application

Diagonal Calculator

Calculate the diagonal of a rectangle from its length and width using the Pythagorean theorem. Perfect for construction, interior design, and screen measurement.

Back to Pythagorean Theorem Calculator

Pythagorean Theorem Tips

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Try an Example

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Ladder Against Wall

A 10-foot ladder leans against a wall with the base 6 feet away. How high does it reach?

Key values: Leg a = 6 ft · Hypotenuse c = 10 ft · Solve for leg b

TV Screen Diagonal

Find the diagonal of a TV with a 16:9 aspect ratio (width 40", height 22.5").

Key values: Width = 40 in · Height = 22.5 in · Find diagonal

Classic 3-4-5 Triangle

Verify that sides 3, 4, and 5 form a right triangle (the most famous Pythagorean triple).

Key values: Sides: 3, 4, 5 · Right triangle · Primitive triple

Documentation

Rectangle Diagonal

The diagonal of a rectangle with width $w$ and height $h$ is a direct application of the Pythagorean theorem:

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

A rectangle has two diagonals that are equal in length and bisect each other.

Common Diagonal Calculations

Shape	Diagonal formula	Example
Square (side s)	$s\sqrt{2}$	10 cm square → 14.14 cm
Rectangle (w × h)	$\sqrt{w^2 + h^2}$	3 × 4 → 5
Cube (side s)	$s\sqrt{3}$	10 cm cube → 17.32 cm
Rectangular box (l × w × h)	$\sqrt{l^2 + w^2 + h^2}$	3 × 4 × 12 → 13

3D Space Diagonal

The Pythagorean theorem extends naturally to three dimensions. The space diagonal of a box goes from one corner to the opposite corner, passing through the interior:

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

Shipping example: Will a 50-inch rod fit inside a 30 × 24 × 36 inch box? Space diagonal = $\sqrt{900 + 576 + 1296} = \sqrt{2772} \approx 52.6$ ". Yes, the rod fits diagonally.

Screen Size from Diagonal

Screens are measured by diagonal length. Given the aspect ratio and diagonal, compute the dimensions:

w = \frac{d \cdot r}{\sqrt{1 + r^2}} \qquad h = \frac{d}{\sqrt{1 + r^2}}

where $r = w/h$ is the aspect ratio (16/9 ≈ 1.778 for widescreen).

Diagonal	Aspect ratio	Width	Height	Area
27"	16:9	23.5"	13.2"	311 in²
32"	16:9	27.9"	15.7"	437 in²
55"	16:9	47.9"	27.0"	1,293 in²

Frequently Asked Questions

How do you calculate the diagonal of a rectangle?

Use the Pythagorean theorem: $d = \sqrt{w^2 + h^2}$ , where $w$ is the width and $h$ is the height. For example, a 3 x 4 rectangle has diagonal $\sqrt{9 + 16} = \sqrt{25} = 5$ .

What is the diagonal of a square?

For a square with side length $s$ , the diagonal is $s\sqrt{2}$ , which is approximately $1.414 \times s$ . A 10 cm square has a diagonal of about 14.14 cm.

How do you find the space diagonal of a box?

Extend the Pythagorean theorem to three dimensions: $d = \sqrt{l^2 + w^2 + h^2}$ . For a 3 x 4 x 12 box, the space diagonal is $\sqrt{9 + 16 + 144} = \sqrt{169} = 13$ .

How are TV and monitor screen sizes measured?

Screen sizes are measured diagonally from one corner to the opposite corner. For a 16:9 widescreen, the width is about 87.2% of the diagonal and the height is about 49.0%. A 55-inch TV is roughly 47.9 inches wide and 27.0 inches tall.

Can I use the diagonal formula to check if something fits in a box?

Yes. The space diagonal $\sqrt{l^2 + w^2 + h^2}$ gives the longest straight object that fits inside a rectangular box. If the object is shorter than the space diagonal, it can be placed diagonally inside the box.

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