Rectangular Pyramid

Rectangular Pyramid is one of the many pyramid structures in Geometry. A pyramid is a three-dimensional structure that has a polygon as its base and triangular faces covering its sides, meeting at a common point known as the apex of the pyramid. In the case of a rectangular pyramid, the base is a rectangle, which is why it is called a rectangular pyramid, with four triangular faces connecting the sides of the rectangle to the apex.

A Rectangular Pyramid can be either right or oblique, depending on the alignment of the apex and the center of the base. If the apex aligns with the center of the base at a right angle, then it is a right rectangular pyramid; if not, then it is an oblique rectangular pyramid.

This article provides a well-rounded description of the geometric solid known as the Rectangular Pyramid, including its definition, shapes, and types. In addition to that, we will also discuss the formulas for surface area and volume for the Rectangular Pyramid.

What is Rectangular Pyramid?

Rectangular Pyramid, also simply known as a ‘Pyramid,’ consists of five faces, one of which is rectangular, while the other four are triangular. These five faces connect with each other at five vertices and eight edges, forming the shape known as a Rectangular Pyramid. In simpler terms, we can say that a Rectangular Pyramid is a pyramid with a rectangle as its base.

Rectangular Pyramid Definition

Rectangular Pyramid is a 3D geometric solid that has a rectangle as its base and triangles as its sides, with these four triangular faces meeting at the apex.

Note: Special case of the Rectangular Pyramid is a Square Pyramid when all the sides of the base of the pyramid become equal.

Rectangular Pyramid in Real Life

There are various examples of Rectangular Pyramid in Real Life, some of those are:

• Tents
• Perfume Bottles
• Roof of the House
• Pyramids of Giza
• Mayan Temples

Types of Rectangular Pyramid

Based on the alignment of apex and the centre of the base, rectangular pyramids can be classified into two types i.e.,

• Right Rectangular Pyramid
• Oblique Rectangular Pyramid

Let’s discuss these types in detail.

Right Rectangular Pyramid

Any rectangular pyramid is a right rectangular pyramid if and only if its apex is aligned with the center of the base, i.e., when viewed from the top of the pyramid, its apex appears to be at the center of the base. This type of rectangular pyramid is the most stable and has been used in many architectural structures throughout history, such as the Great Pyramid of Giza.

Oblique Rectangular Pyramid

On the other hand, when the apex doesn’t align with the center of the base, i.e., when viewed from the top of the pyramid, the center doesn’t appear to be in the center of the base and may possibly be outside the base. Since these pyramids are not very stable in nature, we can’t use them to create structures in real life.

Properties of Rectangular Pyramid

There are various properties of Rectangular Pyramid, 

• Base
• Edge
• Face
• Vertices
• Net

Let’s discuss these properties in detail.

Base of a Rectangular Pyramid

The base of a rectangular pyramid is a rectangle with two pairs of parallel sides. It is often called the “base rectangle.”

Rectangular Pyramid Net

A rectangular pyramid can be unfolded or “netted” to create a flat, two-dimensional representation that can be folded back into its three-dimensional shape. The net of a rectangular pyramid consists of the base rectangle and four triangles that represent its faces which is given in the following illustration:

Rectangular Pyramid: Faces, Edges & Vertices

Faces: A rectangular pyramid has a total of five faces:

• One rectangular base, and
• Four triangular faces.

Edges: A rectangular pyramid has eight edges.

Vertices: A rectangular pyramid has five vertices, where three edges meet at each vertex and four edges meet at the apex.

Read more about Faces, Edges & Vertices.

Slant Height of Rectangular Pyramid

In a rectangular pyramid, the slant height refers to the length of the line segment that connects the apex (the top vertex) of the pyramid to a point on the edge of the base. The formula for slant height of the rectangular pyramid is given by

l = √(l² + w² + h²)

Where,

• L is the slant height,
• h is the height of the pyramid,
• l is the length of rectangular base, and
• w is width of rectangular base.

Surface Area of Rectangular Pyramid

As we know, calculating the surface area of such a pyramid involves determining the total area of all its faces, including both the base and the triangular faces. As there is no curved surface involved here, the surface area of the rectangular pyramid is also referred to as the total surface area of a rectangular pyramid.

Surface Area Formula for Rectangular Pyramid

The surface area of a rectangular pyramid encompasses the total area of all its faces, including both the base and the triangular faces. The formula for calculating the surface area (SA) is as follows:

Surface Area of Rectangular Pyramid = 

Where,

• l is the length of rectangular base,
• w is width of rectangular base, and
• h is the height of the pyramid.

Volume of Rectangular Pyramid

A rectangular pyramid is not just about its surface area; it also encloses a certain volume of space within its structure. Calculating the volume of a rectangular pyramid helps us quantify this space.

Volume Formula for Rectangular Pyramid

The volume of a rectangular pyramid quantifies the space enclosed by the pyramid. To calculate the volume (V), the following formula is employed:

V = (1/3) × l × w × h

OR

V = (1/3) × A × h

Where,

• l is the length of rectangular base,
• w is width of rectangular base, 
• h is the height of the pyramid, and
• A is the area of the base of Rectangular Pyramid.

Rectangular Pyramid Formula

Here is a table summarizing the essential formulas for a rectangular pyramid:

Property	Formula
Surface Area (A)	OR l × w + (1/2) × P × S
Volume (V)	(1/3) × l × w × h
Slant Height (L)	√(l² + w² + h²)
Perimeter of Base (P)	2(l + w)
Height (H)	√(L² – l² – w²)

Where,

• l is the length of the rectangular base,
• w is the width of the rectangular base,
• h is the height of the pyramid, and 
• P is the perimeter of the base of rectangular pyramid.

Read More,

• Polyhedrons
• Regular Polygon
• Polygon Formula

Solved Problems on Rectangular Pyramid

Problem 1: Find the surface area of a rectangular pyramid with the following dimensions:

• Length of the base (L) = 6 units
• Width of the base (W) = 4 units
• Height of the pyramid (H) = 8 units

Solution:

To find the surface area (SA), we’ll use the formula:

SA = L × W + (1/2) × P × S

Calculate the perimeter of the base (P):

P = 2(l + w)

⇒ P = 2(6 + 4)

⇒ P = 2(10) = 20 units

Calculate the slant height (S) using the Pythagorean theorem:

S = √(l² + w² + h²)

⇒ S = √(6² + 4² + 8²)

⇒ S = √(36 + 16 + 64)

⇒ S = √116 ≈ 10.77 units

Now, substitute these values into the surface area formula:

Surface Area = 6 × 4 + (1/2) × 20 × 10.77

⇒ Surface Area = 24 + 107.7 ≈ 131.7 square units`

So, the surface area of the rectangular pyramid is approximately 131.7 square units.

Problem 2: Calculate the volume of a rectangular pyramid with a base length of 5 cm, a base width of 3 cm, and a height of 7 cm.

Solution:

To calculate the volume (V), we’ll use the formula:

V = (1/3) × L × W × H

Substitute the given values into the formula:

V = (1/3) × 5 cm × 3 cm × 7 cm

⇒ V = (1/3) × 105 cm³

⇒ V = 35 cm³

So, the volume of the rectangular pyramid is 35 cubic centimetres.

Problem 3: Calculate the slant height (S) of a rectangular pyramid with a base length of 8 inches, a base width of 6 inches, and a height of 10 inches.

Solution:

To find the slant height (S), we can use the Pythagorean theorem:

S = √(L² + W² + H²)

Substitute the given values into the formula:

S = √(8² + 6² + 10²)

S = √(64 + 36 + 100)

S = √200 ≈ 14.14 inches

The slant height of the rectangular pyramid is approximately 14.14 inches.

Practice Problems on Rectangular Pyramid

Problem 1: Given a rectangular pyramid with a base length of 8 cm, a base width of 5 cm, and a height of 12 cm, calculate its volume.

Problem 2: A rectangular pyramid has a base length of 10 meters, a base width of 6 meters, and a slant height of 8 meters. Calculate its total surface area.

Problem 3: If the volume of a rectangular pyramid is 240 cubic inches, and its base length is 6 inches, and the base width is 4 inches, find the height of the pyramid.

Problem 4: Given a rectangular pyramid with a base length of 12 cm, a base width of 8 cm, and a height of 15 cm, calculate its lateral surface area.

Rectangular Pyramid – FAQs

1. What is a Rectangular Pyramid?

A rectangular pyramid is a three-dimensional geometric shape characterized by a rectangular base and triangular faces converging at a single point known as the apex or vertex.

2. How to Find the Surface Area of a Rectangular Pyramid?

The surface area of a rectangular pyramid can be found using the formula:

Surface Area of a Rectangular Pyramid = l × w + (1/2) × P × S

OR

Surface Area of a Rectangular Pyramid =

Where,

• l and w are the dimensions of the base rectangle,
• P is the perimeter of the base, and
• S is the slant height.

3. How Many Vertices does a Rectangular Pyramid Have?

A rectangular pyramid has five vertices.

4. What is a Rectangular Right Pyramid?

A rectangular right pyramid is a three-dimensional geometric shape with a rectangular base and triangular sides where the apex is aligned with the center of the base.

5. How Many Faces Rectangular Pyramid Have?

A rectangular pyramid has five faces:

• One rectangular base.
• Four triangular faces, which are the sides of the pyramid.

6. What Is Formula for Calculating Volume Rectangular Pyramid?

The formula for calculating the volume (V) of a rectangular pyramid is:

V = (1/3) × l × w × h

Where,

• l is the length of the base,
• w is the width of the base, and
• h is the height of the pyramid.

7. How to Find Volume Rectangular Pyramid?

We can find the volume of Rectangular Pyramid using the above mentioned formula.

8. How many Edges Rectangular Pyramid Have?

A rectangular pyramid has 8 edges.

9. Describe the Shape of Rectangular Pyramid.

Rectangular Pyramid resembles a solid structure with a square or rectangular base and sloping triangular faces converging to a single apex.

10. How to Find Slant Height Rectangular Pyramid?

Slant height can be calculated using the following formula

S= √(l² + w² + h²)

Where,

• l is the length of the base,
• w is the width of the base, and
• h is the height of the pyramid.