How Do You Calculate the Diagonal of a Square?

If you know the side length of a square, you can calculate its diagonal by multiplying the side length by the square root of 2. The formula is:

Diagonal = Side Length × √2

In mathematical form, this is written as:

d = s√2

In this formula, d means the diagonal of the square, and s means the length of one side. Since all four sides of a square are equal, only one side measurement is needed.

This calculation is basic geometry, but it is also useful in practical design work. Display size, for example, is commonly described by diagonal length. When engineers evaluate LCD active area, touchscreen size, cover glass dimensions, or mechanical enclosure space, diagonal measurement becomes an important reference.

What Is the Diagonal of a Square?

The diagonal of a square is the straight line that connects one corner of the square to the opposite corner. A square has two diagonals, and both diagonals have the same length.

Each diagonal divides the square into two right triangles. In each triangle, the two shorter sides are the sides of the square, and the diagonal becomes the hypotenuse. This is why the diagonal of a square can be calculated using the Pythagorean theorem.

For example, if a square has a side length of 10 mm, the diagonal is longer than 10 mm because it crosses the square from corner to corner. However, it is not twice the side length. It is approximately 1.414 times the side length.

Diagonal of a Square Formula

The standard formula for calculating the diagonal of a square is:

d = s√2

Where:

• d = diagonal length
• s = side length of the square
• √2 ≈ 1.41421356

For most practical calculations, √2 can be approximated as 1.414. This is accurate enough for general geometry, early product design estimation, and rough display-size comparison. For mechanical drawings, tooling design, or formal engineering documentation, the required tolerance should be checked from the project specification.

Why Does the Formula Work?

The square diagonal formula comes from the Pythagorean theorem. When a diagonal is drawn inside a square, it creates a right triangle. The two legs of that triangle are equal because they are both sides of the square.

The Pythagorean theorem is:

a² + b² = c²

For a square, both shorter sides are equal to s, so the formula becomes:

s² + s² = d²

This simplifies to:

2s² = d²

Taking the square root of both sides gives:

d = s√2

This is why the diagonal of any square is always the side length multiplied by √2.

Step-by-Step Example

Assume a square has a side length of 50 mm.

Use the formula:

d = s√2

Substitute the side length:

d = 50 × 1.414

The result is:

d ≈ 70.7 mm

So, the diagonal of a square with a 50 mm side length is approximately 70.7 mm.

If you want to calculate diagonal size from width and height, you can use RJY Display’s display parameter calculator for a faster estimate.

Use the Display Parameter Calculator

Square Diagonal Calculation Table

This table shows the fixed relationship between side length and diagonal length. When the side length doubles, the diagonal also doubles. The ratio does not change because all squares have the same geometric proportions.

How to Calculate the Side Length from the Diagonal

Sometimes the diagonal is known first, and the side length needs to be calculated. In that case, the formula can be rearranged:

s = d / √2

For example, if the diagonal of a square is 100 mm:

s = 100 / 1.414

s ≈ 70.7 mm

So, a square with a diagonal of 100 mm has a side length of approximately 70.7 mm.

This reverse calculation is useful when a product is described by diagonal size but the designer needs to estimate the width and height of a square active area, panel opening, or mechanical layout.

How This Relates to Display Size

Display size is usually described by diagonal length. A “7-inch display,” for example, normally refers to the diagonal size of the active display area, not the full outer size of the LCD module, cover glass, frame, or enclosure.

For a square display area, the relationship between side length and diagonal is direct:

Diagonal = Side × √2

For a rectangular display, the calculation is different:

Diagonal = √(Width² + Height²)

This difference matters in display selection. A square 4-inch display and a rectangular 4-inch display can have very different width, height, viewing area, mechanical fit, and user interface layout. The same diagonal size does not always mean the same usable display area.

Square Display Example

Assume a square LCD active area has a side length of 80 mm. To calculate its diagonal:

d = 80 × 1.414

d ≈ 113.1 mm

To convert millimeters to inches, divide by 25.4:

113.1 / 25.4 ≈ 4.45 inches

So, a square active display area with an 80 mm side length has a diagonal of about 4.45 inches.

This type of calculation can help engineers compare a square display concept with standard rectangular TFT LCD options. It can also help product teams estimate whether a display will fit the available mechanical space before requesting a formal drawing or datasheet review.

Active Area, Module Size, and Cover Glass Size Are Different

When calculating display size, it is important to separate several dimensions.

Active area is the part of the display that actually shows the image.

Module outline size is the physical size of the LCD module, including areas outside the active display region.

Cover glass size is the size of the front glass, which may be larger than the LCD module depending on the product design.

Enclosure opening size is the mechanical opening in the customer’s product housing.

The diagonal size normally refers to the active area unless otherwise specified. For engineering projects, relying only on the diagonal can lead to selection mistakes. Width, height, outline dimensions, FPC position, connector location, touch structure, and mounting method should also be reviewed.

Common Mistakes When Calculating a Square Diagonal

One common mistake is to double the side length. For example, if the side length is 50 mm, the diagonal is not 100 mm. It is approximately 70.7 mm.

Another mistake is using the square diagonal formula for a rectangular shape. The formula d = s√2 only applies to a square because both sides are equal. For a rectangle, width and height must be calculated separately.

A third mistake is assuming that display diagonal size fully defines mechanical fit. In display engineering, diagonal size is only one measurement. A complete evaluation also needs active area, outline size, thickness, interface, touch panel structure, cover glass dimensions, brightness requirement, and operating environment.

Square Diagonal vs Rectangular Screen Diagonal

Most TFT LCD modules are rectangular rather than square. However, the square diagonal formula remains useful for concept design, cover glass planning, icon layout, interface design zones, and square mechanical spaces inside a product enclosure.

When Do Engineers Need This Calculation?

The diagonal of a square is useful in many design and engineering situations. It can be used to estimate screen size, compare display formats, plan a square user interface, check whether a square front panel opening is large enough, or convert between side length and diagonal size.

In display product development, this calculation is especially useful during early concept review. Before a project reaches the detailed drawing stage, a simple diagonal calculation can help the engineering team understand whether a square display concept is close to a standard display size or whether it may require a more customized mechanical design.

How to Use the Formula in Display Selection

When selecting a display, start by identifying whether the available space is square or rectangular. If the space is square, the diagonal can be estimated from one side length. If the space is rectangular, measure both width and height and use the rectangular diagonal formula.

After calculating the diagonal, compare it with available display sizes. However, do not stop at the diagonal value. A display with a matching diagonal may still fail to fit if the outline dimensions, FPC design, connector location, touch panel stack, or cover glass requirements do not match the product structure.

For B2B display projects, the better approach is to provide the full mechanical and electrical requirements. This helps the supplier check whether a standard TFT LCD module, a modified display module, a custom cover glass design, or a display with controller board support is more suitable.

Quick Formula Summary

Conclusion

To calculate the diagonal of a square, multiply the side length by √2. The formula is:

d = s√2

This formula comes from the Pythagorean theorem because the diagonal divides the square into two right triangles. If the side length is 50 mm, the diagonal is approximately 70.7 mm. If the diagonal is known, the side length can be calculated by dividing the diagonal by √2.

For display projects, diagonal size is only one part of the selection process. It helps estimate screen size, but it does not replace active area, outline size, touch structure, cover glass dimensions, interface requirements, and mechanical fit. When selecting an LCD or touchscreen module, the diagonal should be reviewed together with the complete display specification.

Need Help Matching a Display Size?

If you are selecting a TFT LCD, touchscreen display, or display module for an embedded product, RJY Display can help review your size, resolution, interface, brightness, touch, cover glass, controller board, and mechanical requirements. Send your project dimensions, target application, and expected display format so our team can help check suitable options for your design.

FAQ

How do you calculate the diagonal of a square?

Multiply the side length of the square by √2. The formula is d = s√2, where d is the diagonal and s is the side length.

What is the diagonal of a 10 cm square?

The diagonal of a 10 cm square is approximately 14.14 cm. The calculation is 10 × 1.414 = 14.14.

Why is the diagonal of a square longer than its side?

The diagonal runs from one corner to the opposite corner, so it crosses the square at an angle. It becomes the hypotenuse of a right triangle, making it longer than either side of the square.

Can I calculate the side length if I know the diagonal?

Yes. Divide the diagonal by √2. The formula is s = d / √2.

Is screen size measured by diagonal?

Yes. Display size is usually described by diagonal length, especially for LCD and touchscreen products. However, engineers should also check active area, module outline size, cover glass size, and mechanical fit.

Is the square diagonal formula the same for a rectangular display?

No. The formula d = s√2 only applies to squares. For rectangular displays, use d = √(width² + height²).