Rectangular Prism Calculator (Cuboid)

Enter any 3 dimensions or known quantities (length, width, height, volume, surface area, or space diagonal) to instantly compute all unknown variables with dynamic 3D WebGL rendering and step-by-step mathematical proofs.

Input Variable

Solve Rectangular Prism by Any Known Value

Unit:

Given Variable

Length (l)cm

Width (w)cm

Height (h)cm

Material Mass EstimatorMass: --

Step-by-Step Mathematical Proofs

Interactive 3D Rectangular PrismWebGL Orbit Controls

Volume

V = l \cdot w \cdot h

--

cm³

Total Surface Area

S = 2(lw+lh+wh)

--

cm²

Space Diagonal

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

--

cm

Lateral Surface Area

S_{\text{lat}} = 2h(l+w)

--

cm²

Base Area

B = l \cdot w

--

cm²

Base Perimeter

P = 2(l+w)

--

cm

Complete Mathematical Analysis

Rectangular Prism Formulas & Definitions

A rectangular prism (also called a rectangular cuboid or orthogonal parallelepiped) is a convex 3D polyhedron bounded by six rectangular faces meeting at perpendicular 90° dihedral angles.

Fundamental Equations

Volume (V):

V = l \cdot w \cdot h

Multiplication of length, width, and height.

Total Surface Area (S):

S = 2(lw + lh + wh)

Sum of all 6 rectangular boundary faces.

Space Diagonal (d):

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

Interior 3D distance between opposing corners.

Lateral Surface Area

S_{\text{lat}}

:

S_{\text{lat}} = 2h(l + w)

Perimeter of base multiplied by prism height.

How to Solve for Missing Dimensions

Case 1: Finding Height given Volume, Length, and Width

Since $V = l \cdot w \cdot h$ , isolate $h$ by dividing both sides by the base area ( $l \cdot w$ ):

h = \frac{V}{l \cdot w}

Case 2: Finding Height given Surface Area, Length, and Width

Expand $S = 2lw + 2lh + 2wh = 2lw + 2h(l + w)$ . Subtract $2lw$ and divide by $2(l+w)$ :

h = \frac{S - 2lw}{2(l + w)}

Case 3: Finding Height given Space Diagonal, Length, and Width

Square both sides of $d = \sqrt{l^2 + w^2 + h^2} \implies d^2 = l^2 + w^2 + h^2$ . Isolate $h$ :

h = \sqrt{d^2 - l^2 - w^2} \quad (d^2 > l^2 + w^2)

Real-World Engineering & Practical Applications

Packaging & Logistics (CBM): Freight forwarders calculate Cubic Meter (CBM) volume as l × w × h to price pallet shipping and air freight containers.
HVAC & Architecture: Room air displacement, cooling load BTU requirements, and radiant heating sizing require accurate volume and wall surface area.
Aquarium & Tank Capacity: Fish tank volume in gallons or liters is derived from the rectangular prism volume of the water column.
Masonry & Concrete Footings: Determining cubic yards of cement needed for building foundations and slab formwork.

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