Volume Of Hexagonal Pyramid Formula

In geometry, a pyramid is a three-dimensional shape whose base is a polygon, and all its triangular faces join at a common point called the apex. A pyramid is formed past joining all the sides of a polygon base of operations to an apex. The pyramids of Arab republic of egypt are real-life examples of pyramids. There are different types of pyramids based on the shape of the base of the pyramid. The dissimilar types of pyramids are triangular pyramids, square pyramids, rectangular pyramids, pentagonal pyramids, etc. The side faces of whatever type of pyramid are triangular, and 1 side of each triangular face merges with the side of the base. The top of the pyramid is called the apex, and the side faces are called the lateral faces of a pyramid.

The volume of a Pyramid

The volume of a pyramid refers to the full space enclosed betwixt all the faces of a pyramid; in simple words, the total space inside a closed pyramid. The formula for the volume of a pyramid is equal to one-third of the product of the base expanse and the top of the pyramid and is usually represented past the letter "V".

The formula for the volume of a pyramid is given as follows,

The book of a Pyramid = i/three × base expanse × acme

V = 1/iii AH cubic units

Where V is the book of the pyramid,

A is the base of operations area of the pyramid and

H is the tiptop or distance of a pyramid.

Derivation of the book of a pyramid

Let'south consider a rectangular pyramid and a prism where the base and tiptop of both the pyramid and the prism are the same. Now accept a rectangular pyramid full of water and pour the h2o into the empty prism. We tin observe that only one-third part of a prism is full. And then, repeat the experiment over again, and we volition discover that there is still some vacant space in the prism. Once once again, repeat the experiment, and this time we tin discover that the prism is filled upward to the brim. Hence, the volume of a pyramid is equal to one-third of the volume of a prism if the base and height of both the pyramid and the prism are the same. And so,

The volume of a prism = 3 × [Volume of a pyramid]

i.eastward.,

The volume of a pyramid = (i/3) × [Book of a prism]

We know that,

The volume of a prism = AH cubic units

Hence,

Book of a pyramid (V) = (one/3) AH cubic units

A is the base area of the pyramid and H is the height or altitude of a pyramid.

The volume of a triangular pyramid

The pyramid that has a triangular base is chosen the triangular pyramid. A triangular pyramid has three triangular faces and 1 triangular base, where the triangular base can be equilateral, isosceles, or a scalar triangle. A triangular pyramid is also referred to as a tetrahedron. The formula for the volume of the triangular pyramid is given,

The volume of the triangular pyramid = ane/three AH cubic units

Where H is the height of the pyramid and A is the area of the base

We know that,

Area of a triangle = 1/ii b × h

Where b is the length of the base of the triangle and h is its height.

Now, the volume of the triangular pyramid (V)= 1/iii (1/2 b × h)H cubic units

5 = 1/6 bhH cubic units

Hence,

The volume of the triangular pyramid (V)= 1/vi bhH cubic units

The volume of a square pyramid

The pyramid that has a square base is chosen the square pyramid. A foursquare pyramid has four triangular faces and one square base of operations. The formula for the volume of the square pyramid is given,

The book of the square pyramid = i/iii AH cubic units

Where H is the meridian of the pyramid and A is the surface area of the base

Area of a square = a^two

Where a is the length of the side of the square.

Now, the volume of the square pyramid (V)= 1/3 (a²) H cubic units

V = (one/3) a²H cubic units

Hence,

The volume of the square pyramid (V)= (1/3) a²H cubic units

The volume of a rectangular pyramid

The pyramid that has a rectangular base is called the rectangular pyramid. A rectangular pyramid has iv triangular faces and 1 rectangular base. The formula for the volume of the rectangular pyramid is given,

The volume of the rectangular pyramid = one/3 AH cubic units

Where H is the height of the pyramid and A is the area of the base of operations

Area of a rectangle = l × w

Where l is the length of the rectangle and w is its width.

At present, the volume of the rectangular pyramid (Five)= 1/3 (l × due west) H cubic units

V = 1/iii (l × w × H) cubic units

Hence,

The book of the rectangular pyramid (Five)= 1/3 (l× west ×H) cubic units

The volume of a pentagonal pyramid

The pyramid that has a pentagonal base is called the pentagonal pyramid. A pentagonal pyramid has five triangular faces and one pentagonal base. The formula for the book of the pentagonal pyramid is given,

The volume of the pentagonal pyramid = ane/3 AH cubic units

Where H is the height of the pyramid and A is the expanse of the base

Area of a pentagon = (five/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 (v/ii S × a) H cubic units

V = 5/6 aSH cubic units

Hence,

The volume of the pentagonal pyramid (V)= v/half dozen aSH cubic units

The volume of a hexagonal pyramid

The pyramid that has a hexagonal base is called the hexagonal pyramid. A hexagonal pyramid has half-dozen triangular faces and one hexagonal base of operations. The formula for the volume of the hexagonal pyramid is given,

The volume of the hexagonal pyramid = one/3 AH cubic units

Where H is the height of the pyramid and A is the surface area of the base

Surface area of a hexagon = 3√three/2 a²

Where a is the length of the side of the hexagon.

Now, the volume of the hexagonal pyramid (V)= ane/3 (3√iii/2 a²) H cubic units

V = √3/two aⁱⁱ H cubic units

Hence,

The volume of the hexagonal pyramid (V)= √3/two aⁱⁱ H cubic units

Sample Problems

Problem 1: What is the volume of a square pyramid if the sides of a base of operations are 6 cm each and the tiptop of the pyramid is 10 cm?

Solution:

Given data,

Length of the side of the base of a square pyramid = vi cm

Summit of the pyramid = x cm.

The volume of a square pyramid (V) = 1/3 × Surface area of square base × Height

Surface area of square base of operations = aⁱⁱ = half-dozen² = 36 cm²

V = i/3 × (36) ×10 = 120 cm³

Hence, the volume of the given square pyramid is 120 cm^three.

Trouble 2: What is the volume of a triangular pyramid whose base surface area and acme are 120 cmⁱⁱand 13 cm, respectively?

Solution:

Given data,

Area of the triangular base = 120 cm²

Height of the pyramid = 13 cm

The volume of a triangular pyramid (5) = i/iii × Area of triangular base × Height

V = 1/3 × 120 × 13 = 520 cm³

Hence, the volume of the given triangular pyramid = is 520 cm³

Problem iii: What is the volume of a triangular pyramid if the length of the base and altitude of the triangular base of operations are 3 cm and 4.5 cm, respectively, and the tiptop of the pyramid is 8 cm?

Solution:

Given information,

Height of the pyramid = 8 cm

Length of the base of the triangular base = 3 cm

Length of the altitude of the triangular base = 4.v cm

Area of the triangular base (A) = 1/ii b × h = 1/two × three × 4.v = six.75‬ cm^two

The volume of a triangular pyramid (V) = 1/three × A × H

V = ane/three × 6.75 × 8 = eighteen cmⁱⁱⁱ

Hence, the book of the given triangular pyramid is 18 cm³

Problem iv: What is the book of a rectangular pyramid if the length and width of the rectangular base of operations are eight cm and 5 cm, respectively, and the height of the pyramid is 14 cm?

Solution:

Given data,

Height of the pyramid = 14 cm

Length of the rectangular base (l) = viii cm

Width of the rectangular base (w) = v cm

Surface area of the rectangular base (A) = l‬ × w = 8 × five = twoscore cm²

We have,

The volume of a rectangular pyramid (V) = 1/3 × A × H

V = one/3 × twoscore × 14 = 560/iii = 186.67 cm³

Hence, the volume of the given rectangular pyramid is 186.67 cm³.

Trouble 5: What is the book of a hexagonal pyramid if the sides of a base are 8 cm each and the pinnacle of the pyramid is fifteen cm?

Solution:

Given data,

Superlative of the pyramid = xv cm

Length of the side of the base of a hexagonal pyramid = half dozen cm

Area of the hexagonal base (A) = iii√3/2 a² = three√3/two (6)² = 54√3 cm²

The volume of a hexagonal pyramid (Five) = one/3 × A × H

Five = 1/iii × 54√three × 15 = 270√3 cm³

Hence, the volume of the given hexagonal pyramid is 270√3 cm³.

Problem 6: What is the volume of a pentagonal pyramid if the base area is 150 cmⁱⁱ and the superlative of the pyramid is 11 cm?

Solution:

Given data,

Expanse of the pentagonal base of operations = 150 cm^two

Elevation of the pyramid = eleven cm

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

V = ane/iii × 150 × xi = 550 cm³

Hence, the volume of the given pentagonal pyramid = 550 cmⁱⁱⁱ