Regular Hexagonal Pyramid Formula

A hexagonal pyramid is a three-dimensional shaped pyramid that has a hexagonal base along with the sides or faces in the shape of isosceles triangles that meet at the apex or the top of the pyramid. A hexagonal pyramid is one of the different types of pyramids, which are classified based on the shape of the base of a pyramid. It is also known as a heptahedron since a hexagonal pyramid consists of 7 faces, which includes a hexagonal base and 6 isosceles triangular lateral faces. It has a total of seven faces, twelve edges, and seven vertices. One of the seven vertices is the apex, which is at the top, and the other six are at the base of the pyramid. Out of the twelve edges, six edges connect the triangle edges that meet at the apex, and the other six are the edges of the base.

 Regular Hexagonal Pyramid

A regular hexagonal pyramid is a pyramid whose hexagonal base is regular and the pyramid is straight, whereas an irregular hexagonal pyramid is a pyramid whose hexagonal base is irregular and the pyramid is oblique. A right regular pyramid is a hexagonal pyramid with a regular hexagonal base and, the apex of the pyramid is right above the center of the base, such that the apex forms a right angle with the center of the base and any other vertex.

Regular Hexagonal Pyramid formula

There are two formulas for a regular hexagonal pyramid, i.e., the surface area of a regular hexagonal pyramid and the volume of a regular hexagonal pyramid. To calculate the surface area or the volume of a regular hexagonal pyramid, we need to know its four major aspects, i.e., the length of the side of the base; the apothem, which is the distance from the center of the base to any point on the side of the base; the height of the pyramid, which is the perpendicular distance from the apex to the center of the base; and finally the slant height of the pyramid, which is the height of the triangular faces or the perpendicular distance from the apex to any point on the boundary of the base of the pyramid.

Lateral surface area (LSA)

The lateral surface area is the region occupied by the lateral surfaces or triangular faces of a regular hexagonal pyramid. The formula to determine the lateral surface area of the regular hexagonal pyramid (LSA) is given as follows,

The lateral surface area of the regular hexagonal pyramid = The sum of areas of the lateral surfaces (triangles) of the pyramid

= 6 × [½ × base × height] =3 (s × l)

Lateral surface area of the regular hexagonal pyramid = 3(s × l)

Where,

“s” is the side length of the base, and

“l” is the slant height of the pyramid. 

Total surface area (TSA)

The total surface area is the total region occupied by all the surfaces of a regular hexagonal pyramid, i.e., the area occupied by the lateral surfaces, or triangular faces, and also is hexagonal base. 

Total surface area of a pyramid (TSA) = Lateral surface area of the pyramid + Base area

The surface area of the hexagonal pyramid can be calculated when we have the slant height of the pyramid which is the height from the apex to any point on the boundary of the base of the pyramid. Hence, let us see both the formula of the hexagonal pyramid – base area and surface area.

Base area = 3as

Where,

“a” is the apothem length, and

“s” is the side length of the base.

TSA = LSA + Base area

TSA = 3sl + 3as

Hence,

Total surface area of the regular hexagonal pyramid (TSA) = 3sl + 3as

Where,

“s” is the side length of the base, 

“l” is the slant height, and 

“a” is the apothem length.

When the apothem of the regular hexagonal pyramid is not mentioned and the triangular faces are equilateral, there is another alternative formula to calculate its surface area, i.e.,

Total surface area of the hexagonal pyramid = 3(s × l) + 3√3/2 (s)²

Where, 

“s” is the side length of the base, and 

“l” is the slant height of the pyramid. 

Area of the hexagonal base = 3√3/2 (s)²

Volume of the regular hexagonal pyramid

The volume is the total space enclosed between all the faces of a regular hexagonal pyramid. The general formula for calculating the volume of a pyramid is equal to one-third of the product of the base area and the height of the pyramid.

Volume (V) = (1/3) × Base area × Height cubic units

Now, by substituting the values of the base area and the height, we get

Volume of the regular hexagonal pyramid = (a × s × h) cubic units

Where,

“a” is the apothem length,

“s” is the side length of the base, and

“h” is the height of the pyramid. 

When the apothem of the regular hexagonal pyramid is not mentioned and the triangular faces are equilateral, there is another alternative formula to calculate its volume, i.e.,

Volume of the regular hexagonal pyramid (V)= (√3/2) × s² × h cubic units

Where,

“s” is the side length of the base, and

“h” is the height of the pyramid. 

Practice Problems based on Regular Hexagonal Pyramid

Problem 1: What is the volume of a regular hexagonal pyramid whose apothem length is 5 cm, length of the side of the base is 10 cm, and height is 13 cm?

Solution: 

Given data,

Apothem length (a) = 5 cm

The length of the side of the base  = 10 cm,

The height of the pyramid = 13 cm

We know that,

The volume of a regular hexagonal pyramid (V) = (a × s × h) cubic units

V = 5 × 10 × 13

Volume = 650 cm³

Therefore, the volume of the given hexagonal pyramid is 650 cu. cm.

Problem 2: What is the surface area of a regular hexagonal pyramid if its apothem length is 6 inches, the length of the side of the base is 8 inches, and the slant height is 15 inches?

Solution: 

Given data,

Apothem length (a) = 6 inches

The length of the side of the base (s)  = 8 inches

The slant height of the pyramid (l) = 15 inches

We know that,

The surface area of the hexagonal pyramid = 3as + 3sl square units

= 3 × 6 × 8 + 3 × 8 × 15

= 144 + 360 = 504 sq. in

Therefore, the surface area of the given pyramid is 504 sq. in.

Problem 3: Find the height of a regular hexagonal pyramid if its volume is 576 cu. cm, the length of the side of the base is 8 cm, and the apothem length is 8 cm.

Solution:

Given data,

Apothem length (a) = 8 cm

The length of the side of the base (s) = 8 cm

Volume = 576 cu. cm

We know that,

The volume of a regular hexagonal pyramid (V) = (a × s × h) cubic units

⇒ 8 × 8 × h = 576

⇒ 64h = 576

⇒ h = 576/64 = 9 cm

Hence, the height of a regular hexagonal pyramid is 9 cm.

Problem 4: What is the volume of a regular hexagonal pyramid if the sides of a base are 7 cm each and the height of the pyramid is 14 cm?

Solution:

Given data,

Height of the pyramid (h) = 14 cm

The length of the side of the base (s) = 7 cm

Area of the hexagonal base (A) = 3√3/2 b² = 3√3/2 (7)² = 147√3/2 sq. cm

The volume of a regular hexagonal pyramid (V) = 1/3 × A × h

V = 1/3 × (147√3/2) × 14 = 594.09 cm³

Hence, the volume of the given pyramid is 594.09 cm³.

Problem 5: Determine the lateral surface area of a regular hexagonal pyramid if the side length of the base is 15 inches and the pyramid’s slant height is 21 inches.

Solution:

Given data,

The length of the side of the base (s) = 15 inches, and

Slant height (l) = 21 inches

The perimeter of the square base (P) = 6s = 6(15) = 90 inches

We know that,

The lateral surface area (LSA) = (½) Pl

= (½ ) × (90) × 21 = 945 sq. in

Therefore, the lateral surface area of the given pyramid is 945 sq. in.

FAQs based on Regular Hexagonal Pyramid

Question 1: What is a Hexagonal Pyramid?

Answer:

A hexagonal pyramid is a 3D shape with hexagonal base combined with 6 triangles faces against each sides of the hexagonal base erected  in such a way to form a pyramid at its apex. These triangles may be either isosceles triangles or equilateral triangles and these triangles are called as lateral faces. A hexagonal pyramid contains 7 vertices, 7 faces, and 12 edges.

Question 2: What is the formula for finding the volume of the Hexagonal Pyramid?

Answer:

The formula for calculating the volume of the hexagonal pyramid is given by,

Volume of Hexagonal Pyramid(V) = (abh) cubic units

where,

a is the apothem of the pyramid,

b is the base, and h is the height.

Question 3: What is the formula for finding the Surface Area of a Hexagonal Pyramid?

Answer:

Formula for finding the surface area of a hexagonal pyramid is given by,

Surface Area of Hexagonal Pyramid (TSA)= (3ab + 3bs) square units,

where,

a is the apothem of the pyramid,

b is the base, and

s is the slant height of the pyramid.