Surface Area of a Square Pyramid: Definition, Formula & Solved Examples

Square Pyramid is a three-dimensional form with five faces, a square base, and four triangular bases that are linked at a vertex, and is defined by a square base. It has a square base and triangular side faces with a shared vertex. The Great Pyramid of Giza is the most recognized example of a square pyramid. The total surface area and lateral surface area of a square pyramid are measured in centimetres and meters in square units. A square pyramid's surface area is equal to the sum of the areas of its four triangular side faces plus the square base area. There are two different kinds of areas in a square pyramid- Lateral Surface Area (LSA) and Total Surface Area (TSA).

Key Terms: Square Pyramid, Surface Area, Slant Height, Total Surface Area, Lateral Surface Area, Base, Square, Pentahedron, Triangular Face, Area

---

What is a Square Pyramid?

[Click Here for Sample Questions]

A pyramid with a square base is what we call a square pyramid. It's a pentahedron, to be precise. The entire area occupied by the surface of a square pyramid is measured by its surface area. To put it another way, it's the sum of the areas of four of its triangular side faces plus the square base area. The two different types of surface areas are lateral surface area (LSA) and the total surface area (TSA) of a square pyramid. We know that a square pyramid has the following characteristics:

• A square base is required.
• Each of the four sides faces is a triangle.
• Each of these triangles is isosceles and congruent, having a side that coincides with a side of the base (square).

Square Pyramid

Also Read: Edges, Faces, and Vertices

---

Surface Area Formula for a Square Pyramid

[Click Here for Sample Questions]

Consider, a square pyramid with a base length of 'a' and a height of '\(l\)' on each side face (this is also known as the slant height). Where 'a' and '\(l\)' stands for the base and height of each of the four triangle faces, respectively.

As a result, the base area of the square pyramid is a x a  = a² , and the area of each triangle face is \({1 \over 2} \times a \times l\). As a result, the total area of all four triangular sides is

 4(\({1 \over 2} \times a \times l\)) = 2 x a x l

Now, using height and slant height, calculate the lateral and total surface area of a square pyramid. 

---

Total Surface Area of a Square Pyramid Using Slant Height

[Click Here for Sample Questions]

A square pyramid's total surface area is the total area covered by its four triangular sides and square base. The formula to calculate the total surface area of a square pyramid with slant height is

Surface area of a square pyramid = a² + 2al

Where,

• a denotes base length of a square pyramid and,
• l denotes the slant height or the height of each side face.

Also Read: Geometry Formula

---

Total Surface Area of a Square Pyramid Using Height

[Click Here for Sample Questions]

Assume, that the pyramid's height is 'h'. Then, using Pythagoras theorem we get,

 l = \(\sqrt{\frac{a^2}{4}+h^2}\)

By substituting this in the previous formula,

The surface area of a square pyramid = a² + 2al = a² + 2a\(\sqrt{\frac{a^2}{4}+h^2}\)

 Therefore, the surface area of a square pyramid will be, a² + 2a(\(\frac{1}{2}\)\(\sqrt{a^2+4h^2}\))

---

Lateral Surface Area of a Square Pyramid

[Click Here for Sample Questions]

A square pyramid's lateral surface area is the area covered by its four triangular sides. The formula may be used to calculate the lateral surface area of a square pyramid with slant height. The lateral surface area of a square pyramid = 2al or,

The lateral surface area of a square pyramid = 2a\(\sqrt{\frac{a^2}{4}+h^2}\)

Where,

a denotes base length of a square pyramid

l denotes the slant height or the height of each side face and,

h denotes height square pyramid

Read More:  

---

How to Calculate the Surface Area of a Square Pyramid?

[Click Here for Sample Questions]

A square pyramid's surface area is equal to the sum of the areas of its four triangular side faces plus the square base area. For a square pyramid, calculating the surface area is simple:

A square pyramid's surface area may be computed by converting the 3D figure into a 2D net. We will obtain one square and four triangles after transforming the 3D figure into a 2D net.

The surface area of a square pyramid is calculated using the steps below:

• To determine the area of the square base a², 'a’ is the base Length.
• To calculate the area of the four triangle faces: The surface area of the four triangular side faces may be calculated as follows: 2a\(l\) , ‘\(l\) ‘ is the slant height. If the slant height isn't specified, we can figure it up using height, 'h,' and the base length as \(l\) =\(\sqrt{\frac{a^2}{4}+h^2}\)
• The overall surface area of a square pyramid is equal to the sum of all the areas, whereas the lateral area is equal to the area of four triangular faces.
• The surface area of a square pyramid is, a² + 2a\(l\) while the lateral surface area is 2a\(l\) represented as square units.

---

Base Area of a Square Pyramid

[Click Here for Sample Questions]

A square pyramid's base is also a square. As a result, we may get the base area by calculating the square of the edge length.

Base area=side*side or edge²

Base and Apex of Square Pyramid

Also Read: Area of Square

---

Things to Remember

• A pyramid with a square base is what we call a square pyramid. The entire area occupied by the surface of a square pyramid is measured by its surface area. 
• The base area of the square pyramid is a²
• The total area of all four triangular sides is 4(\(\frac{1}{2}\)a × l ) = 2a x l
• The surface area of a square pyramid is a² + 2al, while the lateral surface area is 2a x l
• A square pyramid's surface area using height is a² + 2a(\(\frac{1}{2}\)\(\sqrt{a^2+4h^2}\)).
• A square pyramid's surface area using slant height is a² + 2al.

---