Square Pyramid

A square pyramid is a 3D shape in geometry.  It has a square base. It also has four triangular faces. All of them meet at a single point called the apex. Square pyramids have many real-life uses. In this article, we explore many areas of square pyramids.

1.0What is a Square Pyramid?

A Square Pyramid 3D Shape has one square base and four triangular sides. These triangles are called lateral faces. All four triangles join at a single point above the base.

Parts of a Square Pyramid

Here are the main parts of a square pyramid:

• Base: A square at the bottom.
• Lateral faces: Four triangles.
• Apex: The top point where all triangles meet.
• Height (h): The perpendicular distance from the base to the apex.
• Slant height (l): The distance from the apex to the centre of any side of the base

2.0Square Pyramid Faces, Edges, Vertices

Understanding the structure of a square pyramid is easy. The table below shows the number of square pyramid faces, edges, and vertices:

3.0Properties of Square Pyramid

The properties of square pyramid include:

1. It is a polyhedron.
2. The base is always a square.
3. It has 5 faces: 1 square and 4 triangles.
4. It has 5 vertices.
5. It has 8 edges.
6. The triangles may be isosceles if the pyramid is regular.
7. All lateral faces meet at one point (apex).
8. The shape is symmetrical if all faces are equal in size.

4.0Types of Square Pyramid

There are two main types of square pyramid:

Regular Square Pyramid

• All four triangles are congruent.
• The apex lies directly above the centre of the square base.
• It has a symmetrical shape and equal slant heights.

Irregular Square Pyramid

• The triangular faces are not all the same.
• The apex is not directly above the centre.
• It lacks symmetry.

5.0Square Pyramid Formula

To work with a square pyramid in geometry or real life, we use the following square pyramid formula set:

Square Pyramid Formula Volume

The square pyramid formula volume is calculated as: 

$V=\frac{1}{3}{a}^{2}h$

Where:

• a = side of the square base
• h = height (vertical height from base to apex)

Square Pyramid Formula Surface Area

The square pyramid formula for surface area includes the base area and the lateral area.

$\text{ The Surface area of pyramid }=a^2+2al$

Where:

• a = side of the square base
• l = slant height

6.0Square Pyramid Net Diagram

A square pyramid net diagram is a 2D layout that can be folded into the 3D shape. It helps visualise and construct a pyramid. The net of a square pyramid includes:

• One square (the base)
• Four triangles (the lateral faces)

This diagram is often used in school projects and geometry worksheets.

7.0Real-Life Examples of Square Pyramid

There are many examples of square pyramid in the real world. Some common ones include:

• The Great Pyramid of Giza: A famous ancient structure in Egypt.
• Tents: Many have a square base and pointed top.
• Roof tops: Some houses have pyramid-style roofs.
• Trophies: Some awards are in a pyramid shape.
• Paperweights: Decorative items shaped like pyramids.

8.0Solved Example Problems

Example 1: Volume

Problem: Find the volume of a square pyramid with a base side of 6 cm and height of 9 cm.

$V=\frac{1}{3}{a}^{2}h=\frac{1}{3}\times 6^2\times 9=108{\mathrm{cm}}^3$

Example 2: Surface Area

Find the surface area of a square pyramid with base side 5 cm and slant height 8 cm.

Solution: It is given that in a given pyramid, the base side is 5 cm and the slant height is 8 cm. Hence using the formula for Surface area: 

$\begin{array}{rl}& \text{ The Surface area of pyramid }=a^2+2al\\ & \text{ The Surface area of pyramid }=5^2+2\times 5\times 8\\ & \text{ The Surface area of pyramid }=105{\mathrm{cm}}^3\end{array}$

Example 3: Surface Area with Found Slant Height

Find the surface area of the square pyramid with base side 10 cm and slant height 13 cm.

Solution: 

$\begin{array}{rl}& \text{ The Surface area of pyramid }=a^2+2al\\ & \text{ The Surface area of pyramid }=10^2+2\times 10\times 13\\ & \text{ The Surface area of pyramid }=360{\mathrm{cm}}^2\end{array}$