# Square Prism: Volume, Surface Area Formulas, and Examples

## What is a Square Prism?
A **square prism** is a three-dimensional solid with **two equal, parallel square bases** joined by **four rectangular faces** (the lateral faces). Because both ends are squares of side `a` and the prism rises to a height `h`, it keeps the _same square cross-section_ all the way up — that constant cross-section is what makes it a prism rather than a tapering pyramid.

A square prism has **6 faces** (2 squares and 4 rectangles), **12 edges**, and **8 vertices**. When the lateral faces stand perpendicular to the base, it is a **right square prism** — the standard, upright case these formulas describe. If it leans, it is an **oblique** square prism.

A square prism is a close relative of the [rectangular prism](/content/math/geometry/rectangular-prism/index.html) (a cuboid): a rectangular prism has rectangular bases, while a square prism's bases are the special rectangles that are squares. Swap the square base for a triangle and you get a triangular prism; the whole family is covered in the guide to [prisms](/content/math/geometry/prism/index.html). The square base itself is the flat shape you meet in the [square](/content/math/geometry/square/index.html) article, and the whole solid sits among the other 3D geometry shapes you study in solid geometry.

**A cube is a special square prism.** When the height equals the base side (h = a), all six faces become equal squares — that is a [cube](/content/math/formula/cube/index.html). So every cube is a square prism, but most square prisms (the tall, box-like ones) are not cubes.

## Volume of a Square Prism
The **volume** of a square prism is:

V = a²h
  
**Where this comes from:** the volume of any prism is the **area of its base times its height** — stack identical copies of the base, and the total space is one base multiplied by how tall the stack is. The base here is a square of side `a`, so its area is a². Multiply by the height `h`:

V = (base area) × (height) = a² × h = a²h

When h = a (a cube), this becomes a² × a = a³ — the familiar cube-volume formula falls straight out.

**Variable glossary:** V is the volume, a is the side of the square base, h is the height (the distance between the two square bases). Volume comes out in cubic units (cm³, m³).

## Surface Area of a Square Prism
A square prism has two kinds of face: the two square ends and the four rectangular sides.

**Lateral surface area (LSA)** — the four rectangles only:

LSA = 4ah

**Where this comes from:** each of the four side faces is a rectangle of width `a` (a base edge) and height `h`, so each has area a × h. There are four of them:

LSA = 4 × ah = 4ah

**Total surface area (TSA)** — the four sides plus both square bases:

TSA = 4ah + 2a² = 2a² + 4ah

**Where this comes from:** add the two square ends, each of area a², contributing 2a². Add that to the lateral surface 4ah.

The clearest way to see all six faces is the prism's **net**: unfold it flat and you get a row of four rectangles (the sides) with a square attached at each end.

| Quantity | Formula | Cube (h = a) | Units |
| --- | --- | --- | --- |
| Volume | V = a² h | V = a³ | cubic |
| Lateral surface area | LSA = 4 a h | LSA = 4 a² | square |
| Total surface area | TSA = 2 a² + 4 a h | TSA = 6 a² | square |

**Variable glossary:** V is the volume, LSA is the lateral surface area (four sides only), TSA is the total surface area (sides plus both bases), a is the square base side, h is the height. Surface area comes out in square units (cm², m²).

## Examples of the Square Prism
For consistency, every example below uses centimeters throughout.

### Example 1
**Find the volume of a square prism with base side 4 cm and height 9 cm.**

V = a²h

V = 4² × 9

V = 16 × 9

**Final answer:** V = 144 cm³

### Example 2
**A square prism has base side 5 cm and height 8 cm. A student finds the total surface area using 6a². Find the correct total surface area.**

TSA = 2a² + 4ah = 2 × 5² + 4 × 5 × 8

TSA = 2 × 25 + 160 = 50 + 160

**Final answer:** TSA = 210 cm²

### Example 3
**Find the lateral surface area of a square prism with base side 6 cm and height 10 cm.**

LSA = 4ah

LSA = 4 × 6 × 10

**Final answer:** LSA = 240 cm²

### Example 4
**Find the total surface area of a square prism with base side 3 cm and height 7 cm.**

TSA = 2a² + 4ah = 2 × 3² + 4 × 3 × 7

TSA = 18 + 84

**Final answer:** TSA = 102 cm²

### Example 5
**A square prism has volume 245 cm³ and base side 7 cm. Find its height.**

245 = 7² × h

245 = 49h

Divide both sides by 49.

**Final answer:** h = 5 cm

### Example 6
**A square prism has base side 4 cm and height 4 cm. Find its volume and total surface area, and confirm it is a cube.**

Volume:

V = 4² × 4 = 64 cm³

Total surface area:

TSA = 2a² + 4ah = 2 × 16 + 4 × 4 × 4 = 32 + 64 = 96 cm²

**Final answer:** V = 64 cm³, TSA = 96 cm² — and because h = a, every face is a 4 × 4 square, so the solid is a cube.

## Why the Square Box Runs the Warehouse
The square prism is the workhorse of packing, storage, and stacking.

A square base tiles a floor with no gaps, and flat faces sit flush against each other, so square prisms stack into a solid wall with no wasted space — which is exactly why shipping cartons, pillars, and storage columns are built this way rather than rounded.

## Conclusion
- A **square prism** has two equal square bases joined by four rectangular faces, keeping a constant square cross-section all the way up.
- **Volume** is a²h — the square base area a² times the height, the standard "base area times height" prism rule.
- **Lateral surface area** is 4ah (the four rectangular sides); **total surface area** is 2a² + 4ah, adding both square ends.
- A **cube** is the special case h = a: the formulas collapse to a³ and 6a².
- The most common errors are reusing the cube formula on a non-cube and squaring the height instead of the base side.