Diagonal of Rectangle - Formula & Examples

What Is The Diagonal Of A Rectangle?

A diagonal of a rectangle is a straight line joining two opposite (non-adjacent) corners. Every rectangle has two diagonals, and in a rectangle they are always equal in length and bisect each other. The length of each diagonal is:

d = \sqrt{l^2 + w^2}

Here, l is the length and w is the width of the rectangle, and d is the diagonal. The formula comes straight from the Pythagorean theorem, because a diagonal cuts the rectangle into two right triangles.

Why this formula works (a one-line derivation): the diagonal, the length, and the width form a right triangle with the diagonal as the longest side (the hypotenuse). Pythagoras says d^2 = l^2 + w^2, and taking the positive square root gives d = \sqrt{l^2 + w^2}.

A carpenter checking whether a door frame is truly square measures both diagonals - and if the two numbers do not match, the frame is a lopsided parallelogram, not a rectangle.

Examples of Diagonal of Rectangle

These examples build from a clean plug-in to working backwards from the diagonal. Each problem statement is bold; the steps are plain.

Example 1

Find the diagonal of a rectangle with length 444 cm and width 333 cm.

Apply the formula:

d = \sqrt{l^2 + w^2}

d = \sqrt{4^2 + 3^2}

d = \sqrt{16 + 9} = \sqrt{25}

d = 5 , \text{cm}

Final answer: the diagonal is 5 cm.

Example 2

Find the diagonal of a rectangle with length 666 m and width 888 m.

The diagonal is the shortest path across, not the sum of two sides walked around the corner. Walking 666 then 888 is going around the edge; the diagonal cuts straight across, so it must be shorter than 14 m.

d = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 , \text{m}

Final answer: the diagonal is 10 m.

Example 3

A rectangle has length 1212 and width 555. Find its diagonal.

d = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13

Final answer: the diagonal is 13.

Example 4

A rectangle has length 555 cm and width 222 cm. Find the diagonal, rounded to two decimals.

d = \sqrt{5^2 + 2^2} = \sqrt{25 + 4} = \sqrt{29}

29 is not a whole number, so leave it exact or approximate it:

d \approx 5.39 , \text{cm}

Final answer: d \approx 5.39 cm.

Example 5

A rectangle has a diagonal of 1010 cm and a length of 888 cm. Find its width.

Rearrange the formula to solve for w:

d^2 = l^2 + w^2

w^2 = d^2 - l^2 = 10^2 - 8^2

w^2 = 100 - 64 = 36

w = 6 , \text{cm}

Final answer: the width is 6 cm.

Example 6

A rectangular field is 4040 m by 3030 m. How much shorter is the diagonal path than walking two sides?

Diagonal:

d = \sqrt{40^2 + 30^2} = \sqrt{1600 + 900} = \sqrt{2500} = 50 , \text{m}

Walking two sides: 40 + 30 = 70 m.

Saving: 70 - 50 = 20 m.

Final answer: the diagonal path saves 20 m, cutting across at 50 m instead of 70 m.

Why The Diagonal Formula Matters: "Pythagoras Hiding Inside Every Rectangle"

The diagonal formula is really the Pythagorean theorem wearing a rectangle's clothes. The moment you draw one diagonal, the rectangle splits into two right triangles, and the diagonal becomes the hypotenuse. That single insight is why the formula is worth understanding rather than memorising.

• Checking square corners. A rectangle's two diagonals are equal. Builders and carpenters measure both; unequal diagonals mean the corners are not true right angles, so the shape has skewed into a parallelogram.

• Fitting things through gaps. Whether a table fits through a doorway, or a screen's size is a quoted size measured by its diagonal, computed from width and height.

• Shortest routes. Across a rectangular room, field, or park, the diagonal is the shortest straight-line distance between opposite corners, always less than walking two sides.

Common Mistakes With The Diagonal Of A Rectangle

These errors show up as soon as the numbers stop being a clean Pythagorean triple.

Mistake 1: Adding the sides instead of squaring them

Where it slips in: Treating the diagonal as "length plus width."

Don't do this: Writing d = l + w, so a 333 by 444 rectangle gets a diagonal of 777.

The correct way: The diagonal is \sqrt{l^2 + w^2}, not l + w. For 333 by 444 that is \sqrt{9 + 16} = 5, not 777. The rusher who adds the sides forgets the diagonal cuts straight across, which is always shorter than the corner route.

Mistake 2: Forgetting the square root

Where it slips in: Stopping at l^2 + w^2 and calling that the diagonal.

Don't do this: Reporting d = 25 for a 333 by 444 rectangle because 9 + 16 = 25.

The correct way: \sqrt{l^2 + w^2} equals d^2, not d. Take the square root: 25 = 5.

Mistake 3: Squaring the units wrongly or mixing units

Where it slips in: Combining a length in metres with a width in centimetres.

Don't do this: Using l = 2 m and w = 50 cm directly in the formula.

The correct way: Convert to one unit first (2 m = 200 cm), then apply the formula. Keep units consistent throughout the calculation, and the final diagonal carries that same unit.

Conclusion

• The diagonal of a rectangle is d = \sqrt{l^2 + w^2}, the hypotenuse of the right triangle formed by two sides.
• The formula comes from the Pythagorean theorem, so it is worth deriving, not just memorising.
• A rectangle has two equal diagonals that bisect each other.
• Always square the sides, add, then take the square root - dropping the root is the most common slip.
• The diagonal is the shortest straight path across the rectangle and longer than either side.