How this instrument works
A 45-45-90 triangle is the isosceles right triangle: one 90° corner and two 45° corners, which forces the two sides facing those 45° angles — the legs — to be equal. Substitute that equality into the Pythagorean relation and the general a² + b² = c² collapses to leg² + leg² = hyp², or 2·leg² = hyp². Taking the square root of both sides gives hyp = leg × √2 — not a separate rule, just the Pythagorean theorem with a = b.
The same shape is exactly half of a square. Cut a square along one diagonal and each piece is a 45-45-90 triangle: the square's two sides become the equal legs, and the diagonal becomes the hypotenuse. That is why measuring a square's side and multiplying by √2 hands you the diagonal with no separate formula to remember — it is this triangle wearing a different name.
The edge case sits at leg = 0, where the triangle collapses to a single point and the hypotenuse rightly reports zero rather than an error. At the other end, no leg length ever makes the ratio anything but 1 : 1 : √2 — doubling the leg doubles the hypotenuse, and the shape of the triangle, unlike a scalene one, never changes as it grows.
- Enter the length of either leg — the two are equal by definition — into the Leg length field.
- Read the answer straight off the Hypotenuse field; it recalculates as leg × √2 the moment you change the input.
- Treat the result as a diagonal check too: it equals the diagonal of a square whose side matches your leg length.
- A negative entry is rejected outright, since a real triangle cannot have a side of negative length; a zero leg returns a hypotenuse of exactly zero.
Worked example — a leg of 5 units
Set Leg length to 5 and the sheet returns Hypotenuse = 5 × √2 = 7.0710678118654755. The instrument carries √2 to full double precision rather than rounding early, so that long tail of digits is the exact double-precision value, not a display artifact — a carpenter reading it to two decimals sees 7.07 and loses nothing that matters at that scale.
Shrink the same triangle to a leg of 1 and the pattern is easiest to see: the hypotenuse becomes 1.4142135623730951, which is √2 itself. That is the very number Greek geometers examined on the diagonal of a unit square and proved could never be written as a ratio of two whole numbers — the first irrational number anyone had pinned down, discovered on this exact triangle.
Questions
What makes a triangle a 45-45-90 triangle?
Two angles of exactly 45° and one right angle. Because a triangle's angles fix the ratios of its sides, the two sides facing the equal 45° angles — the legs — must themselves be equal in length, which is why this shape is also called the isosceles right triangle.
How is hyp = leg × √2 derived from the Pythagorean theorem?
Start from a² + b² = c² and set both legs equal: leg² + leg² = hyp², so 2·leg² = hyp². Taking the square root of both sides gives hyp = leg × √2 directly. No separate proof is required — the 45-45-90 case is simply the Pythagorean theorem with a = b, so any right-triangle calculator would agree with it exactly.
How does a 45-45-90 triangle differ from a 30-60-90 triangle?
In a 30-60-90 triangle the sides run 1 : √3 : 2 — the hypotenuse is a clean whole-number double of the short leg, and only the long leg carries an irrational √3 factor. In a 45-45-90 triangle the two legs are equal outright, and the irrational factor √2 sits on the hypotenuse instead, since no whole-number step connects an equal-legged triangle's sides.
Why does the same formula give a square's diagonal?
Because a 45-45-90 triangle is exactly half of a square, split along one diagonal. The square's side plays the role of the leg, and its diagonal plays the role of the hypotenuse, so hyp = leg × √2 and diagonal = side × √2 are the identical statement viewed from two different shapes.
What are sin, cos, and tan of a 45° angle?
sin 45° = cos 45° = √2 ⁄ 2 ≈ 0.7071, and tan 45° = 1. Sine and cosine come out equal because the triangle is isosceles — the leg opposite 45° and the leg adjacent to it are the same length — and tan 45° = 1 simply restates that fact as a ratio of the two equal legs.
Can the leg length be zero or negative?
Zero is accepted and collapses the triangle to a single point, so the sheet reports a hypotenuse of exactly 0 rather than an error. A negative leg length is rejected before any arithmetic runs, since no real triangle has a side of negative length.