The two triangles whose sides you can write exactly.
The two with fixed ratios
Answers in surd form as well as decimal.
30-60-90, ratio 1 : √3 : 2
5 : 5√3 : 10
In decimals: 5, 8.660254, 10. The surd form is exact; the decimals are not.
Short leg (30°)
5
exactly 5
Long leg (60°)
8.660254
exactly 5√3
Hypotenuse
10
exactly 10
Area
21.650635
half the product of the legs
Perimeter
23.660254
all three sides
Side ratio
1 : √3 : 2
fixed, whatever the size
The sides are always in the ratio 1 : √3 : 2, with the shortest opposite the 30° angle and the hypotenuse exactly twice it. That doubling is the fact worth remembering; the √3 follows from Pythagoras.
It comes from cutting an equilateral triangle down the middle: the half-base is half the original side, and the hypotenuse is still a full side — hence the 1 : 2.
The long leg is NOT twice the short one. It is √3 ≈ 1.732 times it, and reaching for 2 there is the commonest error with this triangle.
Because the ratios are exact, these two triangles let you write an answer in surd form rather than as a decimal — which is what exam questions usually want.
The surd column is the exact answer; the decimals beside it are rounded, and an exam question usually wants the first.
What this tool shows
30-60-90 is always 1 : √3 : 2. 45-45-90 is always 1 : 1 : √2. Those ratios never change with size, which is what lets you write an exact answer instead of a decimal — and the long leg is √3 times the short one, not twice it.
A 30-60-90 triangle from any one of its sides
A 45-45-90 triangle from a leg or the hypotenuse
Every side in exact surd form
The same sides as decimals
Area and perimeter
Where each ratio comes from
Both triangles Exact surd answers From any side The ratios derived
Updated 7 September 2026 · Works in any browser, no installation
Scale the ratio. A 30-60-90 has sides 1 : √3 : 2, so a short leg of 5 gives a long leg of 5√3 and a hypotenuse of 10. A 45-45-90 has sides 1 : 1 : √2, so a leg of 7 gives a hypotenuse of 7√2. The ratios never change, whatever the size.
At a glance
Formula shown
A 30-60-90 triangle has sides in the ratio 1 : \u221a3 : 2, with the shortest opposite the 30\u00b0 angle and the hypotenuse exactly twice it. A 45-45-90 triangle has sides 1 : 1 : \u221a2, both legs equal and the hypotenuse the diagonal of a square.
Scenario support
A geometry exercise wanting an exact answer; a roof or brace at 30\u00b0 or 45\u00b0; recovering a side from a diagonal.
Educational estimate
Planning support from the values you enter — not professional advice.
Where the ratios come from
Neither ratio is arbitrary. Both fall out of cutting a familiar shape in half.
30-60-90: take an equilateral triangle of side 2 and cut it down the middle. The base is halved to 1, the cut edge is still a full side of 2, and Pythagoras gives the height as √(4 − 1) = √3. That is the ratio 1 : √3 : 2, and it explains the doubling — the hypotenuse is a whole original side while the short leg is half of one.
45-45-90: cut a square of side 1 along its diagonal. Both legs are 1, both non-right angles are 45°, and the diagonal is √(1 + 1) = √2.
Deriving them takes ten seconds and is more reliable than remembering which surd goes where.
The mistake to avoid
In a 30-60-90 triangle, the long leg is not twice the short one. The HYPOTENUSE is twice the short one; the long leg is √3 ≈ 1.732 times it.
The error is easy to make because the 2 is memorable and the √3 is not, and because a factor of 1.732 against 2 produces an answer that looks plausible — about 13% too large, which no sanity check catches.
Another: the 30° angle is opposite the SHORT side. The largest angle faces the longest side, so the 60° angle faces the √3 side and the right angle faces the hypotenuse. That ordering is true of every triangle and worth using as the check.
Why surd form matters
5√3 is exact. 8.66 is not, and 8.660254038 is not either.
Keeping the surd matters for two practical reasons. It never accumulates rounding error through a longer calculation, and it shows the structure — seeing √3 tells a reader a 30-60-90 triangle was involved, which 8.66 does not.
Exams almost always want the surd, and often say “leave your answer in surd form” explicitly. The page gives both columns so you can take whichever the question asks for.
Tidying matters too: √12 should be written 2√3, and 5/√3 is usually written 5√3/3. The Simplify Radicals Calculator does that step.
The exact trig values
These two triangles are where the memorised trigonometric values come from. They are not arbitrary; they are side ratios read off the diagram.
From 30-60-90: sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3, and the 60° values are those swapped — sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3.
From 45-45-90: sin 45° = cos 45° = 1/√2 = √2/2, and tan 45° = 1 because the legs are equal.
If you can draw the two triangles you never need to memorise the table — each value is opposite over hypotenuse, or adjacent over hypotenuse, read straight off the sides.
Where they turn up
Construction. A 45° brace across a rectangular frame has length side√2. A 30° roof pitch gives a rafter of twice the rise.
Paper sizes. A-series sheets have a side ratio of 1 : √2, which is exactly why halving an A4 sheet gives an A5 of the same proportions. The 45-45-90 diagonal is the same number.
Hexagons. A regular hexagon is six equilateral triangles, so every 30-60-90 relationship inside it applies — which is where the √3 in hexagonal packing and nut-and-bolt sizing comes from.
Screens and grids. A square screen’s diagonal is side√2; isometric drawing is built on 30° lines and inherits the √3.
Rationalising the denominator
Answers from these triangles often arrive as 1/√3 or 1/√2, and convention says to move the surd off the bottom.
Multiply top and bottom by the surd: 1/√3 × √3/√3 = √3/3. Likewise 1/√2 becomes √2/2. The value is unchanged; only the form is.
The convention predates calculators, when dividing by 1.732 by hand was harder than dividing by 3. It survives because it gives everyone the same written form, which makes answers comparable.
It is worth knowing both forms are correct. 1/√3 and √3/3 are the same number, and a marker asking for one will usually accept the other with a note.
Sources and methodology
The ratios are curriculum standards and the exact values are tabulated; these are the references.
Method. Whichever side you supply is divided by its place in the ratio to recover the unit, and the other two are multiplied out from there — so every entry point produces the same triangle. The surd form is constructed from the unit rather than reverse-engineered from a decimal. The suite asserts, from all three entry points and at two hundred sizes, that the hypotenuse is exactly twice the short leg, the long leg is exactly √3 times it, and Pythagoras holds on the result. That engine is verified on every change against 76 hand-written assertions, including that the 30-60-90 ratios hold from every entry point across six hundred generated cases, checked against Pythagoras rather than against themselves. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
TriangleSolve any triangle from three measurements — and when two sides and a non-included angle describe TWO triangles, this one shows both instead of picking one.
Pythagorean Theorema² + b² = c² solved for whichever side you leave blank, with the triangle drawn to scale and integer triples reduced to their primitive.
Reference AngleThe acute angle to the x-axis — never the y-axis — with the quadrant rule written out using your own numbers and the sign of each trig function beside it.
Simplify RadicalsSimplest radical form at any index from 2 to 12 with a coefficient in front, shown prime by prime, plus the rationalised denominator.
Square RootThe exact square root first — 72 gives 6 root 2 — then the decimal to as many as sixty places, computed on whole numbers rather than a double.
Heron’s FormulaArea from three sides in the rearrangement that keeps its digits — with the textbook form shown beside it so you can see what the usual order costs.
To tidy a surd answer into simplest form — √12 into 2√3 — the Simplify Radicals Calculator does that step, and the Reference Angle Calculator covers where these exact values apply outside the first quadrant.
Educational use disclaimer
This is an educational tool. The surd column is exact; the decimals beside it are rounded, and an exam answer usually wants the first.
Published both special triangles on one page, since they are the same idea at two ratios and separate pages would have been near-duplicates of each other.
Answers come in exact surd form as well as decimals, because the surd is what makes these two worth memorising and what an exam question asks for.
Names the error the ratio invites: in a 30-60-90 the HYPOTENUSE is twice the short leg, while the long leg is √3 times it — about 13% out, which no sanity check catches.
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