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Rectangular Prism - Volume, Surface Area, Formulas

What Is a Rectangular Prism?

A rectangular prism is a 3D shape with:

• 6 rectangular faces (3 pairs of identical opposite faces)
• 12 edges (4 each of length, width, height)
• 8 vertices (corners)
• All angles at vertices are right angles (90°).

Common everyday examples: a book, a brick, a shoebox, a refrigerator, a swimming pool, a shipping container. The world is full of rectangular prisms — they're the most common 3D shape in human-made objects.

Names you'll see:

• Rectangular prism (most common in US schools)
• Cuboid (more common in UK / Indian schools)
• Rectangular parallelepiped (formal mathematical name)

When all three dimensions are equal, the rectangular prism becomes a cube — a special case where length = width = height.

Volume Formula

The volume of a rectangular prism is:

V=l×w×h

where l = length, w = width, h = height.

In words: volume is the product of the three dimensions.

Units. If dimensions are in centimetres, volume is in cubic centimetres (cm³). Cubic metres (m³) for metric. Cubic inches or feet (in³, ft³) in US customary units.

Worked example. Find the volume of a rectangular prism with l=8 cm, w=5 cm, h=3 cm.

V=8×5×3=120 cm³

Surface Area Formula

The total surface area of a rectangular prism is the sum of the areas of all 6 faces:

S=2(lw+lh+wh)

Worked example. Find the surface area of a rectangular prism with l=8 cm, w=5 cm, h=3 cm.

S=2(8⋅5+8⋅3+5⋅3)=2(40+24+15)=2(79)=158 cm²

Lateral Surface Area

The lateral surface area is the area of the sides only — excluding the top and bottom faces:

LSA=2h(l+w)

Length of the Diagonal

The space diagonal of a rectangular prism:

d=l²+w²+h²

This is the 3D Pythagorean theorem. For a cube with side s:

d=s√3.

Three Worked Examples — Quick, Standard, Stretch

Quick — Volume

A box has dimensions 10×6×4 cm. Find its volume.

V=10×6×4=240 cm³.

Standard — Surface Area

Find the surface area of a brick with l=20 cm, w=10 cm, h=7 cm.

S=2(20⋅10+20⋅7+10⋅7)=2(200+140+70)=2(410)=820 cm²

Stretch — Diagonal

A shipping container measures 12×8×5 metres.

d=√(12²+8²+5²)≈15.26 m

Properties of a Rectangular Prism

• 6 faces, all rectangles.
• 12 edges.
• 8 vertices.
• 3 pairs of parallel faces.
• All face diagonals lie on a face; the space diagonal runs through the interior.
• A cube is a special case where l=w=h.

Why Does the Rectangular Prism Matter?

Rectangular prisms appear in nearly every applied geometry context:

• Construction and architecture.
• Shipping and packaging.
• Water tanks and pools.
• Manufacturing.
• Computer graphics.

Key Takeaways

• A rectangular prism (cuboid) has 6 rectangular faces, 12 edges, 8 vertices.
• Volume: V=l×w×h.
• Surface area: S=2(lw+lh+wh).
• Space diagonal: d=√(l²+w²+h²).
• A cube is the special case where all three dimensions are equal.