Book A Free Math Class

# 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.