Square Pyramid Formula (original) (raw)

Last Updated : 23 Jul, 2025

A square pyramid is a three-dimensional geometric shape with a square base and four triangular faces that meet at a single point called the apex. Several formulas are associated with a square pyramid, including those for the surface area, volume, and slant height.

In this article, we will learn about the square pyramid definition, related formulas, examples and others in detail.

Table of Content

What is a Square Pyramid?

A square pyramid is also known as a pentahedron as it has five faces, including a square base and four triangular faces that meet at a point at the top called the apex. The Great Pyramid of Giza is the best example of a square pyramid. A square pyramid is a pyramid that has a square base and four triangular faces (lateral faces). It has five (5) faces four (4) triangular faces, a square base, five (5) vertices, and eight (8) edges.
Square pyramids can be distinguished depending on the lengths of their edges, the position of the apex, and so on. The different types of square pyramids are equilateral square pyramids, right square pyramids, and oblique square pyramids.

Square Pyramid

Square Pyramid

Square Pyramid Formulae

There are two types of surface areas, namely, the lateral surface area and the total surface area. A pyramid's total surface area is calculated by adding the areas of its base, side faces, and lateral surfaces, while the lateral surface area of a pyramid is calculated by adding the sum of its lateral surfaces, or side faces.

Regular Square Pyramid

Regular Square Pyramid

Surface Area of Square Pyramid

Surface Area of Square Pyramid is calculated using the formula:

**Lateral Surface Area of Regular Square Pyramid (LSA)= 2al square units

**Total Surface Area of Regular Square Pyramid(TSA) = 2al + a 2 square units

where,

Slant height of the pyramid (l) = √[(a/2)2 + h2]

Now,

**Lateral Surface Area of Square Pyramid (LSA)= 2a√(a 2 /4 + h 2 ) square units

**Total Surface Area of Square Pyramid (TSA) = a 2 + 2a√(a 2 /4 + h 2 ) square units

**Volume of Regular Square Pyramid

Formula to determine the volume of a pyramid is given as,

**Volume of a Pyramid = 1/3×Ah cubic units

where,

Here, as the base of the pyramid is a square,

Base Area = a2

Now,

**Volume of Regular Square Pyramid (V)= (1/3)a 2 h cubic units

where,

**Also Check:

Examples on Square Pyramid Formula

**Example 1: Calculate a square pyramid's total surface area if the base's side length is 20 inches and the pyramid's slant height is 25 inches.

**Solution:

Given,

Perimeter of the square base (P) = 4a = 4(20) = 80 inches

Lateral surface area of a regular square pyramid = (½) Pl

LSA = (½ ) × (80) × 25 = 1000‬ sq. in

Now, the total surface area = Area of the base + LSA

= a2 + LSA

= (20)2 + 1000‬

= 400 + 1000 = 1400 sq. in

Hence, the total surface area of the given pyramid is 1400 sq. in.

**Example 2: Calculate the slant height of the regular square pyramid if its lateral surface area is 192 sq. cm and the side length of the base is 8 cm.

**Solution:

Given data,

Slant height (l) = ?

We know that,

Lateral surface area of a regular square pyramid = (½) Pl

Perimeter of the square base (P) = 4a = 4(8) = 32 cm

⇒ 192 = ½ × 32 × l

⇒ l = 12 cm

Hence, the slant height of the square pyramid is 12 cm.

**Example 3: What is the volume of a regular square pyramid if the sides of a base are 10 cm each and the height of the pyramid is 15 cm?

**Solution:

Given data,

Volume of a regular square pyramid (V) = 1/3 × Area of square base × Height

Area of square base = a2 = (10)2 = 100 sq. cm

V = 1/3 × (100) ×15 = 500 cu. cm

Hence, the volume of the given square pyramid is 500 cu. cm.

**Example 4: Calculate the lateral surface area of a regular square pyramid if the side length of the base is 7 cm and the pyramid's slant height is 12 cm.

**Solution:

Given,

Perimeter of the square base (P) = 4a = 4(7) = 28 cm

Lateral surface area of a regular square pyramid = (½) Pl

LSA = (½ ) × (28) × 12 = 168‬ sq. cm

Hence, the lateral surface area of the given pyramid is 168 sq. cm.

**Example 5: Calculate the height of the regular square pyramid if its volume is 720 cu. in. and the side length of the base is 12 inches.

**Solution:

Given,

Height (H) =?

We know that,

Volume of a regular square pyramid (V) = 1/3 × Area of square base × Height

Area of square base = a2 = (12)2 = 144 sq. in

⇒ 720 = 1/3 × 144 × H

⇒ 48H = 720

⇒ H = 720/48 = 15 inches

Hence, the height of the square pyramid is 15 inches.

**Example 6: Calculate the volume of a regular square pyramid if the base's side length is 8 inches and the pyramid's height is 14 inches.

**Solution:

Given data,

Volume of a regular square pyramid (V) = (1/3)a2h cubic units

V = (1/3) × (8)2 ×14

= (1/3) × 64 × 14

= 298.67 cu. in

Hence, the volume of the given square pyramid is 298.67 cu. in.

**Example 7: Find the surface area of a regular square pyramid if the base's side length is 15 units and the pyramid's slant height is 22 units.

**Solution:

Given,

We know that,

Total surface area of a regular square pyramid (TSA) = 2al + a2 square units

= 2 × 15 × 22 + (15)2

= 660 + 225= 885 sq. units

Hence, the total surface area of the given pyramid is 885 sq. units.