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# Volume of a Pentagonal Pyramid

A pentagonal pyramid is a three-dimensional geometric figure that has a pentagonal base with five triangular faces that meet at a point. The meeting point is known as the apex, which joins the triangular faces and the pentagonal base. It has six faces (five triangular faces and a pentagonal base), six vertices, and ten edges. There are three types of pentagonal pyramids: a regular pentagonal pyramid, an irregular pentagonal pyramid, and a right pentagonal pyramid. A regular pentagonal pyramid has a regular pentagonal base, and its lateral faces are equilateral triangles. The triangular faces of an irregular pentagonal pyramid are either isosceles or scalene. A right pentagonal pyramid has the apex aligned exactly above the center of the base.

Pentagonal Pyramid

## Pentagonal pyramid’s real-life applications:

In real life, pentagonal pyramids can be found in many different shapes and forms such as architectural designs, lamps, sculptures, and so on. They are used for both functional and decorative purposes. The volume of a pentagonal pyramid can be used to calculate the amount of material needed to create a specific object or structure. Additionally, it can be used to determine the amount of space that an object or structure will occupy. Pentagonal pyramids have real-life applications in various fields such as architecture, construction, and engineering. Some of the common uses are:

Building design: Pentagonal pyramids are often used in building design and construction as structural supports and decorative elements.

Geometry education: The volume of a pentagonal pyramid can be used to teach basic geometry concepts in schools and colleges.

Surveying and mapping: The shape of a pentagonal pyramid can be used in surveying and mapping to determine the elevation of land and the slope of hills.

Art and sculpture: Pentagonal pyramids can also be used in art and sculpture as interesting shapes to create unique and visually appealing designs.

## Volume of a Pentagonal Pyramid

The volume of a pentagonal pyramid is the amount of space enclosed by it, and it is measured in terms of cubic units. The formula to find the volume of a pyramid is one-third the product of its base area and height. The formula to find the volume of the pyramid is given as follows:

V = (1/3) A × h cubic units

where,

• V is the volume
• A is the area of base
• h is the height

We know, Area of a pentagonal base is given by, (5/2) s × a, where “s” is the length of the side of a pentagon and “a” is its apothem length.

Now, the volume of the pentagonal pyramid (V) = 1/3 (5/2 × s × a) h cubic units

Therefore, the volume of the pentagonal pyramid becomes,

V = (5/6) ash cubic units

where
a is the apothem length
s is the base length
h is the height of the pyramid

## How to Find the Volume of the Pentagonal Pyramid?

Let’s take an example to understand how to calculate the volume of a pentagonal pyramid.

Example: Calculate the volume of a pentagonal pyramid whose apothem length is 4 cm, base length is 5 cm, and height is 9 cm.

Step 1: Note the values of the given dimensions. Here, the apothem length is 4 cm, the base length is 5 cm, and the height is 9 cm.

Step 2: We know that the formula to find the volume of a pentagonal pyramid is (5/6) ash cubic units. Now, substitute the given values in the formula.

Step 3: Thus, the volume of a pentagonal pyramid is calculated as, (5/6) × 4 × 5 × 9 = 150 cu. cm.

## Solved Examples on Pentagonal Pyramid

Example 1: What is the volume of a pentagonal pyramid if the base area is 125 cm2 and the height of the pyramid is 10 cm?

Solution:

Given data,

Area of the pentagonal base = 125 cm2

Height of the pyramid = 10 cm

We know that,

The volume of a pentagonal pyramid (V) = 1/3 × Area of pentagonal base × Height

V = 1/3 × 125 × 10 = 416.67 cm3

Hence, the volume of the given pentagonal pyramid is 416.67 cm3.

Example 2: What is the volume of a pentagonal pyramid whose apothem length is 5 in, base length is 10 in, and height is 12 in?

Solution:

Given data,

Apothem length (a) = 5 in

Base length (s) = 10 in,

Height (h) = 12 in,

We know that,

The volume of a pentagonal pyramid (V) = (5/6) ash cubic units

= (5/6) × 5 × 10 × 12

= 500 cu. in

Therefore, the volume of this pentagonal pyramid is 500 cu. in.

Example 3: Find the volume of a pentagonal pyramid if its base area is 135 cm2 and the height of the pyramid is 13 cm.

Solution:

Given data,

Area of the pentagonal base = 135 cm2

Height of the pyramid = 13 cm

The volume of a pentagonal pyramid (V) = 1/3 × Area of pentagonal base × Height

V = 1/3 × 135 × 13 = 585 cm3

Hence, the volume of the given pentagonal pyramid is 585 cm3.

Example 4: Find the height of a pentagonal pyramid if its base area is 150 m2 and the volume is 600 m3.

Solution:

Given data,

Area of the pentagonal base = 150 m2

The volume of a pentagonal pyramid (V) = 600 m3

We know that,

The volume of a pentagonal pyramid (V) = 1/3 × Area of pentagonal base × Height

⇒ 600 = (1/3) × 150 × h

⇒ 50h = 600

⇒ h = 600/50 = 12 m

Hence, the height of the given pentagonal pyramid is 12 m.

Example 5: Calculate the volume of a pentagonal pyramid whose apothem length is 7 cm, base length is 8 cm, and height is 15 cm.

Solution:

Given data,

Apothem length (a) = 7 cm

Base length (s) = 8 cm,

Height (h) = 15 cm,

We know that,

The volume of a pentagonal pyramid (V) = (5/6) ash cubic units

= (5/6) × 7 × 8 × 15

= 700 cu. cm

Therefore, the volume of this pentagonal pyramid is 700 cu. cm.

## FAQs on Pentagonal Pyramid

Question 1: What is a Pentagonal Pyramid?

A pentagonal pyramid is a three-dimensional geometric figure that has a pentagonal base with five triangular faces that meet at a point. The meeting point is known as the apex, which joins the triangular faces and the pentagonal base. It has six faces (five triangular faces and a pentagonal base), six vertices, and ten edges.

Question 2: What shape does the base of a pentagonal pyramid have?

As its name implies, the shape of the base of a pentagonal pyramid is a pentagon, which is a five-sided polygon.

Question 3: What is the formula to calculate the volume of a pentagonal pyramid?

The volume of a pentagonal pyramid is the amount of space enclosed by it, and it is measured in terms of cubic units.

V = (5/6) ash cubic units

where
a” is the apothem length,
s” is the base length,
h” is the height of the pyramid.

Question 4: How many faces does a pentagonal pyramid have?