How do you find the diagonal of a face (not the space diagonal) of a cuboid?

In one line

The rule

The diagonal of a rectangle (face of a cuboid) with sides a and b is given by the formula: d = √(a² + b²)

Where:

• d = diagonal of the face
• a = length of one side of the face
• b = length of the other side of the face

Let's understand it simply

Imagine you have a matchbox (a cuboid). Look at just one flat side — like the top. That side is a rectangle. If you draw a line from one corner to the opposite corner, that is the face diagonal. It is the longest line you can draw on that flat face. To find its length, you need to know the lengths of the two sides of that rectangle.

How it works

1. Pick the face you want (for example, the top face: length and width).
2. Note the lengths of the two sides of that face (say, l and w).
3. Use the formula: d = √(l² + w²).
4. Square both side lengths.
5. Add the squares together.
6. Take the square root of the sum to get the diagonal.

The story behind it

This formula comes from the Pythagoras theorem. Pythagoras was a Greek mathematician who lived around 2500 years ago. He found that in a right-angled triangle, the square of the longest side (hypotenuse) equals the sum of the squares of the other two sides. The diagonal of a rectangle forms a right-angled triangle with the sides of the rectangle.

Where you see it in real life

• TV or mobile screen: The diagonal is the size of the screen (measured from one corner to the opposite).
• Tabletop: The longest stick you can lay flat across a rectangular table is its diagonal.
• Books: The diagonal is the longest distance across the cover.
• Tiles: When tiling a floor, the diagonal helps check if the tiles are square.
• Cricket pitch boundary: The longest straight shot you can hit across a rectangular field is along the diagonal.

Why it matters

• Used in construction to check if a rectangle is really straight (not tilted).
• Helps in designing screens, windows, and doors.
• Needed in packaging to check if an object will fit inside a box.
• Used in geometry problems in exams and higher studies.

Worked example

Let's say a cuboid has length = 5 cm, width = 3 cm, and height = 4 cm. Find the diagonal of the top face (length × width).

Given: length (l) = 5 cm, width (w) = 3 cm
Step 1: Write the rule         d = √(l² + w²)
Step 2: Put in the numbers     d = √(5² + 3²)
Step 3: Calculate squares      d = √(25 + 9)
Step 4: Add                    d = √34
Step 5: Find the square root   d ≈ 5.83 cm

Common mistakes

• Mixing up the space diagonal (corner to farthest corner through the box) with the face diagonal (just across one face).
• Forgetting to use the correct two sides for the face you are looking at (for example, using height instead of width).
• Not squaring the numbers before adding.

Remember it

Face diagonal = square root of (side 1 squared + side 2 squared).