Introduction to Pyramids
The history of pyramids dates back a few thousand years. The word pyramid comes from the Greek word “pyramis”. The ancient Egyptians first built pyramids around 2700 BCE until 1700 BCE. The pyramids are huge structures that are built of bricks, stones, and other building materials.
The outer surface of the pyramid has triangular sides and these surfaces gradually make a single sharp vertex. The base of pyramids are triangular, quadrilateral in shape or it could be in the shape of any polygon. The pyramids of Egypt are on the list of the Seven Wonders of the World. These pyramids are the largest construction projects in the world. But the world’s largest pyramid by volume is the great pyramid of Cholula in Mexico.
What are the Pyramids?
Pyramids are three dimensional shapes or structures whose base is a polygon and all the lateral surfaces are triangular in shape. The triangular faces meet at the top and make a vertex. Pyramids may have different polygonal bases, for example, a square base, a triangular base, or even a hexagonal base. If the base of the pyramid is a regular polygon then the pyramid is said to be a regular pyramid.
The vertex is called the apex of a pyramid. The height of the pyramid is the perpendicular distance from the apex to the base of the pyramid. The height of the triangular faces is called the slant height of the pyramid.
If there are N numbers of sides in the base of the pyramid then the number of faces will be N + 1, the number of edges will be 2N, and the number of vertices will be N + 1 in that pyramid.
What is the Volume of a Pyramid?
The volume of a pyramid refers to the space or air enclosed within its surfaces. It is measured in cubic units like cubic meters, cubic inches or cubic feet. To find the volume of the prism we take a rectangular pyramid and a prism of the same height and base. If we fill the pyramid with water and later pour that amount of water into the rectangular prism of the same height and base, then we can observe that the prism will be filled to one thirds of its height.
There is also another way to guess the volume of a pyramid. We first take a solid rectangular prism and then can cut this prism into three pyramids of equal volume.
Thus, we can say that the volume of any pyramid is one-third of the volume of a prism whose base is congruent to the base of the pyramid and height is equal to the height of the pyramid.
Read More:
Formula for the Volume of Pyramid
The volume (V) of a pyramid is one-third the product of the area of the base and the height of the pyramid.
The algebraic formula for the volume of a pyramid is,
\( V=\frac{1}{3}Bh \)
Where ‘B’ is the area of the base and ‘h’ is the height of the pyramid.
There are a few special types of pyramids such as square pyramids, rectangular pyramids and so on. We can directly find the volume of pyramids using the formulas as mentioned in the given table to calculate the volume of different types of pyramids.
Volume of a Pyramid Examples
1. Find the volume of a pyramid whose height is 9 inches and base area is 45 square inches.
Solution:
Volume of pyramid \( (V)=\frac{1}{3}\times B\times h \)
\( =\frac{1}{3}\times 45\times 9 \) [Substitute the values of B and h]
\( =135 \) [Multiply]
The volume of the pyramid is about 135 cubic inches.
2. Find the volume of the pyramid in the image provided here.
Solution:
Volume of pyramid \( (V)=\frac{1}{3}\times B\times h \)
\( =\frac{1}{3}\times 21\times 6 \) [Substitute the values of B and h]
\( =42 \) [Multiply]
The volume of the pyramid is 42 cubic feet.
3. Find the volume of this pyramid.
Solution:
Volume of pyramid \( (V)=\frac{1}{3}\times B\times h \)
\( =\frac{1}{3}\times 27\times 5 \) [Substitute the values of B and h]
\( =45 \) [Multiply]
The volume of the pyramid is 45 cubic centimeters.
4. Gary has built two sand castles in the form of pyramids as shown in the figure. Find out which castle among the two required more sand to be built.
Solution:
Two castles are made of sand. The more the volume of the sand castle, the more sand is required to make the sand castle.
Therefore the volume of pyramid 1 is,
Volume of pyramid \( (V)=\frac{1}{3}\times B\times h \)
\( =\frac{1}{3}\times (30)\times 6 \) [Replace B with 30 and h with 6]
= 60 [Simplify]
The volume of pyramid 2 is,
Volume \( (V)=\frac{1}{3}\times B\times h \)
\( =\frac{1}{3}\times (24)\times 8 \) [Replace B with 30 and h with 6]
= 64 [Simplify]
The volume of the second spire is more. Therefore, more sand is required to build the second sand castle.
5. Find the volume of the tetrahedron given in the image.
Solution:
Volume of tetrahedron \( (V)=\frac{a^3}{6\sqrt{2}} \)
\(=\frac{(12)^3}{6\sqrt{2}}\) [Substitute the values of a]
\( \approx 203.67 \) [Multiply]
The volume of the tetrahedron is 203.67 cubic inches.
The volume of a pyramid is the space occupied by the pyramid. The volume of the pyramid whose height is ‘h’ and Base area ‘B’ is (1/3)×B×h Cubic units.
A tetrahedron is a three dimensional solid that has four triangular faces. It is also a triangular pyramid. It has four triangular faces, four vertices, and six edges.