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.
Definition of Pentagonal Pyramid
A pentagonal pyramid is a 3D object with the base of a pentagon upon which five triangle-shaped faces meet at the apex. This apex forms at the top of the pentagonal pyramid and it combines the triangular faces and the pentagonal base together. In a regular pentagonal pyramid, the base is a regular pentagon with lateral faces shaped like equilateral triangles.
Surface Area of a Pentagonal Pyramid
The surface area of a pentagonal pyramid has two types of surface areas, i.e., a lateral surface area and a total surface area, which are measured in terms of square units like m2, cm2, in2, ft2, etc.
Lateral Surface Area
The lateral surface area of a pentagonal pyramid is equal to the sum of the areas of its lateral faces, i.e., the five triangular faces. We know that the general formula to find the lateral surface area of a pyramid is:
Lateral Surface Area (LSA) = ½ × P × l
where,
“P” is the perimeter of the base,
“l” is the slant height of the pyramid.
The perimeter of the pentagonal base = s + s + s + s + s = 5s
We know that the slant height of the pyramid is “l”.
So, the formula to find the lateral surface area of the pentagonal pyramid is given as follows:
Lateral Surface Area (LSA) = 5⁄2 (s × l) square units
Where,
“s” is the side length of the base,
“l” is the slant height of the pyramid.
Total Surface Area
The total surface area of a pentagonal pyramid is equal to the total area covered by its five triangular side faces and the pentagonal base. We know that the general formula to find the total surface area of a pyramid is:
Total Surface Area (TSA) = LSA of the pyramid + Area of the base
Area of the pentagonal base = 5⁄2 (a × s)
So, the total surface area = 5⁄2 (s × l) + 5⁄2 (a × s) = 5⁄2 × s × (l + a)
Total Surface Area = 5⁄2 × s × (a + l)
where,
“s” is the side length of the base,
“a” is apothem length of the base, and
“l” is the slant height of the pyramid.
Formula of the Surface Area in terms of the Height of the Pyramid
We know that,
Slant height of the pyramid (l) = √[(s/2)2 + h2]
Now,
Lateral Surface Area of the Pentagonal Pyramid (LSA) = 5⁄2 × s × √[(s2/4) + h2]
Total Surface Area of the Pentagonal Pyramid (TSA) = 5⁄2 × s × [a + √(s2/4) + h2)]
How to find the Surface Area of a Pentagonal Pyramid?
Let’s take an example to understand how to calculate the surface area of a pentagonal pyramid.
Example: Find the surface area of a pentagonal pyramid whose apothem length is 5 inches, base length is 7 inches, and slant height is 11 inches.
Step 1: Note the values of the given dimensions. Here, the apothem length is 5 inches, the base length is 7 inches, and the slant height is 11 inches.
Step 2: We know that the formula to find the surface area of a pentagonal pyramid is 5⁄2 × s × (a + l) square units. Now, substitute the given values in the formula.
Step 3: Thus, the surface area of a pentagonal pyramid is calculated as (5⁄2) × 7 × (5 + 11) = 280 sq. in.
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 = 5⁄2 (s × l) square units
= 5⁄2 × 5 × 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) = √(s2/4 + h2)
= √(144/4 + 64) = √(36 + 64)
= √100 = 10 cm
The surface area of the pentagonal pyramid = 5⁄2 × s × (a + l) square units
= 5⁄2 × 12 × (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 = 5⁄2 (s × l) square units
= 5⁄2 × (15 × 20)
= 5⁄2 × 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 = 5⁄2 × s × (a + l) square units
= 5⁄2 × 7 × (4 + 9)
= 5⁄2 × 7 × 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 = 5⁄2 (s × l)
240 = 5⁄2 × s × 12
30s = 240
s = 240/30 = 8 cm
Hence, the base length of the pentagonal pyramid is 8 cm.
FAQs on Pentagonal Pyramid
Question 1: What is a Pentagonal Pyramid?
Answer:
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.
Question 2: What is meant by a regular pentagonal pyramid?
Answer:
A regular pentagonal pyramid is a pyramid that has a regular pentagonal base, and its lateral faces are equilateral triangles.
Question 3: What is the general formula to find the lateral surface area of a pyramid?
Answer:
The general formula to find the lateral surface area of a pyramid is equal to half the product of the perimeter of the polygonal base and the slant height of the pyramid.
Lateral surface area (LSA) = ½ × P × l
where “P” is the perimeter of the base and “l” is the slant height of the pyramid.
Question 4: What is the formula to find the total surface area of a pentagonal pyramid?
Answer:
The formula to find the total surface area of the pentagonal pyramid is given as follows:
The total surface area of the pentagonal pyramid = 5⁄2 × s × (a + l)
where “s” is the side length of the base,
“a” is apothem length of the base, and
“l” is the slant height of the pyramid.
Question 5: What is the formula to find the lateral surface area of a pentagonal pyramid in terms of the height of the pyramid?
Answer:
The formula to determine the lateral surface area of a pentagonal pyramid in terms of the height of the pyramid is given as follows:
The lateral surface area of the pentagonal pyramid (LSA) = 5⁄2 × s × √[(s2/4) + h2]
where “s” is the side length of the base, and
“h” is the height of the pyramid.
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