How to Calculate the Volume and Surface Area of 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.
(image will be uploaded soon)
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 on Rectangular Pyramid Explained: Definition, Formulas & Uses
1. What is a rectangular pyramid?
A rectangular pyramid is a three-dimensional geometric shape that has a rectangular base and four triangular faces. These triangular faces meet at a single point above the base, which is known as the apex or vertex. It is a type of polyhedron, specifically a pyramid with a rectangular base.
2. What are the main properties of a rectangular pyramid in terms of its faces, edges, and vertices?
A rectangular pyramid has distinct properties that define its structure. For any rectangular pyramid, you will find:
Faces: It has a total of 5 faces – one rectangular face (the base) and four triangular faces (the lateral faces).
Edges: It has a total of 8 edges – four edges form the rectangular base, and the other four edges connect the corners of the base to the apex.
Vertices: It has a total of 5 vertices (or corners) – four vertices are located at the corners of the rectangular base, and the fifth is the apex.
3. What are the essential formulas used for a rectangular pyramid?
There are two primary formulas used to describe a rectangular pyramid:
Volume (V): The formula to calculate the space it occupies 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 perpendicular height of the pyramid.
Surface Area (SA): The total surface area is the sum of the area of the base and the area of the four triangular faces. The formula is SA = (l × w) + l√( (w/2)² + h² ) + w√( (l/2)² + h² ).
4. How is a 'right' rectangular pyramid different from an 'oblique' one?
The difference lies in the position of the apex relative to the base. In a right rectangular pyramid, the apex is located directly above the centre of the rectangular base. This means the imaginary line representing its height forms a perpendicular (90-degree) angle with the base. In an oblique rectangular pyramid, the apex is not directly above the centre, causing the pyramid to appear slanted or tilted to one side.
5. What are some real-world examples of rectangular pyramids?
While the classic pyramids of Giza have square bases, the rectangular pyramid shape can be seen in various real-world structures and objects. Examples include certain types of building roofs (hipped roofs), the top part of some monuments, and specific crystal formations. Many paperweights, tents, and architectural designs also use this strong and visually appealing shape.
6. How does a rectangular pyramid differ from a rectangular prism?
A rectangular pyramid and a rectangular prism are fundamentally different, although both have a rectangular base. A rectangular prism (like a shoebox) has two identical rectangular bases and four rectangular side faces. In contrast, a rectangular pyramid has only one rectangular base and four triangular side faces that converge to a single apex. Consequently, a prism has 6 faces, 12 edges, and 8 vertices, while a pyramid has 5 faces, 8 edges, and 5 vertices.
7. Why is the concept of 'slant height' important in a rectangular pyramid?
The slant height is the height of each of the triangular faces, measured from the midpoint of a base edge to the apex. It is different from the pyramid's perpendicular height. The slant height is crucial for calculating the lateral surface area (the combined area of the triangular faces). Since a rectangular base has unequal length and width, a rectangular pyramid has two different slant heights, one for the triangles on the length sides and one for the triangles on the width sides. You cannot find the surface area without first calculating these slant heights.
