How this instrument works
A 30-60-90 triangle is one of geometry's two 'special' right triangles (the other being the 45-45-90), so named because its side lengths always sit in the exact, fixed ratio 1 : √3 : 2, regardless of the triangle's overall size. Given just the short leg (opposite the 30° angle), the long leg (opposite the 60° angle) is short leg × √3, and the hypotenuse is exactly double the short leg.
This fixed ratio comes from cutting an equilateral triangle exactly in half along one of its own heights: the resulting right triangle has half the original base as its short leg, the full height as its long leg, and one of the original equal sides as its hypotenuse — the 1:√3:2 ratio is simply what falls out of that bisection, provable directly with the Pythagorean theorem.
Because the ratio is fixed, a 30-60-90 triangle needs only ONE known measurement to be completely solved — a genuine shortcut over a general right triangle, which needs two independent pieces of information (two sides, or a side and an angle) before the Pythagorean theorem or ordinary trigonometry can pin down the rest.
- Enter the triangle's short leg (the side opposite the 30° angle) into the Short leg field.
- Read Long leg and Hypotenuse: the sheet applies the fixed 1:√3:2 ratio directly.
- Double the short leg to confirm both other sides double too, since the ratio never changes.
Worked example — a short leg of 5
A 30-60-90 triangle has a short leg of 5. Its long leg is 5√3 ≈ 8.66, and its hypotenuse is exactly 10 — the full triangle solved from that single starting measurement, with the fixed 1:√3:2 ratio doing all the work.
A larger 30-60-90 triangle with a short leg of 10 has a long leg of 10√3 ≈ 17.32 and a hypotenuse of 20 — the identical ratio, simply scaled up, confirming the relationship holds at any size.
Questions
What is the side ratio of a 30-60-90 triangle?
1 : √3 : 2 — the short leg (opposite 30°), the long leg (opposite 60°), and the hypotenuse (opposite 90°) always sit in that exact fixed ratio, regardless of the triangle's overall size.
Where does the 1:√3:2 ratio come from?
Cutting an equilateral triangle exactly in half along one of its own heights produces a 30-60-90 triangle directly — the half-base, the full height, and one original side become the short leg, long leg, and hypotenuse, and the Pythagorean theorem confirms the resulting 1:√3:2 relationship.
Why does only one measurement solve the whole triangle?
Because the angles (30°, 60°, 90°) are already fixed, the triangle's SHAPE never varies — only its size does. A single known side determines that size completely, and the fixed ratio fills in the other two automatically.
How is this different from the 45-45-90 triangle?
The 45-45-90 triangle is the OTHER special right triangle, formed by cutting a square in half along its diagonal instead of an equilateral triangle. Its side ratio is 1:1:√2, handled by this site's separate Isosceles Right Triangle Hypotenuse page.
Can I start from the long leg or hypotenuse instead of the short leg?
Yes — since the ratio is fixed, any one known side determines the other two through simple division or multiplication by the appropriate ratio factor; this page is scoped to starting from the short leg specifically, the most common starting measurement.