Solved Examples on Pentagonal Pyramid
Example 1: Calculate the lateral surface area of a pentagonal pyramid whose base length is 5 cm and slant height is 7 cm.
Solution:
Given data,
Base length (s) = 5 cm
Slant height (l) = 7 cm
We know that,
The lateral surface area of the pentagonal pyramid = 5β2 (s Γ l) square units
= 5β2 Γ 5 Γ 7
= 87.5 sq. cm
Hence, the lateral surface area of the pentagonal pyramid is 87.5 sq. cm.
Example 2: Calculate the surface area of a pentagonal pyramid if the base length is 12 cm, the apothem length is 7 cm, and the height is 8 cm.
Solution:
Given data,
Base length (s) = 12 cm
Apothem length (a) = 7 cm
Height (h) = 8 cm
We know that,
slant height (l) = β(s2/4 + h2)
= β(144/4 + 64) = β(36 + 64)
= β100 = 10 cm
The surface area of the pentagonal pyramid = 5β2 Γ s Γ (a + l) square units
= 5β2 Γ 12 Γ (7 + 10) = 510 sq. cm
Hence, the surface area of the pentagonal pyramid is 510 sq. cm.
Example 3: Find the lateral surface area of a pentagonal pyramid if the base length is 15 inches and the slant height is 20 inches.
Solution:
Given data,
Base length (s) = 15 inches
Slant height (l) = 20 inches
We know that,
The lateral surface area of the pentagonal pyramid = 5β2 (s Γ l) square units
= 5β2 Γ (15 Γ 20)
= 5β2 Γ 300 = 750 sq. in
Hence, the lateral surface area of the pentagonal pyramid is 750 sq. in.
Example 4: Find the total surface area of a pentagonal pyramid if the base length is 7 cm, the apothem length is 4 cm, and the slant height is 9 cm.
Solution:
Given data,
Base length (s) = 7 cm
Apothem length (a) = 4cm
Slant height (l) = 9 cm
We know that,
The surface area of the pentagonal pyramid = 5β2 Γ s Γ (a + l) square units
= 5β2 Γ 7 Γ (4 + 9)
= 5β2 Γ 7 Γ 13 = 227.5 sq. cm
Hence, the total surface area of the pentagonal pyramid is 227.5 sq. cm.
Example 5: What is the base length of a pentagonal pyramid if its slant height is 12 cm, and the lateral surface area is 240 sq. cm?
Solution:
Given data,
Slant height (l) = 12 cm
The Lateral Surface Area = 240 sq. cm
We know that,
The lateral surface area of the pentagonal pyramid = 5β2 (s Γ l)
240 = 5β2 Γ s Γ 12
30s = 240
s = 240/30 = 8 cm
Hence, the base length of the pentagonal pyramid is 8 cm.
Related Resources:
Area of a Pentagonal Pyramid
In geometry, a pentagonal pyramid is a three-dimensional figure with a pentagonal base upon which five triangular faces are erected and meet at a meeting point called the apex. It has six faces, i.e., a pentagonal base and five triangular faces, six vertices, and ten edges. In a pentagonal pyramid, each edge of the pentagonal base is connected to the apex, and thus the five triangular/lateral faces are formed. A regular pentagonal pyramid is a pyramid that has a regular pentagonal base, and its lateral faces are equilateral triangles. Based on the shape of the polygonal base of a pyramid, every pyramid has a different formula. In this article, we will discuss the surface area of a pentagonal pyramid in detail.
Table of Content
- Definition of Pentagonal Pyramid
- Surface Area of a Pentagonal Pyramid
- Lateral Surface Area
- Total Surface Area
- Formula of the Surface Area in terms of the Height of the Pyramid
- How to find the Surface Area of a Pentagonal Pyramid?
- Solved Examples on Pentagonal Pyramid
- Related Resources: