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 $4$ cm and width $3$ 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 $6$ m and width $8$ m.

Your first instinct might be to add the sides directly, reasoning that the diagonal "goes across both," and write $6 + 8 = 14$ m. Let's test that against the geometry.

The diagonal is the shortest path across, not the sum of two sides walked around the corner. Walking $6$ then $8$ is going around the edge; the diagonal cuts straight across, so it must be shorter than $14$.

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

The straight cut ($10$ m) is indeed shorter than the corner route ($14$ m), which confirms adding the sides was wrong.

Final answer: the diagonal is $10$ m.

Example 3

A rectangle has length $12$ and width $5$. 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 $5$ cm and width $2$ cm. Find the diagonal, rounded to two decimals.

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

$\sqrt{29}$ is not a whole number, so leave it exact or approximate it:

$$d \approx 5.39 \text{ cm}$$

Final answer: $\sqrt{29} \approx 5.39$ cm.

Example 5

A rectangle has a diagonal of $10$ cm and a length of $8$ 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 $40$ m by $30$ 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 ("$32$-inch TV") is quoted — those are diagonal measurements, 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.

Screen sizes are the everyday version of this: a "$50$-inch" television, measured by its display diagonal, is $50$ inches along the diagonal, not across the width. Shoppers who assume $50$ inches is the width are reading the wrong measurement - the same slip as adding the two sides instead of applying $d = \sqrt{l^2 + w^2}$. The diagonal is a single number that packages both dimensions through Pythagoras.

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 $3$ by $4$ rectangle gets a diagonal of $7$.

The correct way: The diagonal is $\sqrt{l^2 + w^2}$, not $l + w$. For $3$ by $4$ that is $\sqrt{9 + 16} = \sqrt{25} = 5$, not $7$. 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 $3$ by $4$ rectangle because $9 + 16 = 25$.

The correct way: $l^2 + w^2$ equals $d^2$, not $d$. Take the square root: $\sqrt{25} = 5$. The memorizer who recalls "square and add" but drops the final root ends up with the diagonal squared.

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.

Practise What You Have Learned

Work through these to test your understanding: find the diagonal of a $9$ by $12$ rectangle (Answer to Question 1: $15$); find the width of a rectangle whose diagonal is $13$ and length is $5$ (Answer to Question 2: $12$); and compute the diagonal of a $7$ by $7$ square, leaving it exact. To learn this with a teacher, explore Bhanzu's geometry tutor, middle school math tutor, or math classes online. Want to watch the diagonal update as you resize a rectangle? Book a free demo class.

Read More

• Diagonal of Rhombus - how diagonals differ in a rhombus.

• Is a Square a Rectangle - the special rectangle whose diagonal simplifies.

• Difference Between a Square and a Rectangle - where their diagonals agree and differ.

• Square - the diagonal of the equal-sided rectangle.