Square Prism: Volume, Surface Area Formulas, and Examples
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 (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. The square base itself is the flat shape you meet in the square 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. 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.