Diagonal of Square Calculator

Finding the Diagonal of a Square

The diagonal of a square connects two opposite corners, passing through the center. It's one of the most fundamental measurements in geometry and appears frequently in construction, design, and mathematics.

The Formula

The diagonal of a square with side length s is:

$$d = s\sqrt{2}$$

Where √2 ≈ 1.41421356...

To find the side from the diagonal:

$$s = \frac{d}{\sqrt{2}} = \frac{d\sqrt{2}}{2}$$

Why √2?

Pythagorean Theorem Proof

The diagonal creates a right triangle with two sides of the square:

$$d^2 = s^2 + s^2 = 2s^2$$

Taking the square root:

$$d = \sqrt{2s^2} = s\sqrt{2}$$

The 45-45-90 Triangle

A square's diagonal divides it into two congruent 45-45-90 triangles. In any 45-45-90 triangle:

• The legs are equal (both = s)
• The hypotenuse = leg × √2

This is why the ratio √2 appears in the diagonal formula.

Related Measurements

Area from Diagonal

You can calculate area directly from the diagonal:

$$A = \frac{d^2}{2}$$

This is because A = s² and s = d/√2, so:

$$A = \left(\frac{d}{\sqrt{2}}\right)^2 = \frac{d^2}{2}$$

Perimeter from Diagonal

$$P = 4s = 4 \times \frac{d}{\sqrt{2}} = 2d\sqrt{2}$$

Circumradius (Circumscribed Circle)

The radius of a circle passing through all four corners:

$$R = \frac{d}{2}$$

Inradius (Inscribed Circle)

The radius of a circle fitting inside the square:

$$r = \frac{s}{2}$$

Practical Applications

Construction and Carpentry

When squaring a foundation or frame:

1. Measure both diagonals
2. If they're equal, the shape is a rectangle
3. If sides are also equal, it's a square

Example: For a 10×10 foot deck, the diagonal should be 10√2 ≈ 14.14 feet.

Screen Sizes

TV and monitor sizes are measured diagonally:

• A 50" TV might have roughly 43.5" × 24.5" dimensions
• For a square display, a 50" diagonal means ~35.4" sides

Tile and Flooring

When laying tiles diagonally:

• A 12" tile placed diagonally spans 12√2 ≈ 17" corner to corner
• Coverage calculations must account for increased cut waste

Photography

The "rule of thirds" grid creates squares, and understanding diagonals helps with composition and focal length calculations.

Common Diagonal Values

Side Length	Diagonal
1	1.414
2	2.828
3	4.243
4	5.657
5	7.071
6	8.485
8	11.314
10	14.142
12	16.971
15	21.213
20	28.284

Relationship to Other Shapes

Rectangle

For a rectangle with sides a and b:

$$d = \sqrt{a^2 + b^2}$$

A square is a special case where a = b.

Cube

The space diagonal of a cube with side s:

$$d_{space} = s\sqrt{3}$$

The face diagonal (on any face) uses the same formula as a square.

Regular Polygons

For a regular n-gon, the relationship between side and longest diagonal varies:

• Square (n=4): d = s√2
• Pentagon (n=5): d = s × φ (golden ratio)
• Hexagon (n=6): d = 2s

The Irrational Nature of √2

A Famous Discovery

The ancient Greeks discovered that √2 cannot be expressed as a fraction—it's irrational. This was revolutionary and allegedly caused a crisis in Pythagorean philosophy.

Decimal Approximation

√2 = 1.41421356237309504880168872420969807856967187537694...

For most practical purposes, 1.414 or 1.4142 provides sufficient accuracy.

Rational Approximations

Close fractions for √2:

• 99/70 ≈ 1.4143
• 239/169 ≈ 1.41420
• 577/408 ≈ 1.414216

Working with Units

Converting Between Units

When the side is in different units:

• Side in inches, diagonal in feet: d = (s × √2) / 12
• Side in cm, diagonal in meters: d = (s × √2) / 100

Mixed Unit Example

A room is 12 feet square. What's the diagonal in feet and inches?

• d = 12 × √2 = 16.97 feet
• 16.97 feet = 16 feet, 11.6 inches

Solving for Side Length

Given Diagonal

If you know the diagonal and need the side:

$$s = \frac{d}{\sqrt{2}} \approx 0.7071 \times d$$

Example: A TV has a 55" diagonal (assuming square). Side ≈ 55 × 0.7071 ≈ 38.9"

Given Area

If you know the area:

$$s = \sqrt{A}$$ $$d = \sqrt{2A}$$

Summary

Key formulas for square diagonals:

1. Diagonal from side: d = s√2
2. Side from diagonal: s = d/√2
3. Area from diagonal: A = d²/2
4. Circumradius: R = d/2
5. Inradius: r = s/2

The √2 relationship is fundamental to understanding squares, rectangles, and 45-45-90 triangles in geometry.