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
• Length of the side of the base  = 10 cm
• Height of the pyramid = 13 cm

We know that,

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
• Length of the side of the base (s)  = 8 inches
• 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
• Length of the side of the base (s) = 8 cm

Volume = 576 cu. cm

We know that,

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
• Length of the side of the base (s) = 7 cm

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

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,

• Length of the side of the base (s) = 15 inches
• Slant height (l) = 21 inches

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

We know that,

Lateral surface area (LSA) = (½) P.l

= (½ ) × (90) × 21 = 945 sq. in

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

A hexagonal pyramid is a three-dimensional pyramid that has a hexagonal base along with 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 include a hexagonal base and 6 isosceles triangular lateral faces.