What is a rectangular prism?

A 3D shape with 6 rectangular faces (cuboid/box). All angles are right angles. Cube is a special case when length = width = height.

What is the volume of a rectangular prism?

V = l × w × h. Length times width times height.

What is the surface area?

A = 2(lw + lh + wh). Sum of areas of all six faces.

What is the space diagonal?

d = √(l² + w² + h²). The longest line inside the box; 3D Pythagorean theorem.

What is lateral area?

Lateral area = 2(l + w)h — the four vertical sides only, excluding top and bottom.

Where are rectangular prisms used?

Boxes, rooms, shipping containers, phones, packaging. The most common 3D shape in daily life.

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GEOMETRY3D GeometryMathematics Calculator

📦

Rectangular Prism — Boxes, Rooms, Containers

Compute volume, surface area, space diagonal, base area, and lateral area from length, width, and height. Six rectangular faces, right angles—the most common 3D shape.

Concept Fundamentals

V = l × w × h

Volume

A = 2(lw + lh + wh)

Surface Area

d = √(l² + w² + h²)

Space Diagonal

A_b = l × w

Base Area

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Cube is a special case: l = w = h Space diagonal = longest line inside the box Lateral area = 4 sides only; total = lateral + 2×base Face diagonal on one face: √(l²+w²); space: √(l²+w²+h²) Volume for capacity; surface area for material (cardboard, wrap)

Key quantities

V = l × w × h

Volume

Key relation

A = 2(lw + lh + wh)

Surface Area

Key relation

d = √(l² + w² + h²)

Space Diagonal

Key relation

A_b = l × w

Base Area

Key relation

Ready to run the numbers?

Why: Rectangular prisms (cuboids) are everywhere—boxes, rooms, shipping containers, phones. Volume for capacity, surface area for material, space diagonal for 'will it fit?' questions.

How: Enter length, width, and height. Volume = lwh. Surface area = 2(lw+lh+wh)—sum of all six face areas. Space diagonal = √(l²+w²+h²)—3D Pythagorean theorem. Base area = lw. Lateral = 2(l+w)h.

Cube is a special case: l = w = hSpace diagonal = longest line inside the box

Sources:Wolfram MathWorld — CuboidKhan Academy — 3D Geometry

Run the calculator when you are ready.

Rectangular Prism CalculatorEnter length, width, and height to compute all prism properties

📦

3D GEOMETRYCuboid / Box

Rectangular Prism — Boxes, Rooms, Containers

Six rectangular faces, right angles. Volume = lwh, SA = 2(lw+lh+wh), diagonal = √(l²+w²+h²).

Cube →Pyramid →

📦 Sample Examples — Click to Load

Prism Dimensions

Length (l)

Width (w)

Height (h)

rectprism_calc.sh

CALCULATED

$ calculate_rectprism --l=33 --w=18 --h=12

Volume

7,128

Surface Area

2,412

Space Diagonal

39.4588

Base Area

594

Lateral Area

1,224

Rectangular Prism Properties

l=33, w=18, h=12

7,128 units³

📐 SA: 2412.00↗️ Diagonal: 39.46

3D Visualization

Property Radar

Property Comparison

Volume vs SA Breakdown

📐 Calculation Breakdown

INPUT

Length (l)

Width (w)

Height (h)

AREAS

Base Area (A_b)

594

l × w = 33 × 18

Lateral Area (A_L)

1,224

2(l+w)h = 2(33+18)×12

Total Surface Area (A)

2,412

2(lw+lh+wh)

VOLUME

Volume (V)

7,128

l × w × h = 33 × 18 × 12

DIAGONAL

Space Diagonal (d)

39.4588

√(l²+w²+h²)

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

📦

A cube is a special rectangular prism where l = w = h

— Geometry

🚚

Standard shipping containers are 20 or 40 ft—rectangular prisms optimized for stacking

— Logistics

📐

Space diagonal is the longest line inside the box—useful for "will it fit?" questions

— Packing

🏗️

Rooms are rectangular prisms—volume determines HVAC capacity

— Architecture

📋 Key Takeaways

• A rectangular prism (cuboid/box) has 6 rectangular faces, 12 edges, 8 vertices
• Volume = $l \times w \times h$ — length × width × height
• Surface area = $2(lw + lh + wh)$ — sum of all 6 face areas
• Space diagonal = $\sqrt{l^2 + w^2 + h^2}$ — 3D Pythagorean theorem

💡 Did You Know?

📦A cube is a special rectangular prism where l = w = hSource: Geometry

🚚Standard shipping containers are 20 or 40 ft — rectangular prisms optimized for stackingSource: Logistics

📐Space diagonal is the longest line inside the box — useful for "will it fit?" questionsSource: Packing

🏗️Rooms are rectangular prisms — volume determines HVAC capacitySource: Architecture

📱Phone dimensions are often given as l×w×h — a thin rectangular prismSource: Consumer Tech

🧊Ice cubes and dice are cubes — rectangular prisms with all edges equalSource: Everyday Objects

📖 How Rectangular Prism Calculations Work

Three dimensions define a rectangular prism: length (l), width (w), height (h).

Volume

$V = lwh$ — base area × height. How much space inside.

Surface Area

$A = 2(lw + lh + wh)$ — each face pair (front/back, left/right, top/bottom) contributes twice.

Space Diagonal

$d = \sqrt{l^2 + w^2 + h^2}$ — corner to opposite corner through the interior. 3D Pythagoras.

🎯 Expert Tips

💡 Lateral vs Total SA

Lateral = 2h(l+w) — four sides only. Total = Lateral + 2×Base.

💡 Face vs Space Diagonal

Face diagonal on one face: √(l²+w²). Space diagonal: √(l²+w²+h²).

💡 Packaging

Volume for capacity; surface area for material (cardboard, wrap).

💡 Will It Fit?

Space diagonal = max length of rigid object that fits inside.

⚖️ Comparison

Feature	This Calculator	Manual	Basic Online
All 5 properties + diagonal	✅	❌	⚠️
Step-by-step	✅	✅	❌
Charts & 3D viz	✅	❌	❌
Examples	✅	❌	❌
Copy & share	✅	❌	❌
AI analysis	✅	❌	❌

❓ FAQ

Prism vs cuboid vs box?

Same shape — 6 rectangular faces, right angles. Terms used interchangeably.

Is a cube a rectangular prism?

Yes — a cube is a rectangular prism with l = w = h.

Face diagonal vs space diagonal?

Face diagonal lies on one face (√(l²+w²)). Space diagonal goes through the interior (√(l²+w²+h²)).

Lateral vs total surface area?

Lateral = 4 side faces only. Total = lateral + top + bottom.

Units?

Use same unit for l, w, h. Volume in cubic, area in square units.

📊 Prism Stats

Faces

Edges

Vertices

Space diagonals

📚 Sources

Wolfram MathWorld — Cuboid ↗

Mathematical reference

Khan Academy — 3D Geometry ↗

⚠️ Disclaimer: Results are mathematically precise. Verify independently for critical applications.

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⬅️Jump in and explore the concept!

Rectangular Prism — Boxes, Rooms, Containers

Rectangular Prism — Boxes, Rooms, Containers

📦 Sample Examples — Click to Load

Prism Dimensions

3D Visualization

Property Radar

Property Comparison

Volume vs SA Breakdown

📐 Calculation Breakdown

🧮 Fascinating Math Facts

📋 Key Takeaways

💡 Did You Know?

📖 How Rectangular Prism Calculations Work

Volume

Surface Area

Space Diagonal

🎯 Expert Tips

💡 Lateral vs Total SA

💡 Face vs Space Diagonal

💡 Packaging

💡 Will It Fit?

⚖️ Comparison

❓ FAQ

Prism vs cuboid vs box?

Is a cube a rectangular prism?

Face diagonal vs space diagonal?

Lateral vs total surface area?

Units?

📊 Prism Stats

📚 Sources

Related Calculators

Cube Calculator

Rectangular Prism Box Calculator

Cylinder Calculator

Sphere Calculator

Ellipsoid Volume Calculator

Generic Surface Area Calculator

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Rectangular Prism — Boxes, Rooms, Containers

Rectangular Prism — Boxes, Rooms, Containers

📦 Sample Examples — Click to Load

Prism Dimensions

3D Visualization

Property Radar

Property Comparison

Volume vs SA Breakdown

📐 Calculation Breakdown

🧮 Fascinating Math Facts

📋 Key Takeaways

💡 Did You Know?

📖 How Rectangular Prism Calculations Work

Volume

Surface Area

Space Diagonal

🎯 Expert Tips

💡 Lateral vs Total SA

💡 Face vs Space Diagonal

💡 Packaging

💡 Will It Fit?

⚖️ Comparison

❓ FAQ

Prism vs cuboid vs box?

Is a cube a rectangular prism?

Face diagonal vs space diagonal?

Lateral vs total surface area?

Units?

📊 Prism Stats

📚 Sources

Related Mathematics Calculators

Related Calculators

Cube Calculator

Rectangular Prism Box Calculator

Cylinder Calculator

Sphere Calculator

Ellipsoid Volume Calculator

Generic Surface Area Calculator

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