Rectangle Diagonal Calculator
Calculate rectangle diagonal from length and width using d = √(l²+w²).
Returns aspect ratio, area, perimeter, and 3D box diagonal if height is given.
The diagonal of a rectangle is the straight line connecting opposite corners. Because the diagonal, length, and width form a right triangle, the Pythagorean theorem applies directly.
The formula:
d = √(l² + w²)
A rectangle has two diagonals and they are always equal in length. If you measure both diagonals of a frame and they differ, you have a parallelogram (slanted), not a true rectangle. Carpenters use this exact test when checking that a floor frame, door opening, or cabinet is square.
Square special case:
d = s√2 ≈ 1.414 × s
For a square (l = w = s), the diagonal is about 41% longer than each side. The √2 ratio is the simplest geometric irrational number and the one the Pythagoreans famously found unsettling, since it appears in the most ordinary geometric figure.
3D box (rectangular prism) diagonal:
d₃ = √(l² + w² + h²)
Each additional dimension adds a squared term under the radical. The longest line you can draw inside a rectangular box runs corner-to-corner along this space diagonal. This is what you check when asking “will this long object fit in this box?” The answer is yes only if the object’s length is ≤ d₃.
Worked example, TV size: A TV screen measures 48 inches wide × 27 inches tall (a 16:9 aspect ratio at 48 × 27 = 1.778, exactly 16:9).
d = √(48² + 27²) = √(2304 + 729) = √3033 = 55.1 inches
That is why this display is sold as a “55-inch TV”. The diagonal is the industry-standard measurement. Note that two TVs both labeled “55 inches” can have noticeably different active surface areas if one is widescreen and the other is closer to square, but 16:9 is now universal for new sets.
Worked example, room layout: You have a 5 m × 4 m room and want to know the longest item you can carry diagonally across the floor.
d = √(5² + 4²) = √(25 + 16) = √41 = 6.40 m
That is your absolute maximum for a single rigid object on the floor. Furniture moving guides assume you can lift a 6.40 m × 6.40 m square (impossible), but a 6.40 m beam slid diagonally fits exactly.
Worked example, fitting a long object into a box: A shipping box is 1.0 m × 0.8 m × 0.6 m. Can you ship a 1.4 m hockey stick in it?
d₃ = √(1.0² + 0.8² + 0.6²) = √(1.00 + 0.64 + 0.36) = √2.00 = 1.414 m
Yes, 1.40 m < 1.414 m, just barely. The stick has to lie corner-to-corner along the longest space diagonal, but it does fit.
Quick reference, common rectangles:
| Dimensions | Diagonal |
|---|---|
| 3 × 4 | 5 (exact, the classic 3-4-5 Pythagorean triple) |
| 5 × 12 | 13 (Pythagorean triple) |
| 8 × 15 | 17 (Pythagorean triple) |
| 9 × 16 (16:9 TV at “1 unit”) | 18.36 ≈ 1.147 × longer side |
| 4 × 3 (4:3 old TV) | 5.0 |
| 1 × 1 (square) | 1.414 (√2) |
| 1 × 2 (domino) | 2.236 (√5) |
| Golden rectangle (1 × 1.618) | 1.902 |
Why TVs are sold by diagonal, not area: Manufacturers settled on diagonal because it is a single number that grows with both dimensions, while the aspect ratio (length:width) varies by format. A 55-inch 16:9 TV has different length and width than a 55-inch 4:3 monitor would, but both share the same diagonal, making same-diagonal comparisons consistent across formats. The downside is that a wider aspect ratio has less active surface area for the same diagonal: a 16:9 55" TV has about 947 in², while a 4:3 55" set (rarely made now) would have ~1,151 in², over 20% more glass for the same “size.”
Checking squareness in construction: The 3-4-5 rule is just the diagonal formula in disguise. Carpenters check that a corner is exactly 90° by measuring 3 units along one wall, 4 units along the other, and confirming the diagonal is exactly 5 units. If it is, the angle is square. Off by even a few millimeters at small scale becomes large errors at the corner of a foundation, so this verification is non-negotiable in framing and concrete work.