What Are the Types of Pyramids?
As we can see in the image there are multiple types of pyramids. Starting with the triangular pyramid, with a base and the four faces that are triangles. It has four vertices and six edges.
Next, is the most popular type of pyramid as seen in Giza, Egypt. This is a square pyramid with a square base, five faces, five vertices and eight edges. The third type of pyramid is a rectangular pyramid with the same number of faces, vertices and edges as a square pyramid. It needs to be seen that the number of sides of the base of a pyramid can be increased such as a pentagonal or a hexagonal pyramid, with a corresponding increase in the number of faces, vertices and edges.
What Is a Cross Section?
We often see slices of bread, and each slice is actually a cross section of the solid loaf of bread. A three dimensional loaf of bread when sliced results in a two dimensional shape, usually in the form of a square. So the term cross section is the two dimensional shape obtained when a plane intersects a three dimensional solid. Let’s take the example of a plane slicing through a cone.
If the plane intersects the cone perpendicular to the base, then we obtain a triangle as the cross section. Rotating the plane such that it is parallel to the base results in circles of varying sizes. Rotating the plane either towards or away from the base results in the elliptical cross sections.
What Is the Cross Section of Pyramids?
We can start this section by observing the two types of planes that can intersect a pyramid. The vertical plane intersects the pyramid to result in the formation of triangles as cross sections. A horizontal plane intersects the pyramid to form a square, or a polygon similar to the base of the pyramid.
It must be noted that as the plane is moved on either side of the apex when vertically aligned with the pyramid, the shape of the cross section keeps changing and will result in a trapezoid as shown in the image.
Solved Examples
Example 1: Identify the type of pyramid and the shape of the cross section in the image.
Solution:
In the image provided in the problem, there are six sides to the polygon that forms the base of the pyramid.
Hence, this is an example of a hexagonal pyramid.
The cross section depicts a flat shape that has two parallel sides, which is a trapezoid.
Example 2: A cross section is made to a rectangular pyramid as shown in the image.
Given the dimensions of the cross section, what is the area and perimeter of the cross section?
Solution:
As shown in the image, the dimensions of cross section is 3.5 mm and 2 mm.
The shape of the cross section is a rectangle.
To calculate area:
\(Area\text{ }of\text{ }rectangle\text{ }A = l \times w\) [Write formula for area of rectangle]
\(Area = 3.5 \times 2 = 7\text{ }mm^{2}\) [Substitute and simplify]
Hence, the area of the cross section is 7 square millimeters.
To calculate perimeter:
Perimeter = 2(l + w) [Write formula for the perimeter of rectangle]
Perimeter = 2(3.5 + 2) = 11 mm [Substitute and simplify]
Hence the perimeter of the cross section is 11 millimeters.
Example 3: A plane intersects a square pyramid as shown in the image. What is the shape of the cross section? Select the correct option.
Solution
As the plane intersecting the pyramid is neither perpendicular nor parallel to the base, the cross section will neither be a triangle or a square.
In this case the cross section will be a trapezoid.
Hence option B is the correct answer.
When a plane intersects a sphere, we always obtain a circle shaped cross section.
The difference between regular and irregular pyramids depends on the base of either type of pyramid. In the case of a regular pyramid the base is a regular polygon with equal sides. But in the case of an irregular pyramid, the polygon forming the base is an irregular polygon.
The formula to calculate the volume of a pyramid is one thirds the base area multiplied by the height of the pyramid.
The difference between the right and oblique types of pyramids is dependent on the position of the apex of each of these. In a right pyramid the apex is directly above the center of the base. However, the apex is not directly above the center of the base in an oblique pyramid.