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Diagonal of a Rectangular Prism

The space diagonal runs through the interior between opposite corners. It is longer than any single edge and follows d = √(l² + w² + h²).

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Rectangular prism diagrams and dimension notes for geometry guides

Quick Answer

Space diagonal d = √(l² + w² + h²).

Formula

  • Face diagonal on lw face: √(l² + w²)
  • Space diagonal uses all three edges

Introduction

Rectangular Prism Calculator helps you calculate volume, surface area, and the space diagonal from length, width, and height.

The space diagonal runs through the interior between opposite corners. It is longer than any single edge and follows d = √(l² + w² + h²).

Face diagonals lie on one rectangle only; mixing the two types is a frequent source of lost points.

This article defines the space diagonal, compares it to face diagonals, and walks through worked values you can check on the home page.

Main Content

What is it?

Build the space diagonal by applying the Pythagorean theorem twice: first along a base diagonal √(l² + w²), then through height to √(c² + h²), which collapses to √(l² + w² + h²).

Engineering sketches sometimes use it for brace or cable length estimates in idealized box frames, though real installations add clearance and offsets.

The diagonal uses all three dimensions. It is not the same as height alone, width alone, or a face diagonal on one side.

Report the result in the same length unit as the edges after the square root.

  • Space diagonal uses l, w, and h together
  • Face diagonal on the lw face: √(l² + w²) only
  • Cube space diagonal: s√3
  • Square edge lengths before adding under the radical
  • Round at the end unless instructions say otherwise

Formula

  • d = √(l² + w² + h²)
  • Base diagonal c = √(l² + w²)
  • Then d = √(c² + h²)

Squaring edges before the root avoids order mistakes inside the radical. Add the three squares, then take one square root.

If a problem gives a face diagonal and height, you may use the two-step path instead of memorizing the three-term line.

Compare your manual root with the home tool when you want a quick decimal check.

Symbol and unit reminders for all targets appear in rectangular prism formulas; step order for full calculations is in how to calculate a rectangular prism.

Step-by-step guide

Use this order for homework, site checks, packaging quotes, or classroom labs.

  1. Square l, w, and h Write three squared terms with the same unit basis.
  2. Add the squares Sum under a single radical before rooting.
  3. Take the square root Use a calculator for decimals; keep radicals exact when required.
  4. Label face versus space State which diagonal the problem requested in your final sentence.
  5. Verify alongside volume and area Enter the same triple in Rectangular Prism Calculator to see d with V and TSA.

Example

Problem: Room model 6 m by 4 m by 2.5 m. Find the space diagonal.

  1. Square: 36, 16, and 6.25.
  2. Sum: 36 + 16 + 6.25 = 58.25.
  3. d = √58.25 ≈ 7.63 m.

Space diagonal about 7.63 m for the idealized prism model.

FAQ

Can a face diagonal equal the space diagonal?
Only in degenerate cases; for a solid box the space diagonal is generally longer than a face diagonal.
Do I need volume to find a diagonal?
No. Diagonal problems use edge lengths only.
Why square edges first?
The Pythagorean theorem uses squares of legs, not the legs themselves added.
Does the calculator show face diagonals?
The home tool reports the space diagonal d alongside volume and surface area for the same inputs.

Conclusion

Label whether the problem wants a face diagonal or the space diagonal before you substitute.

Keep edge units consistent through squaring and rooting.

The home calculator returns d alongside volume and surface area for the same three inputs.

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