Hexagonal Pyramid Formula

Geometry has a major domain in the field of mathematics. It is a study specifically for shapes and structures. Geometry describes the shapes, their dimensions, properties, parameters and also provides formulas that make the calculation of these parameters convenient.

The given article discusses a solid shape hexagonal pyramid. The content of the article consists of the definition of the hexagonal pyramid and its properties. The article also contains the formulas for the calculation of the parameters along with solved mathematics problems for the understanding of the way of calculation. 

Hexagonal pyramid

A hexagonal pyramid is a type of pyramid which has a hexagonal base and the isosceles triangles combine at the apex and form the pyramid. A hexagonal pyramid is a three-dimensional structure with a hexagon base and sides build-up of 6 isosceles triangles.

Hexagonal pyramid is also sometimes known as Heptahedron as it has seven faces, 12 edges, and 7 vertices. Summing up, a hexagonal pyramid is a three-dimensional solid structure with a six-sided base and faces formed of six isosceles triangles meeting at the apex.

Hexagonal Pyramid

Properties of the hexagonal pyramid

• A hexagonal pyramid has a hexagon as a base which is a six-sided polygon.
• It consists of faces made up of six isosceles triangles.
• The isosceles triangles forming the face stands erect and meet at the apex point.
• A hexagonal pyramid has 7 faces, 12 edges, and 7 vertices.

Hexagonal Pyramid Formula

Formula of volume for Hexagonal Pyramid

The volume of a hexagonal pyramid is given by the apothem (length from the center of the base to any point on the base), length of base, and height of the pyramid. The height of the pyramid is measured from the apex to the base.

Hence, the volume formula for the calculation of the volume of the hexagonal pyramid is given by the product of apothem, length of base, and height.

Mathematically, the formula is written as,

The volume of Hexagonal Pyramid = a × b × h 

Where,

a is the apothem of the pyramid

b is the base

h is the height

There is also an alternate formula for the calculation of the volume of the pyramid in case of the absence of apothem and the given triangles of the pyramid are equilateral.

The formula is given by

The volume of hexagonal Pyramid = (√3/2) × a²× h

 where,

 a is the side of the base 

and h is the height of the hexagonal pyramid

Formula of surface area for the Hexagonal pyramid

The surface area of a hexagonal pyramid is given by the apothem of the pyramid, base, and slant height of the pyramid. The slant height of the pyramid is measured from the apex to any point on the boundary of the base of the pyramid.

For the calculation of the surface area of the hexagonal pyramid, there is also a need to look into the formula of the base area. Hence, the formulas for base area and surface area are mentioned below respectively.

Base Area of Hexagonal Pyramid = 3ab

Surface Area of Hexagonal Pyramid = (3ab + 3bs)

Where,

a is the apothem of the pyramid

b is the base

s is the slant height of the pyramid

Sample Problems

Question 1: Calculate the volume of a hexagonal pyramid with apothem length of 3cm, base length as 4cm, and height as 5cm.

Solution:

Given:

Apothem length is 3cm.

Base length is 4cm 

Height is 5cm

Now,

Hexagonal pyramid volume = a × b × h

=> Volume = 3 × 4 × 5

=> Volume = 60cm³

Question 2: Calculate the volume of a hexagonal pyramid with apothem length of 9cm, base length as 16cm, and height as 25cm.

Solution:

Given:

Apothem length is 9cm.

Base length is 16cm

Height is 25cm

Now,

Hexagonal pyramid volume = a × b × h

=> Volume = 9 × 16 × 25

=> Volume = 3600cm³

Question 3: Calculate the base area and surface area of a hexagonal pyramid, if the apothem length is 4cm, base length is 8cm and slant height is 12cm?

Solution:

Given:

Apothem length is 4cm.

Base length is 8cm.

Slant height is 12cm.

Now,

Base area = 3ab

=> Base area = 3 × 4 × 8

=> Base area = 96cm²

Then,

Surface area = (3ab + 3bs)

=> Surface area = (96 + 3 × 8 × 12)

=> Surface area = 96 + 288

=> Surface area = 384cm²

Question 4: Calculate the base area and surface area of a hexagonal pyramid, if the apothem length is 6cm, base length is 9cm and slant height is 12cm?

Solution:

Given:

Apothem length is 6cm.

The base length is 9cm.

The slant height is 12cm.

Now,

Base area = 3ab

=> Base area = 3 × 6 × 9

=> Base area = 162cm²

Then,

Surface area = (3ab + 3bs)

=> Surface area = (96 + 3 × 9 × 12)

=> Surface area = 162 + 324

=> Surface area = 486cm²

Question 5: Calculate the base area and surface area of a hexagonal pyramid, if the apothem length is 12cm, base length is 20cm and slant height is 25cm?

Solution:

Given:

Apothem length is 12cm.

The base length is 20cm.

The slant height is 25cm.

Now,

Base area = 3ab

=> Base area = 3 × 12 × 20

=> Base area = 720cm²

Then,

Surface area = (3ab + 3bs)

=> Surface area = (720 + 3 × 20 × 25)

=> Surface area = 162 +1500

=> Surface area = 1662cm²

Question 6: Calculate the volume of a hexagonal pyramid with apothem length 4cm, base length as 16cm, and height as 24cm.

Solution:

Given:

Apothem length is 4cm.

Base length is 16cm

Height is 24cm

Now,

Hexagonal pyramid volume = a × b × h

=> Volume = 4 × 16 × 24

=> Volume = 1536cm³