be whole numbers.
No. The fixed ratios do the whole job. Trigonometry is only needed once the angles stop being 30, 45 or 60 degrees.
This tool is provided for educational use. Exact radical answers are given in the conventional rationalised form; check the format your course requires before submitting work.
