Diagonal Of Rectangle Calculator - Pythagorean Formula and Sides

A diagonal of rectangle calculator finds the length of the line that connects the two opposite corners of a rectangle, using the Pythagorean theorem. It is the everyday geometry answer to the question, "how long is the diagonal of a rectangle with sides a and b?" Type the length and the width, or the area and one side, and the page returns the diagonal, the squared diagonal, and the right triangle the diagonal cuts across, so you can both check a homework problem and trust the answer.

• • Homework and textbook problems: Confirm a problem that asks for the diagonal of a rectangle, the value of d in d^2 = a^2 + b^2, or the missing side from a given diagonal.
• • Carpentry, screen, and packaging measurements: Find the diagonal of a board, a TV, a monitor, a box, or a sheet of plywood so you can size a cut, a screen, a sleeve, or a shipping carton.
• • Layout and design grids: Compute the corner-to-corner distance of a room, a desk pad, a piece of paper, or a UI mockup to check fit, alignment, or aspect ratio.
• • Pair with rectangle area or perimeter: Use the diagonal alongside the area or the perimeter of the rectangle to back-solve a missing side when only one value is known.

The diagonal of rectangle calculator is intentionally narrow: it answers the d = sqrt(l^2 + w^2) question for the standard rectangular case, where the two adjacent sides are the legs of a right triangle and the diagonal is the hypotenuse. If you need the space diagonal of a cuboid or the area of the rectangle, pair this page with one of the peer calculators below.

If you need to back-solve a missing side from the area, the perimeter, or the diagonal, the Length Of A Rectangle Calculator page handles all six input pairs in one form.

The page implements the Pythagorean theorem applied to a rectangle. It takes the two adjacent sides, squares and sums them, and takes the square root to return the diagonal.

• l, w: Length and width of the rectangle, the two legs of the right triangle.
• d: Diagonal of the rectangle, the hypotenuse of the right triangle.
• l^2 + w^2: Sum of the squared sides, equal to d^2, the value under the square root.
• d = sqrt(l^2 + w^2): Final diagonal length, the only output that depends on a square root.

The same formula works for any positive length and width, including the square case where l = w and the diagonal reduces to l * sqrt(2). Squaring removes the sign of l and w, so the diagonal is always non-negative.

According to Wolfram MathWorld, the diagonal of a rectangle with side lengths a and b is d = sqrt(a^2 + b^2), which is the Pythagorean theorem applied to the right triangle formed by the two adjacent sides.

If you want to solve for any one leg or the hypotenuse of the right triangle the diagonal cuts across, the Pythagorean Theorem Solver page works the same arithmetic in a more general form.

Four ideas explain why the formula is what it is, what the result means, and how the rectangle relates to a right triangle.

These four ideas are the building blocks for the rest of the rectangle geometry chain. If you have the diagonal, the length, and the width, you can recover the area, the perimeter, the aspect ratio, or the missing side of any other rectangle in the same family.

When the two sides form a Pythagorean triple, the diagonal comes out to a clean integer, and the Pythagorean Triples Calculator page is the next click for spotting side pairs that give integer results.

Five short steps cover both input modes, from the clean 3-4-5 textbook example to the area-plus-width shortcut.

1. 1 Pick the input mode: Use Length + Width when you know both sides, or Area + Width when you know the area and one side and want the calculator to recover the missing side first.
2. 2 Enter the length and the width: Type the two adjacent sides. The defaults are 4 and 3, so the example starts as the 3-4-5 right triangle the moment the page loads.
3. 3 Or enter the area and one side: Switch to Area + Width and type the area in the matching square unit and one side in the linear unit. The length is recovered as area / width before the diagonal is computed.
4. 4 Read the diagonal: The primary output is the diagonal d, updated as you type. For a 3 by 4 rectangle the answer is 5, and for a 6 by 6 square the answer is 6 * sqrt(2) ≈ 8.4853.
5. 5 Verify and reset: Read the squared length, squared width, and the sum to confirm the addition. Click Reset to return to the default 3 by 4 example. If you need the area or perimeter, open the Length Of A Rectangle Calculator in a new tab.

Try length = 5 and width = 12. The calculator gives l^2 = 25, w^2 = 144, l^2 + w^2 = 169, d = 13, and the rectangle is not a square. The result 13 confirms that (5, 12, 13) is a Pythagorean triple, this time for a 5 by 12 rectangle.

Once you have the diagonal, the length, and the width, the Right Triangle Calculator page can fill in any missing leg, hypotenuse, or angle of the right triangle those values describe.

These benefits matter most when you are working a problem by hand and need a quick, trustworthy check.

• • Skip the arithmetic mistakes: Manual diagonal problems are easy to get wrong on the squaring step. The calculator handles the squaring and the square root so you can focus on setting up the problem.
• • See the full step-by-step math: The page shows l^2, w^2, the sum l^2 + w^2, and the final d. That makes it a good way to check your own work, not just a way to get a number.
• • Switch between length+width and area+width: Sometimes only the area and one side are known. The mode selector recovers the missing length and runs the same Pythagorean step, so you do not have to do the back-solve yourself.
• • Get an is-square flag: The result panel reports whether the length and width are equal, which is a fast way to see the special case where the diagonal reduces to l * sqrt(2).
• • Connect to the rest of rectangle geometry: If your next step is to find the missing side, the area, or the perimeter, the page links out to the relevant peer calculator in the same category.

The page is most useful as a check, not as a replacement for understanding the formula. Use it to confirm a homework answer, sanity-check a measurement, or pre-validate a side pair before you hand it to a longer script.

If you would rather enter two corner coordinates on a grid and get the distance between them, the 2D Distance Calculator page runs the same sqrt of squared sum on coordinate pairs.

The formula is the same in every case, but a few factors change how the result should be read.

• • This page is the planar, 2D rectangle case only. For the space diagonal of a cuboid, the formula needs an extra height term and the dedicated cuboid surface area page handles that directly.
• • The formula assumes the two given sides are the two adjacent sides of a rectangle. If one value is actually the diagonal, the answer will be too large.
• • If the inputs are inconsistent (for example area and width that would imply a negative length), the calculator shows a validation error rather than a misleading number.

According to Encyclopaedia Britannica, a rectangle is a quadrilateral with four right angles and two pairs of equal opposite sides, and its two diagonals are equal in length and bisect each other.

When you also need the area or the perimeter of the same rectangle, the Surface Area Of A Rectangle Calculator page reads the same length and width and returns the total surface area alongside the diagonal.