How this instrument works
A rectangle's diagonal cuts it into two identical right triangles, with the length and width standing in as the legs and the diagonal itself as the hypotenuse. That is the entire derivation: d = √(l² + w²) is the Pythagorean theorem read off a rectangle's own corners rather than a triangle drawn in isolation. Nothing about the shape needs measuring beyond the two sides that already define it.
Rectangles carry a property triangles alone do not: both diagonals are always equal, and they cross at the shape's exact center. That single fact is what lets a carpenter square a doorframe, a deck, or a foundation without a protractor — four matching side lengths can still rack into a lopsided parallelogram, but stretch a tape between opposite corners both ways, and unequal readings expose the skew instantly. Equal diagonals are the only proof that every corner is truly 90°.
The formula has a quiet limit worth noticing. Set length to zero and the rectangle collapses to a bare line, with the diagonal equal to whatever width remains — no special-casing required, since √(0² + w²) is simply w. At the other extreme, a square is just the case l = w, and the diagonal reduces to l√2, an irrational multiple that stays fixed no matter how large the square grows.
- Enter the rectangle's Length into the Length field.
- Enter the rectangle's Width into the Width field, using the same unit as Length.
- Read Diagonal for the corner-to-corner distance, computed to eight decimal places.
- To check a physical frame for square, measure both diagonals by hand and compare each to this sheet's figure — a match confirms 90° corners.
Worked example — a 3-by-4 gate frame
A gate frame is built 3 units wide and 4 units tall — enter Length = 3 and Width = 4. Squaring each gives 9 and 16; their sum is 25, and the square root is exactly 5, so Diagonal reads 5.0 with no rounding anywhere in the chain. This is the smallest whole-number right triangle there is, which is exactly why a 3-4-5 frame is the standard classroom demonstration of squaring by diagonal.
That 5.0 is also the number a builder would stretch a tape to, corner to corner, across the finished frame. Measure the other diagonal too: if it also reads 5.0, the frame's four corners are genuinely square. Read anything else — 5.2, say — and the frame has racked out of true even though its four side lengths never changed.
Questions
What is the formula for the diagonal of a rectangle?
d = √(l² + w²), where l and w are the rectangle's length and width. The diagonal is the hypotenuse of the right triangle formed by any two adjacent sides, so this is the Pythagorean theorem applied directly to a rectangle's own corners rather than a triangle considered on its own.
Why do carpenters measure both diagonals when squaring a frame?
Because four matching side lengths do not guarantee right angles — a frame can rack sideways into a parallelogram with those same four sides but slanted corners. A true rectangle's two diagonals are always equal, so measuring corner to corner both ways and comparing the two readings catches a skewed frame that measuring the sides alone would miss entirely.
Does the formula still work for a square?
Yes — a square is a rectangle with l = w, so the formula reduces to d = l√2. Enter equal length and width and the sheet returns that same irrational multiple: a 1-by-1 square gives a diagonal of √2 ≈ 1.41421356, a ratio fixed for every square regardless of size.
What happens if the length is entered as zero?
The rectangle collapses into a straight segment equal to the width, and the formula holds without adjustment: √(0² + w²) = w. This sheet accepts zero for either side and returns the remaining side unchanged, since nothing in the underlying identity breaks at that limit.
Are a rectangle's two diagonals really the same length?
Yes, always. Both diagonals connect opposite corners and cross at the rectangle's exact center, and by the shape's symmetry each one bisects the other there. That is a property of rectangles specifically, not of triangles in general, and it is the reason computing one diagonal effectively tells you both.
How does this relate to a circle drawn around the rectangle?
All four corners of a rectangle sit on one circle, centered where the diagonals cross, and that circle's diameter is exactly the rectangle's diagonal. A 3-by-4 rectangle therefore fits perfectly inside a circle of diameter 5 — the same 5 this sheet returns for that pair of sides.