Surface Area of a Square Pyramid

In this section, we will learn about the surface area of a square pyramid. A pyramid is a 3-D object whose all side faces are congruent triangles and whereas its base can be any polygon. One side of each of these triangles coincides with one side of the base polygon. A square pyramid is a pyramid whose base is a square. The pyramids are named according to the shape of their bases. Just like other three-dimensional shapes, a square pyramid also has two types of areas.

• Total Surface Area (TSA)
• Lateral Surface Area (LSA)

Let us learn about the surface area of a square pyramid along with the formula and a few solved examples here. You can find a few practice questions in the end.

The word "surface" means " the exterior or outside part of an object or body". So, the total surface area of a square pyramid is the sum of the areas of its lateral faces and its base. We know that a square pyramid has:

• a base which is a square.
• 4 side faces, each of which is a triangle.

All these triangles are isosceles and congruent, each of which has a side that coincides with a side of the base (square).

So, the surface area of a square pyramid is the sum of the areas of four of its triangular side faces and the base area which is square.

Let us consider a square pyramid whose base's length (square's side length) is 'a' and the height of each side face (triangle) is 'l' (this is also known as the slant height). i.e., the base and height of each of the 4 triangular faces are 'a' and 'l' respectively. So the base area of the pyramid which is a square is a × a = a² and the area of each such triangular face is 1/2 × a × l. So the sum of areas of all 4 triangular faces is 4 ( ½ al) = 2 al. Let us now understand the formulas to calculate the lateral and total surface area of a square pyramid using height and slant height.

Total Surface Area of Square Pyramid Using Slant Height

The total surface area of a square pyramid is the total area covered by the four triangular faces and a square base. The total surface area of a square pyramid using slant height can be given by the formula,
Surface area of a square pyramid = a² + 2al
where,

• a = base length of square pyramid
• l = slant height or height of each side face

Total Surface Area of a Square Pyramid Using Height

Let us assume that the height of the pyramid (altitude) be 'h'. Then by applying Pythagoras theorem (you can refer to the below figure),

\(l = \sqrt{\dfrac{a^{2}}{4}+h^{2}}\)

Substituting this in the above formula,

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

Note: \(\sqrt{\dfrac{a^{2}}{4}+h^{2}}\) can be simplified as \(\dfrac 1 2 \sqrt{a^2+4h^2}\). Thus, the formula of surface area of a square pyramid can be written as a²+ 2a \(\left(\dfrac 1 2 \sqrt{a^2+4h^2}\right)\) = a²+ a\( \sqrt{a^2+4h^2}\).

Lateral Surface Area of a Square Pyramid

The lateral surface area of a square pyramid is the area covered by the four triangular faces. The lateral surface area of a square pyramid using slant height can be given by the formula,
Lateral surface area of a square pyramid = 2 al
or,
Lateral surface area of a square pyramid = 2a\(\sqrt{\dfrac{a^{2}}{4}+h^{2}}\)
where,

• a = base length of square pyramid
• l = slant height or height of each side face
• h = height of square pyramid

The surface area of a square pyramid can be calculated by representing the 3D figure into a 2D net. After expanding the 3D figure into a 2D net we will get one square and four triangles.

The following steps are used to calculate the surface area of a square pyramid :

• To find the area of the square base: a², 'a' is the base length.
• To find the area of the four triangular faces: The area of the four triangular side faces can be given as: 2al, 'l' is the slant height. If slant height is not given, we can calculate it using height, 'h' and base length as, \(l = \sqrt{\dfrac{a^{2}}{4}+h^{2}}\)
• Add all the areas together for the total surface area of a square pyramid, while the area of 4 triangular faces gives the lateral area of the square pyramid.
• Thus, the surface area of a square pyramid is a² + 2al and lateral surface area as 2al in squared units.

Now, that we have seen the formula and method to calculate the surface area of a square pyramid, let us have a look at a few solved examples to understand it better.

What Is the Surface Area of the Square Pyramid?

The surface area of a square pyramid is the sum of the areas of all its 4 triangular side faces with the base area of the square pyramid. If a, h, and l are the base length, the height of the pyramid, and slant height respectively, then the surface area of the square pyramid = a²+ 2al (or) a²+2a \(\sqrt{\dfrac{a^{2}}{4}+h^{2}}\).

How Do You Find the Lateral Area of a Square Pyramid?

To find the lateral area of a square pyramid, find the area of one side face (triangle) and multiply it by 4. If a and l are the base length and the slant height of a square pyramid, then the lateral area of the square pyramid = 4 (½ × a × l) = 2al.

If h is the height of the pyramid, then the lateral area = 2a \(\sqrt{\dfrac{a^{2}}{4}+h^{2}}\).

What Is the Area of One of the Triangular Faces of a Square Pyramid?

If a and l are the base length and the slant height of a square pyramid, then the area of one of the 4 triangular side faces is, ½ × a × l.

How Do You Find the Lateral Area and Surface Area of a Square Pyramid?

The lateral area of a square pyramid is the sum of the areas of the side faces only, whereas the surface area is the lateral area + area of the base. The lateral area of a square pyramid = 2al (or) 2a\(\sqrt{\dfrac{a^{2}}{4}+h^{2}}\).

To get the total surface area, we need to add the area of the base (which is a²) to each of these formulas. The total surface area of a square pyramid = a² + 2al (or) a² + 2a\(\sqrt{\dfrac{a^{2}}{4}+h^{2}}\).
where,

• a = Length of the base (square)
• l = Slant height
• h = Height of the pyramid

How To Calculate Surface Area of a Square Pyramid Without Slant Height?

We know, slant height of a square pyramid is given in terms of height and base length by the formula, \(l = \sqrt{\dfrac{a^{2}}{4}+h^{2}}\). We can calculate the slant height from the given height and base length and apply the formula for surface area of square pyramid as,
LSA of pyramid = 2a\(\sqrt{\dfrac{a^{2}}{4}+h^{2}}\)
TSA of pyramid = a² + 2a\(\sqrt{\dfrac{a^{2}}{4}+h^{2}}\)
where,

• a = Length of the base (square)
• l = Slant height
• h = Height of the pyramid

What Is the Base Area of a Square Pyramid?

The base of a square pyramid is square-shaped. Thus, the base area of square pyramid can be calculated using the formula, Base Area of Pyramid = a², where, a is the length of the base of square pyramid.

How Many Bases Does a Square Pyramid Have?

A square pyramid is a pyramid with a square-shaped base. A square pyramid thus has only one base.

Which Two Shapes Make up a Square Pyramid?

The base of a square pyramid is a square and its side faces are triangles. So the two shapes that make up a square pyramid are square and triangle.