Regular Right Pyramid with Square Base

In geometry, a square pyramid is a shape with a square base and lateral faces, which are four in number. It is a 3-D structure. It has five faces; therefore, it is called a pentahedron. The Great Pyramid of Giza is a famous example of a square pyramid. The triangular Pyramid formula is used to deduce other essential formulas. There are four triangular sides of a square pyramid, which meet at a point known as the apex. Besides the regular right pyramid with a square base, there are also other types of square pyramids. 

Square Pyramid 

A three-dimensional structure with a square-shaped base and triangular sides, which are four in number joined at an apex, is known as a square pyramid. It has three elements:

1. Apex – It is the top point of a square pyramid. 

2. Base – The square at the button is called the base.  

3. Faces – The triangular-shaped sides are the faces.   

Varieties of Square Pyramid 

There are three types of square pyramids. They are

• Right Square Pyramid 

• Oblique Square Pyramid 

• Equilateral Square Pyramid

When the apex is such that a perpendicular is formed with respect to the base, it is called a right square pyramid. In an oblique pyramid, the apex is not in alignment with the central point of the square base. When the four triangular faces have the same edges, the square pyramid is an equilateral square pyramid. 

Properties of a regular triangular pyramid with a square base

The properties of a regular triangular pyramid with a square base are: 

• Five faces

• Square base 

• Four triangular sides 

• Eight edges 

• Five vertices

Right Square Pyramid

When the apex is such that a perpendicular is formed with respect to the base, then it is called a right square pyramid. The lateral edges in a square right pyramid have the same length, and the triangular faces are congruent isosceles triangles. 

Some triangular pyramid formulas are: 

1. Volume= ⅓ l² X h  

2. Slant height= √ h² + l²/2

3. Area= l² + l √ l² + (2h)²

4. Lateral edge length= √ h² + l²/2 

Square Pyramid Formulas 

There are some triangular pyramid formulas with a square base. These formulas are volume, area of the base, surface area, and height.  

• Area of Base 

Since the base of a regular triangular pyramid with a square base is a square, the formula of its base is the same as the area of a square. 

Area of base = (edge)²

• Surface Area  

The surface area of a square pyramid is of two types: the total surface area and the lateral surface area. 

The sum of the areas of all the surfaces of a square pyramid makes up the total surface area. 

Total Surface Area = a² + 2a√[(a²/4) + h²]

Where h= height 

a= base edge.

There is another formula for the total surface area when the slant height is given.

TSA= a² + 2al 

Here l= slant height and a= base edge

Let us take an example to understand the total surface area more clearly.

Suppose the slant height of a square pyramid is given as 13 units and its base area is 12 units, then the total surface area can be calculated as given below. 

TSA= a² + 2al = (12)²+ 2(12)(13) = 144+ 312= 456 units²

The sum of only the side faces of a square pyramid makes up the lateral surface area. The formula of the lateral surface area is:

LSA= 2a√[(a²/4)+ h²]  

Here a= base edge 

h= height 

Another formula of LSA is:

LSA=  2al. Here l= slant height

• Volume 

In a square pyramid, the volume is the space occupied between its five faces. It is one-third of the base area and height of the square pyramid. 

Let a be the length of the edge and h be the height. Then the formula of volume is: 

V= ⅓ a²h

Volume is measured in cubic units. 

Net of a Square Pyramid 

A flat view of every face and the square base of a regular triangular pyramid with a square base is called its net. It is laid out horizontally to define each face of a square pyramid.  

When one folds the net back to its original shape, a square pyramid is obtained in its original 3-D form. The net helps in determining the surface area of any 3-D structure with ease. To make sure that a net correctly forms the 3-D structure, one must ensure that the number of faces in the solid and the net is equal, and the shapes of the net should match the shapes of the solid. 

Conclusion

A square pyramid is a shape with a square base and four lateral faces. The properties of a regular triangular pyramid with a square base are 5 faces, a square base, four sides that are triangular in shape, 8 edges, and 5 vertices. The triangular pyramid formula for volume is one-third of the base area and height of the square pyramid. A flat view of every face and the square base of a regular triangular pyramid with a square base is called its net.