Square Pyramid
A pyramid is a polyhedron that has a base and 3 or greater triangular faces that meet at a point above the base (the apex). Interestingly, pyramids are named after their base, such as
 Rectangular pyramid
 Triangular pyramid
 Square pyramid
 Pentagonal pyramid
 Hexagonal pyramid
A square pyramid characterized by a square base is a threedimensional shape having five faces, thus called a pentahedron. The most famous example of such a square pyramid is the Great Pyramid of Giza.
1.  What Is Square Pyramid? 
2.  Properties of a Square Pyramid 
3.  Types of Square Pyramids 
4.  Square Pyramid Formula 
5.  FAQs on Square Pyramid 
What Is Square Pyramid?
A square pyramid is a threedimensional geometric shape that has a square base and four triangular bases that are joined at a vertex. It is a polyhedron (pentahedron) with five faces. A square pyramid consists of a square base and four triangles connected to a vertex. Its base is a square and the side faces are triangles with a common vertex.
A square pyramid has three components.
 The top point of the pyramid is called the apex.
 The bottom square is called the base.
 The triangular sides are called faces.
Properties of a Square Pyramid
Let us list out the properties we have explored in the above image. All these properties are derived from the definition of pyramid.
 It has 5 faces.
 It has 4 side faces that are triangles.
 It has a square base.
 It has 5 vertices.
 It has 8 edges.
Types of Square Pyramids
We can distinguish the square pyramids on the basis of the lengths of their edges, position of the apex, and so on. Let us discuss the different types of square pyramids.
Right square pyramid
If the apex of the square pyramid is right above the center of the base, it forms a perpendicular with the base. Such a square pyramid is called the right square pyramid.
Oblique square pyramid
If the apex of the square pyramid is not aligned right above the center of the base, the pyramid is called an oblique square pyramid.
Equilateral square pyramid
If all the triangular faces of a square pyramid have equal edges, then the square pyramid is called an equilateral square pyramid.
Square Pyramid Formula
Base Area of a Square Pyramid
Since the square pyramid has a square base, we can calculate its base area using the same formula as the area of square, which is side × side or base edge^{2}.
Example: Assume that the base edge of a square pyramid is given as 7 units. Then, the base area of the square pyramid is: BA = 7 × 7 = 49 square units
Volume of a Square Pyramid
The formula to determine the volume of a square pyramid is: V = [(1/3)a^{2}h]. Here, a is the length of the base and h is the perpendicular height.
Example: Assume that the height (h) and the length of the base edge (a) are 9 units and 5 units, respectively. Then, the volume of the square pyramid is:
Volume = 1/3 x 5^{2} x 9 = 1/3 x 25 x 9 = 75 cubic units
Surface Area of a Square Pyramid
There are two types of surface areas, one is TSA (Total Surface Area), and the other is LSA (Lateral Surface Area). When we talk about its surface area, we generally refer to its total surface area (which is the sum of areas of all faces), whereas the lateral surface area is the sum of the areas of the side faces only. Consider a square pyramid of base edge 'a', height 'h', and slant 'l'.
 The formula to calculate the surface area of a square pyramid when its height h and base edge a are given: Surface Area = a^{2} + 2a√[(a^{2}/4) + h^{2}]
 The formula to calculate the surface area of a square pyramid when its slant height l and base edge a are given: Surface Area = a^{2} + 2al
 The formula to calculate the curved surface area or lateral surface area of a square pyramid is given as: 2a√[(a^{2}/4)+ h^{2}] or 2al
Example: Assume that the height h and the length of the base edge a are 9 units and 5 units, respectively.
Then, the surface area of the square pyramid is:
Surface area = (5)^{2} + 2 x 5 √[(5^{2}/4) + 9^{2}]
= 25 + 10 √[(25/4) + 81]
= 25 + 10√(349/4)
= 25 +10 x 9.34
= 25 + 93.4
= 118.4 square units
Challenging Question
A vessel is in the form of an inverted regular square pyramid of height 20 inches and base edge 15 inches. The depth of the liquid inside the vessel is 10 inches. How much will the surface rise when 2.5 pints of the liquid is added?
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Solved Examples on Square Pyramid

Example 1: Identify the square pyramid.
Solution:
Pyramids are named as per their bases. Thus, the names of the shapes are as follows.
 The base of the first pyramid is a triangle. Thus, it is a triangular pyramid.
 The base of the second pyramid is a square. Thus, it is a square pyramid.
 The base of the third pyramid is a pentagon. Thus, it is a pentagonal pyramid.
 The base of the fourth pyramid is a hexagon. Thus, it is a hexagonal pyramid.
Answer: The second one is a square pyramid.

Example 2: Daniel is constructing a closed square pyramidshaped aquarium in his backyard. The base edge of the square pyramid is 10 inches and the slant height is 15 inches. Help Daniel determine the total surface area of the aquarium.
Solution:
Given, the slant height l of the square pyramid is 15 inches, the base edge a of the square pyramid is 10 inches.
Thus, The total surface area of the aquarium is given by: a^{2 }+ 2al
= 10^{2 }+ 2 × 10 × 15
= 100 + 300 = 400 in^{2}
Answer: Surface Area = 400 square inches

Example 3: Sophia has a vessel in the form of an inverted regular square pyramid that has to be filled with water. The altitude of the vessel is 10 inches and the base edge is 7 inches. What is the volume of water Sophia can fill in the vessel?
Solution:
Given, the height h of the vessel is 10 inches and the base edge a of the vessel is 7 inches.
Thus, the volume of the vessel is given by: [(1/3)a^{2}h]
= [(1/3) × (7)^{2 }× (10)]
= 490/3 = 163.33 inch^{3}
Answer: The volume of vessel =163.33 inch^{3}
Practice Questions on Square Pyramid
FAQs on Square Pyramid on Square Pyramid
What Is a Square Pyramid in Geometry?
A square pyramid refers to a pyramid having a square base and four triangular bases connected to a vertex. Thus, it is a polyhedron with five faces.
What Is the Formula for Surface Area of a Square Pyramid?
The formula to calculate the surface area of a square pyramid is, surface area = a^{2} + 2a√[(a^{2}/4) + h^{2}], where h is the height h and a is the base edge of the square pyramid. The surface area of the square pyramid can also be calculated as, surface area = a^{2} + 2al, where l is the slant height and a is the base edge of the square pyramid.
What Is the Volume of a Square Pyramid?
The volume of a square pyramid can be determined using the formula: V = [(1/3)a^{2}h]. Here, a is the length of the base and h is the perpendicular height.
How Many Faces Does a Square Pyramid Have?
As a square pyramid is a threedimensional geometric shape with a square base and four triangular bases that are joined at a vertex. Thus, we can clearly conclude that it has five faces and is therefore called a pentahedron
Is a Square Pyramid a Prism?
A prism is a polyhedron that has congruent top and bottom faces. A pyramid has only one base that is connected to a single vertex through line segments from each vertex of the base. Therefore, a square pyramid is not a prism.
What Is a Square Pyramid Called?
A square pyramid refers to a pyramid having a square base. It has a total of 5 faces, one square base, and four triangular faces. Thus, it is called a pentahedron because it has five faces.
Is a Rectangular Pyramid the Same as a Square Pyramid?
A rectangular pyramid has a rectangular base whereas a square pyramid has a square base. All square pyramids are rectangular pyramids but all rectangular pyramids are not square pyramids. A rectangular pyramid is the same as a square pyramid only when the length and breadth of the rectangular base are equal.