Rectangular Pyramid

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What is a Rectangular Pyramid?

Pyramids are considered to be three-dimensional structures that have triangular faces and contain an encompassing polygon shape in its base. In cases where the bottom of the pyramid is rectangular, then the pyramid is known as a rectangular pyramid. In a rectangular pyramid, the base is in the shape of a rectangle, but the sides of the pyramid are triangular in shape. So, a pyramid looks like a triangle from every side to the naked eyes. The pyramid's shape helps a student determine surface area and volume of the pyramid.


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Definition

As discussed earlier, a rectangular pyramid is a type of pyramid that has a rectangular shape in the base. When this type of pyramid is looked from the bottom, it looks like a rectangle. Hence, the opposite sides of the base are parallel and equal to each other.


A pyramid is crowned on the top of the base at a point which is termed as the apex. A rectangular pyramid can be of two types, namely the right pyramid and oblique pyramid. In the case of a right pyramid, the apex is located right over the centre of the base, but in an oblique pyramid, the apex doesn't lie over the centre of the base but at the same angle from the centre.


Types of Pyramids

Apart from the rectangular pyramid, there are some other types of a pyramid, which are classified on the basis of the shape of their bases. Some of these pyramids are as follows:

  • Triangular Pyramid.

  • Square Pyramid.

Faces, Edges and Vertices

The key features of a pyramid are its faces, edges and vertices. Let's discuss these three features of the pyramid in brief so that students can have a clearer view:


Faces - A rectangular pyramid consists of a total of five faces. Among these, one of the faces has a shape of a rectangle, and the other four faces are triangular shaped. All the triangular faces in this rectangular pyramid are congruent to its opposite triangular face.


Vertex - A rectangular pyramid consists of a total of five vertices. The point where the edges meet or intersect is termed as vertices. One of the vertices is present at the top right above the base; this is the point where the triangular faces of the pyramid meet. The remaining four vertices lie at the corners of the rectangular-shaped base.


Edges - A rectangular pyramid consists of a total of eight edges. Each edge gets formed when two faces or surfaces intersect with each other. Among these eight edges, four are located at the rectangular base while the other four edges form slopes right above the rectangular base that meets at the peak point which is known as the vertex of the pyramid.


Rectangular Pyramid Formula

The rectangular pyramid has different formulas which students have to understand thoroughly in order to secure good marks in the exams. Formulas are considered as the base for every geometrical chapter. The formulas of the rectangular pyramid are as follows:


Surface Area of a Rectangular Pyramid:

\[A = lw + l \sqrt{(w/2)^{2} + h^{2}} + w \sqrt{(l/2)^{2} + h^{2}}\]


Where,

l =  Length of the rectangular base.

w = Width of the rectangular base.

h = Height of the pyramid.

The above is considered as the rectangular pyramid surface area formula.


Volume of a Rectangular Pyramid:

\[v = (lwh)/3\]


Where,

l = Length of the rectangular base.

w =Width of the rectangular base.

h = Height of the pyramid.

The above formula is for the volume of a rectangular based pyramid.


Lateral Area of a Rectangular Pyramid

\[LA = 1/2 (ps)\]


Where,

p = Perimeter of the rectangular base.

s = Slant height.


Solved Problems

1. Evaluate the surface area of a rectangular pyramid if :

l = 10

w = 5

h =10


Solution:

\[A = lw + l \sqrt{(w/2)^{2} + h^{2}} + w \sqrt{(l/2)^{2} + h^{2}}\]

\[A = (10*5) + 10 \sqrt{(5/2)^{2} + (10)^{2}} + 5 \sqrt{(10/2)^{2} + (10)^{2}}\]

\[A = 50 + 10(25) + 5(11.20)\]

A = 356


2. Evaluate the volume of a rectangular pyramid if:

l = 10

w =5

h =10


Solution:

\[v = (lwh)/3\]

v = (10*5*10) / 3

v = 166.66


3. Evaluate the surface area and volume of a rectangular pyramid, if:

l = 20

w =10

h =15


Solution:

Surface area of a rectangular pyramid

\[A = lw + l \sqrt{(w/2)^{2} + h^{2}} + w \sqrt{(l/2)^{2} + h^{2}}\]

\[A = (20*10) + 20 \sqrt{(10/2)^{2} + (15)^{2}} + 10 \sqrt{(20/2)^{2} + (15)^{2}}\]

A = 200 + 316.2 + 179.4

A =  695.6


Volume of a rectangular pyramid

\[v = (lwh)/3\]

v = (20*10*15) / 3

v = 1000

FAQs (Frequently Asked Questions)

Q1. What are the Characteristics of a Rectangular Pyramid?

Ans: Every structure in geometry has its characteristics. Similarly, rectangular pyramid being a geometrical figure, also has some characteristics which are:

  • The base of the rectangular pyramid has the shape of a rectangle and the remaining sides are of triangle shape.

  • The rectangular pyramid consists of a total of five faces, among which one has a rectangular shape and the other four have a triangular shape.

  • The rectangular pyramid consists of a total of five vertices, among which one is at the vertex, and the other four are at the point where the triangular faces meet.

  • The rectangular pyramid consists of a total of eight edges.

Q2. Define the Triangular Pyramid.

Ans: In Geometry, a triangular pyramid is also known as a tetrahedron. A triangular pyramid is composed of four faces which are triangular in shape. This structure has a total of six straight edges and four vertex corners. The triangular pyramid is considered as the simplest of all the ordinary convex polyhedra that are in geometry. A triangular pyramid is the only polyhedra that have fewer than five faces. The triangular faces are equilateral and are congruent to the opposite triangular shape. Some shapes with this similar shape are octahedron, triangular prism, square pyramid and icosahedron. A triangular pyramid is an important structure.

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