# Square Pyramid - Definition, Formula, and Examples

A square pyramid is a 3D solid with a square base and four triangular faces meeting at an apex, giving 5 faces, 8 edges, and 5 vertices. Its volume is V=\(\frac{1}{3}a^2h\) and its surface area is \(a^2 + 2a\ell\). This article covers the definition, properties, net, formulas, and worked examples.

## The Shape That Has Held Its Corners For 4,500 Years

The Great Pyramid of Giza still stands because a square base spreads a huge load evenly onto the ground while four sloping faces carry that load up to a single point. Long before anyone wrote a volume formula, builders trusted that a square-based pyramid holds its shape better than almost any other solid.

## What Is A Square Pyramid?

A **square pyramid** is a three-dimensional solid with a **square base** and **four triangular faces** that rise from the base edges and meet at a single point called the **apex**.

The **apex** is the top point where all four triangular faces meet. The **base** is the square at the bottom. When the apex sits directly above the centre of the base, the solid is a **right square pyramid**; when it leans to one side, it is an **oblique square pyramid**.

## What Are the Faces, Edges, and Vertices of a Square Pyramid?

A square pyramid has:

- **Faces: 5** 
- **Edges: 8** 
- **Vertices: 5**

These satisfy **Euler's formula** for any convex polyhedron, V−E+F=2.

## What Is the Net of a Square Pyramid?

The **net** is the flat pattern you get by unfolding the solid. For a square pyramid, the net is a central **square** (the base) with a **triangle** attached to each of its four sides.

## What Are the Square Pyramid Formulas?

Every formula comes from the "square base plus four triangles" picture:

| Quantity | Formula | Where it comes from |
| -------- | ------- | ------------------- |
| Volume | \(V=\frac{1}{3}a^2h\) | A third of the prism \(a^2h\) on the same base |
| Slant height | \(\ell=\sqrt{h^2 + (\frac{a}{2})^2}\) | Right triangle: height and half a base side |
| Lateral surface area | \(L=2a\ell\) | Four triangles, each  \(\frac{1}{2}a\ell\) |
| Total surface area | \(A=a^2 + 2a\ell\) | Base \(a^2\) plus the four triangles |

## Examples of Square Pyramid

### Example 1

**A square pyramid has base side a = 6 cm and height h = 4 cm. Find its volume.**

Using \(V=\frac{1}{3}a^2h\) :

\(V=\frac{1}{3} \times 6^2 \times 4\)  
Final answer: the volume is **48 cm³**.

### Example 2

**A square pyramid has base side a = 6 cm and height h = 4 cm. Find the correct surface area.**

First find the slant height:
\(\ell=\sqrt{h^2 + (\frac{a}{2})^2} = 5 cm\)

Then total surface area:  
\(A = a^2 + 2a\ell = 96 cm^2\)
Final answer: the surface area is **96 cm²**.

### Example 3

**Find the slant height of a square pyramid with base side a = 10 cm and height h = 12 cm.**

Final answer: the slant height is **13 cm**.

### Example 4

**A square pyramid has base side a = 8 cm and slant height ℓ = 5 cm. Find its lateral surface area.**

Final answer: the lateral surface area is **80 cm²**.

### Example 5

**A square pyramid has volume 100 cm³ and base side a = 5 cm. Find its height.**

Final answer: the height is **12 cm**.

### Example 6

**A square pyramid has base side a = 6 cm and height h = 4 cm. Find its total surface area.**

Final answer: total surface area **96 cm²**;

## Conclusion

- A **square pyramid** has a square base, four triangular faces, and one apex.

- Volume is \(V=\frac{1}{3}a^2h\); total surface area is \(A=a^2 + 2a\ell\).

- The triangular faces use the slant height \(\ell\), not the vertical height \(h\).