What are Pyramids?
A pyramid is a three-dimensional structure with a polygon-shaped base.
You must have heard about the pyramids that have been built by civilizations all over the world. For example, the Great Pyramid of Cholula, in the Mexican state of Puebla, is the largest pyramid by volume amongst the pyramids found in the world.
The pyramids have been the largest man-made structures on Earth for thousands of years. First, it was the Red Pyramid in the Dahshur Necropolis, and later, the Great Pyramid of Khufu. Both are in Egypt, and the latter is the only one of the Seven Wonders of the Ancient World still standing.
What is the Shape of a Pyramid
A pyramid is a three-dimensional structure with triangular outer surfaces that converge to a single point at the top, forming a shape like the pyramids of ancient times. The base of a pyramid can be a triangle, a quadrilateral, or any other polygonal shape. As a result, a pyramid must have at least three triangular outer surfaces, or at least four faces that include the base.
A pyramid that has a base in the shape of a regular polygon is called a regular pyramid. If the base is in the shape of a non-regular polygon, the pyramid is known as a non-regular pyramid.
Parts of a Pyramid
Apex: A pyramid is a three-dimensional shape with a polygonal base and triangle-shaped faces on its side that meet at a point called the apex, or vertex.
Altitude: The altitude or height of the pyramid is the perpendicular distance between the apex and the center of the base.
Slant height: The slant height is the length of the perpendicular drawn from the apex to the base of one of the triangular side faces of a pyramid. The slant height of a regular pyramid is the height of one of its lateral faces. The slant height is not defined in a non-regular pyramid.
Base: This refers to the shape of the pyramid’s base. A triangular pyramid, for example, has a triangle-shaped base, while a hexagonal pyramid has a hexagonal base.
Face: The faces of a pyramid include the base and the outer lateral faces. The lateral faces are triangular in shape.
A pyramid that has a base in the shape of a regular polygon is called a regular pyramid. If the base is in the shape of a non-regular polygon, the pyramid is known as a non-regular pyramid.
The Net of a Pyramid
The net of a square pyramid, as shown in the figure, can be obtained if we imagine unfolding the figure by separating the lateral faces from the apex. On flattening the lateral faces, we can observe that the net of this pyramid consists of one square on the bottom of the pyramid, and 4 triangles as the sides of the pyramid . We can also obtain the net for a triangular pyramid, a rectangular pyramid, a pentagonal pyramid, and so on.
A pyramid that has a base in the shape of a regular polygon is called a regular pyramid. If the base is in the shape of a non-regular polygon, the pyramid is known as a non-regular pyramid.
Types of a Pyramid and their Nets
Pyramids are classified on the basis of the shape of their base (triangular, rectangle, pentagon, and hexagonal pyramids).
Triangular pyramid:
The base of a triangular pyramid is a triangle, and the lateral faces are triangles as well. Its net is made up of four triangles. The triangular pyramid has 4 faces, 4 vertices, and 6 edges.
A pyramid that has a base in the shape of a regular polygon is called a regular pyramid. If the base is in the shape of a non-regular polygon, the pyramid is known as a non-regular pyramid.
The net of a triangular prism is as shown in the diagram.
Rectangular pyramid:
The base of a rectangular pyramid base is a rectangle, and the lateral faces are triangles. Its net is made up of one rectangle and four triangles. It has 5 faces, 5 vertices, and 6 edges.
This is the net of the rectangular pyramid.
Pentagonal pyramid:
The base of a pentagonal pyramid is a pentagon, and the lateral faces are triangles. Its net is made up of one pentagon and five triangles. It has 6 faces, 6 vertices, and 10 edges.
This is the net of the pentagonal pyramid.
Hexagonal pyramid:
The hexagon serves as the base of a hexagonal pyramid, while triangles serve as its lateral faces. Its net is made up of one hexagon and six triangles. It has 7 faces, 7 vertices, and 12 edges.
The diagram shown here is the net of the hexagonal pyramid.
Square pyramid:
The base of a square pyramid is a square, and the lateral faces are triangles. Its net is made up of four triangles and one square. It has 5 faces, 5 vertices, and 6 edges.
This is the net of a square pyramid.
What is Surface Area of a Pyramid?
The surface area of a pyramid:
The total area occupied by all faces of a pyramid is its surface area. It is measured in square units like \(m^2,cm^2,in^2,ft^2\), and so on.
The term used to denote the sum of the areas of the side faces of a pyramid (triangles) is the lateral surface area or LSA in the abbreviated form.
The total surface area or the TSA of a pyramid is equal to the LSA of the pyramid, in addition to the area of the base.
There are some specialized formulas to obtain the lateral surface area and total surface area for a regular pyramid. Consider a regular pyramid, which has a square base and triangular sides with a base perimeter of “p units”, a base area of “B unit square”, and a slant height of “l unit”.
Then, the lateral surface area of the pyramid (LSA)\(=\frac{1}{2}\times p\times l\)
The total surface area of the pyramid (TSA)\(=LSA+base~area=\left(\frac{1}{2}\times p\times l\right)+B\)
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The Surface Area of a Pyramid using Nets
The net of a pyramid can also be used to determine its surface area.
Let us consider a square pyramid with a slant height of 5 m and a square base with a side of 3 m. Find its total surface area using a net.
Unfold and flatten the pyramid as shown.
A pyramid that has a base in the shape of a regular polygon is called a regular pyramid. If the base is in the shape of a non-regular polygon, the pyramid is known as a non-regular pyramid.
Bottom face: \(3\times 3=9\)
Side face: \(\frac{1}{2}\times 3\times 5=7.5\)
Side face: \(\frac{1}{2}\times 3\times 5=7.5\)
Side face: \(\frac{1}{2}\times 3\times 5=7.5\)
Side face: \(\frac{1}{2}\times 3\times 5=7.5\)
Find the sum of the areas of the faces,
The surface area of the square pyramid = Area of the bottom + Area of a side + Area of a side + Area of a side + Area of a side.
s = 9 + 7.5 + 7.5 + 7.5 + 7.5
s = 39
So, the surface area is 39 square meters.
Surface Area of a Pyramid Examples
Example 1: Determine the surface area of the square pyramid as shown in the diagram.
Solution: Use a net to find the area of each face.
A pyramid that has a base in the shape of a regular polygon is called a regular pyramid. If the base is in the shape of a non-regular polygon, the pyramid is known as a non-regular pyramid.
Bottom face: \(8\times 8=64\)
Side face: \(\frac{1}{2}\times 8\times 10=40\)
Side face: \(\frac{1}{2}\times 8\times 10=40\)
Side face: \(\frac{1}{2}\times 8\times 10=40\)
Side face: \(\frac{1}{2}\times 8\times 10=40\)
Find the sum of the areas of the faces.
The surface area of the square pyramid = Area of the bottom + Area of a side + Area of a side + Area of a side + Area of a side.
s = 64 + 40 + 40 + 40 + 40
s = 224
So, the surface area is 224 square meters.
Example 2: Determine the surface area of the triangular pyramid given in the diagram.
Solution: Use a net to find the area of each face.
Bottom face: \(\frac{1}{2}\times 9\times 6=27\)
Side face: \(\frac{1}{2}\times 9\times 11=49.5\)
Side face: \(\frac{1}{2}\times 9\times 11=49.5\)
Side face: \(\frac{1}{2}\times 9\times 11=49.5\)
Side face: \(\frac{1}{2}\times 9\times 11=49.5\)
Find the sum of the areas of the faces.
The surface area of the square pyramid = Area of the bottom + Area of a side + Area of a side + Area of a side + Area of a side.
s = 27 + 49.5 + 49.5 + 49.5 + 49.5
s = 225
So, the surface area is 225 square inches.
Example 3: A toy block is in the shape of a regular pyramid with a square base. The manufacturer wants its entire surface to be painted blue. How many square centimeters will be painted?
Solution: The surface area of the pyramid will give the area in square centimeters to be painted. Hence, we will calculate the total surface area of the pyramid. Start with using a net to find the area of each face.
Bottom face: \(4\times 4=16\)
Side face: \(\frac{1}{2}\times 4\times 7=14\)
Side face: \(\frac{1}{2}\times 4\times 7=14\)
Side face: \(\frac{1}{2}\times 4\times 7=14\)
Side face: \(\frac{1}{2}\times 4\times 7=14\)
Find the sum of the areas of the faces.
The surface area of the square pyramid = Area of the bottom + Area of a side + Area of a side + Area of a side + Area of a side.
s = 16 + 14 + 14 + 14 + 14
s = 72
So, 72 square centimeters of the area of the toy pyramid will need to be painted.
The lateral surface area of a regular pyramid is equal to the sum of the areas of its lateral faces. For a regular pyramid, the total surface area is equal to the sum of the areas of its lateral faces and the area of the base. However, for non-regular pyramids, the slant height is unknown, so there is no formula for calculating their surface area.
The difference between a square pyramid and a triangular pyramid is that the base of a square pyramid is a square and hence has 4 lateral faces. In comparison, the base of a triangular pyramid is triangular and, hence, has 3 lateral faces.