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Rectangular Prism Calculator

Volume equals length times width times height

Solution

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Calculate Rectangular Prism Volume

Use this form when the three side lengths are known and you need the enclosed volume — moving boxes, fish tanks, packaging, storage containers.

V = l × w × h

Calculate Rectangular Prism Surface Area

Use this form to compute the total surface area of all six rectangular faces — wrapping paper, paint coverage, sheet-metal estimates.

S = 2(lw + lh + wh)

Calculate Rectangular Prism Space Diagonal

Use this form for the longest straight line that fits inside the prism, corner to opposite corner — checking whether an object fits diagonally into a box.

d = √(l² + w² + h²)

Calculate Rectangular Prism Length

Use this rearrangement when volume, width, and height are known and you need the missing length.

l = V / (w × h)

Calculate Rectangular Prism Width

Use this rearrangement when volume, length, and height are known and you need the missing width.

w = V / (l × h)

Calculate Rectangular Prism Height

Use this rearrangement when volume, length, and width are known and you need the missing height.

h = V / (l × w)

How It Works

This rectangular prism (cuboid, box) calculator solves V = l·w·h for volume, S = 2(lw + lh + wh) for total surface area, and d = √(l² + w² + h²) for the space diagonal — the longest straight line that fits inside the prism. Inverse solves cover any one missing dimension given the volume and the other two. Pick the unknown with the solve-for toggle and enter the remaining three values.

Example Problem

A moving box measures 4 m long by 3 m wide by 2 m tall. What are its volume, surface area, and space diagonal?

  1. Knowns: l = 4 m, w = 3 m, h = 2 m
  2. Volume: V = l × w × h = 4 × 3 × 2 = 24 m³
  3. Surface area: S = 2(lw + lh + wh) = 2(12 + 8 + 6) = 2(26) = 52 m²
  4. Space diagonal: d = √(l² + w² + h²) = √(16 + 9 + 4) = √29 ≈ 5.385 m
  5. Sanity check (inverse): from V = 24, w = 3, h = 2, l = V/(wh) = 24/6 = 4 m, recovering the original length.

The space diagonal is the limit on the longest rigid object you can fit inside the box — useful for shipping irregular items.

When to Use Each Variable

  • Solve for Volume — when all three side lengths are known — sizing storage containers, fish tanks, moving boxes.
  • Solve for Surface Area — when you need the area of all six faces — wrapping paper, paint, sheet metal.
  • Solve for Diagonal — when you need the longest straight line that fits inside the prism — checking object fit.
  • Solve for Length — when volume, width, and height are known and the length is the unknown.
  • Solve for Width — when volume, length, and height are known and the width is the unknown.
  • Solve for Height — when volume, length, and width are known and the height is the unknown.

Key Concepts

A rectangular prism (also called a cuboid or box) is a six-faced polyhedron with all faces rectangular and all angles right angles. Three numbers — length l, width w, and height h — fully determine its geometry. Every property (volume, surface area, space diagonal, edge length) follows from those three. A rectangular prism with l = w = h is a cube — a special case where every face is a square. The space diagonal d = √(l² + w² + h²) is a 3D extension of the Pythagorean theorem.

Applications

  • Storage and shipping: size moving boxes, freight containers, shipping crates
  • Construction: estimate concrete volume for foundations, brick volume for walls, drywall surface area for rooms
  • Aquariums and tanks: compute water volume and glass-pane area
  • Packaging design: optimize box dimensions for a target volume or to fit a specific product

Common Mistakes

  • Mixing length units across dimensions (e.g., l in feet, w in meters) without converting first
  • Forgetting the factor of 2 in S = 2(lw + lh + wh) — three pairs of faces, six faces total
  • Confusing space diagonal (corner to opposite corner) with face diagonal (corner to corner of one face)
  • Computing surface area when you actually want only outside faces — e.g., open-top boxes need the bottom + four sides, not all six

Frequently Asked Questions

How do you calculate the volume of a rectangular prism?

Multiply length × width × height: V = l × w × h. For a 4 × 3 × 2 m box, V = 24 m³.

What is the formula for the surface area of a rectangular prism?

S = 2(lw + lh + wh). Compute each of the three pairs of identical faces and double the result. For a 4 × 3 × 2 m box, S = 2(12 + 8 + 6) = 52 m².

What is the space diagonal of a rectangular prism?

d = √(l² + w² + h²). It's the longest straight line that fits inside the prism, going from one corner to the opposite corner. For a 4 × 3 × 2 m box, d = √29 ≈ 5.385 m.

How is a rectangular prism different from a cube?

A cube is a special rectangular prism where all three side lengths are equal (l = w = h). All cubes are rectangular prisms, but only the prisms with l = w = h are cubes.

How do you find the missing dimension of a rectangular prism given the volume?

Divide the volume by the product of the other two dimensions: l = V/(wh), w = V/(lh), or h = V/(lw). The unknown dimension is the volume divided by the area of the face perpendicular to it.

What's the difference between space diagonal and face diagonal?

The face diagonal is the diagonal of one rectangular face (only two dimensions involved, e.g., √(l² + w²) for the bottom face). The space diagonal goes through the interior of the prism from one corner to the opposite corner, involving all three dimensions: √(l² + w² + h²).

How many edges and vertices does a rectangular prism have?

A rectangular prism has 12 edges (4 of each length l, w, h), 8 vertices (corners), and 6 rectangular faces. Euler's formula V − E + F = 8 − 12 + 6 = 2 confirms it's a valid polyhedron.

How do I find the volume in gallons or liters?

Compute volume in cubic meters or cubic feet first, then pick the desired unit from the ResultTable — 1 m³ ≈ 264.17 US gallons, 1 ft³ ≈ 7.4805 US gallons, 1 m³ = 1000 liters. The calculator handles all conversions automatically.

Reference: Weisstein, Eric W. "Cuboid." MathWorld — A Wolfram Web Resource. https://mathworld.wolfram.com/Cuboid.html

Worked Examples

Moving Box

How much does a 4 × 3 × 2 m moving box hold?

A standard moving box measures 4 m × 3 m × 2 m. Compute its volume to plan a load.

  • Knowns: l = 4 m, w = 3 m, h = 2 m
  • Formula: V = l × w × h
  • V = 4 × 3 × 2 = 24 m³

Volume = 24 m³ (about 6,340 US gallons)

Real moving boxes carry 1.5–3 m³ each; this example is more like a small shipping container.

Gift Wrap

How much wrapping paper covers a 0.5 × 0.3 × 0.2 m box?

A gift box measures 0.5 m × 0.3 m × 0.2 m. Compute the surface area to size the wrapping paper.

  • Knowns: l = 0.5 m, w = 0.3 m, h = 0.2 m
  • Formula: S = 2(lw + lh + wh)
  • S = 2(0.15 + 0.10 + 0.06) = 2 · 0.31 = 0.62 m²

Surface area = 0.62 m² (about 6.7 ft²)

Add 10–20% for overlap and folding when buying actual wrapping paper.

Inverse Solve

What length does a 100 L box of fixed footprint need?

A storage box must hold 0.1 m³ (100 L), with a fixed base of 0.5 m × 0.4 m. Find the required length (height).

  • Knowns: V = 0.1 m³, w = 0.5 m, h = 0.4 m
  • Formula: l = V / (w × h)
  • l = 0.1 / (0.5 · 0.4) = 0.5 m

Length = 0.5 m (final box: 0.5 × 0.5 × 0.4 m)

If 'height' is the constrained vertical dimension, swap which dimension you call l/w/h — the formula is symmetric.

Rectangular Prism Formulas

A rectangular prism is defined by three side lengths: l, w, and h. From those, volume, surface area, and space diagonal all derive:

Volume equals length times width times heightSurface area equals two times the quantity l w plus l h plus w hSpace diagonal equals the square root of l squared plus w squared plus h squaredLength equals V divided by w hWidth equals V divided by l hHeight equals V divided by l w
Rectangular prism with labeled length l, width w, and height hlwh

Where:

  • V — volume (m³, L, gal, ft³)
  • S — total surface area of all six faces
  • d — space diagonal (corner to opposite corner, through the interior)
  • l — length (m, cm, in, ft, yd)
  • w — width
  • h — height

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