Square Pyramid Formula

In geometry, a pyramid is a three-dimensional shape with a polygonal base and three or more triangular faces that meet at a common point above the base, known as the apex or vertex. A pyramid is a polyhedron that is classified according to the shape of its polygonal base, such as

• Triangular pyramid
• Rectangular pyramid
• Square pyramid
• Pentagonal pyramid
• Hexagonal pyramid

The apex is the meeting point of a pyramid’s lateral surfaces or side faces. The perpendicular distance from the center of the base to the apex is called the height of a pyramid, while the slant height of a pyramid is defined as the perpendicular distance between the apex and the base of a lateral surface.

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 upon 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.

• Equilateral square pyramid: An equilateral square pyramid is a square pyramid where all the triangular faces of a square pyramid have equal edges.
• Right square pyramid: A right square pyramid is a square pyramid with the apex just above the center of the base, such that a straight line from the apex cuts the base perpendicularly. 
• Oblique square pyramid: An oblique square pyramid is a square pyramid where the apex is not aligned right above the center of the base.

Regular 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.

Lateral surface area of the regular square pyramid (LSA)= 2al square units

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

Where “a” is the base edge, and “l” is the slant height.

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

Now, 

Lateral surface area of the square pyramid (LSA)= 2a√(a²/4 + h²) square units

Total surface area of a regular square pyramid (TSA) = a² + 2a√(a²/4 + h²) square units

Volume of Regular Square Pyramid

The formula to determine the volume of a pyramid is given as,

The volume of a pyramid = 1/3×Ah cubic units

Where h is the height of the pyramid and A is the area of the base

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

Base area = a²

Now, 

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

Where “a” is the base edge, and “h” is the height of the pyramid.

Practise Problems based on Regular Square Pyramid Formula

Problem 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,

The side of the square base (a) = 20 inches, and

Slant height, l = 25 inches

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

The 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.

Problem 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,

Length of the side of the base (a) = 8 cm

The lateral surface area of a regular square pyramid = 192 sq. cm

Slant height (l) = ?

We know that,

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

The 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.

Problem 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,

Length of the side of the base (a)= 10 cm

Height of the pyramid (h) = 15 cm.

The 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.

Problem 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,

The side of the square base (a) = 7 cm

Slant height, l = 12 cm

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

The 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.

Problem 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,

The side of the square base (a) = 12 cm

Volume = 720 cu. in

Height (H) =?

We know that,

The 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.

Problem 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,

Length of the side of the base of a square pyramid = 8 inches

Height of the pyramid = 14 inches.

The 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.

Problem 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,

The side of the square base (a) = 15 units, and

Slant height, l = 22 units.

We know that,

The 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.

Frequently Asked Questions on Square Pyramid

Question 1: What is a square pyramid?

Answer:

Square pyramid is a 3-D figure with a square base and four triangular faces joined at a vertex.

Question 2: Mention the few properties of a square pyramid?

Answer:

Few properties of square pyramid are:

• Square Pyramid has the base of a square.
• It has five vertices and four triangular faces.
• Square Pyramid has 8 edges.

Question 3: What is the formula for the volume of a square pyramid?

Answer:

Formula for the volume of a square pyramid is 

Volume = (⅓)×(Base area of a square)×(Height of the square pyramid) cubic units.