What are Pyramids?
Pyramids are three-dimensional figures with a polygon shaped base and triangular lateral faces. The lateral faces meet at the common point known as the apex of the pyramid. The lateral triangular faces connect the edges of the base to the apex.
One of the most popular examples of pyramids is the Great Pyramid of Giza.
Properties of a Pyramid
- There are no parallel faces in a pyramid.
- The height of each lateral triangle is known as the slant height of the pyramid.
A pyramid with an ‘n-sided’ base will have:
- n + 1 faces
- n + 1 vertices
- 2n edges
What are the Different Types of Pyramids?
Pyramids can be classified based upon the shape of their base. As already mentioned the base of a pyramid is a polygon, so if the base is a triangle then the pyramid is known as a triangular pyramid. Similarly if the base is a square then the pyramid is known as a square pyramid and so on.
- Triangular pyramid – The pyramid has a triangular shaped base (a base with three sides). The triangular pyramid has 4 faces, 4 vertices, and 6 edges.
- Square pyramid – The pyramid has a square shaped base (a base with four sides). The square pyramid has 5 faces, 5 vertices, and 8 edges.
- Pentagonal pyramid – The pyramid has a pentagon shaped base (a base with five sides). The pentagonal pyramid has 6 faces, 6 vertices, and 10 edges.
Right Pyramid vs Oblique Pyramid
In a right pyramid, the apex lies exactly over the center of the base.
In an oblique pyramid, the apex lies slightly away from the center of the base.
Regular vs Irregular Pyramid
Any pyramid having a base in the shape of a regular polygon is termed as a regular pyramid whereas a pyramid with an irregular polygon shaped base is termed as an irregular pyramid.
[Note: All of the faces of a regular pyramid are equilateral triangles. ]
What is the Surface Area of a Pyramid?
The surface area of a pyramid is the sum of the area of the base and the areas of the lateral faces.
The formula for calculating the total surface area of a pyramid is:
Total surface area of the pyramid = Area of the base + Areas of the lateral faces.
[Note: Since the shape of the lateral faces are triangular their area can be determined by applying the formula for the area of a triangle.]
What is the Volume of a Pyramid?
The space occupied within the faces of a pyramid is referred to as its volume. The volume is measured in cubic units and the formula is given by:
Volume of a pyramid = \(\frac{1}{3}\) × Base area × Height
The height is measured from the base to the apex.
Solved Pyramid Examples
Example 1: Determine the total surface area of a square pyramid with slant height of 18 cm if each side of the base measures 15 cm.
Answer:
The surface area of the pyramid, S = Area of the base + Areas of the lateral faces.
Area of the square base = 15 x 15 = 225 square cm
Area of the triangular lateral face: \(\frac{1}{2}\) x 15 x 18 = 135 square cm
Hence,
S = 225 + 135 + 135 + 135 + 135 [There are four identical lateral faces]
S = 765
Therefore, the total surface area of the square pyramid is 765 square cm.
Example 2: A replica of the Pyramid of Giza was installed in Alice’s neighborhood. The dimensions of the replica pyramid are 54 square inches base and its height is 15 in. Find the volume of the replica pyramid.
Answer: Here, base area = 54 square inches and height = 15 in.
Volume of a pyramid = \(\frac{1}{2}\) × Base area × height
= \(\frac{1}{2}\) x 54 x 15 [Substitute the given values]
= 54 x 5
= 270
Thus the volume of the replica pyramid is 270 cubic inches.
A pyramid can be classified as a triangular pyramid, square pyramid, pentagonal pyramid, and so on, depending on the shape of the pyramid.
A tetrahedron pyramid is a kind of pyramid where all its faces along with its base are triangular in shape.It is also known as a triangular pyramid.
A 7 sided pyramid is called a heptagonal pyramid where the base is heptagonal in shape and the remaining faces are triangles.
The net of a pyramid can be obtained when it is flattened and opened up on a two-dimensional surface. For example the net of a square pyramid will show a square shaped base along with four triangular faces. The net of a pyramid can be used to determine its surface area.
