Square Pyramid

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The name pyramid brings in our minds the images of the “Great Pyramids of Egypt”. The pyramids of Giza are from more than 2500 BC and are a perfect example of a square pyramid as they have a square base.

Pyramids are characterized by a unique geometric shape whose shape allows its weight to be distributed evenly across the structure. A pyramid is a shape that can fit all shapes inside of it (triangle, square, rectangle, etc.). In geometry, a pyramid is also called a polyhedron. It is made by connecting a polygon base (it is a 2-D shape with 3 or more straight sides) with an apex (a point at the top).

A pyramid can have a base of any shape; square, triangle, rectangle, and other shapes. The interesting thing about a pyramid is that each of its sides (each base edge and its apex) form a triangle.

What is a Square Pyramid

As discussed above, a pyramid can have a base of different shapes; the shape of its base gives the pyramid its name. So you can now figure out that Square pyramids are also polyhedrons (more specifically, a pentahedron i.e. a polyhedron with 5 sides) like other pyramids with a square base.

Details of a square pyramid face edges vertices are as follows:

• It has five faces.

• Four of its side faces are triangles.

• It has five vertices or corner points.

• It has eight edges.

Definition of a Square-based Pyramid:

It is a three-dimensional (3D) geometrical figure with a square base and four triangular faces or sides that meet at the apex or a single point.

• Equilateral square pyramid - If all the four triangular faces have the same edges, then such a square pyramid is termed as an equilateral square pyramid

• Right angle square pyramid - If the top point of the pyramid (apex) is right above the centre of its base such that a straight line from the apex cuts the base perpendicularly, then it is called a right angle square pyramid.

Formulas for a Square Pyramid

We have mentioned here some of the important formulas related to a square pyramid:

• Surface area = Base area + ½ * perimeter * slant height = x2 + 2x * $\sqrt{(x^{2}/4 + h^{2})}$

• Volume = ⅓ * base area * height = ⅓ * x2 * h

• Slant height = $\sqrt{h^{2} + x^{2}/4}$

In the equation above: x  = length of the side of the base square

h = height or altitude of the pyramid

Square Pyramid Net

A net diagram gives a flattened view of any solid shape by showing each face and base with all of their dimensions.

The pyramid net of a square-based pyramid is its 2D faces laid out, that can be folded to create its 3D shape. You can imagine it as unfolding a pyramid and making it completely flat. The image below shows how it would look like. The four sides can be folded here to recreate the pyramid. A pyramid net helps you see each face of the pyramid with clarity.

A square pyramid net is helpful in teaching 3D shapes in mathematics.

How to Draw a Square Pyramid

Since pyramids are 3-D structures, they could be a little difficult to draw on paper. It requires accuracy of angles so that it looks like a 3-D shape on paper. Here we will look at the process of drawing a square pyramid step-wise. Let us first see what all we need to draw the pyramid:

• A ruler

• A pencil

• A rubber eraser

1. Step 1 - Choose the size of the pyramid you want to make, for this example let it be a square with side 5 cms.

2. Step 2 - Draw a 5 cm line at the base of the pyramid. Use your compass to measure from the point to the pencil at 5 cms.

3. Step 3 - Now put your compass at one end of the baseline and draw a circle. Repeat the same thing at the other end of the baseline. If this is done correctly then you should see both the lines crossing in the middle.

4. Step 4 - Connect the two ends from the baseline to the top so that you end up with a triangle as shown in the image below:

5. Step 5 - Now rub out the lines you drew in step 3.

6. Step 6 - To one side of this triangle, draw an extension making sure the baseline of that side is above the first baseline you drew. You can get clarity with the image below:

Q1: What is Johnson's Solid?

Ans: In a square based pyramid, if all of its triangular sides are equilateral, i.e., all the sides and angles of the triangles are equal, then all the edges of the square pyramid will also be equal. This is called a Jhonson’s solid, which in geometry is a convex polyhedron. The name is given by Norman Jhonson, who published a list of 92 such solids in 1966. Johnson’s solid applies to other geometric figures also provided they are convex polyhedrons that are not uniform but have regular polygon faces.

Q2: Apart from Square Pyramids, What are the Other Types of Pyramids and their Features?

Ans: The other types of pyramids apart from the square pyramid are:

Triangular Pyramids - Triangular pyramids have a triangular base as well as triangular faces. Its key features are:

• It has 4 faces

• Three of the side faces are triangles.

• It has a triangular base.

• It has 4 vertices or common points.

• It has 6 edges.

• It is a tetrahedron.

Pentagonal Pyramids - A pentagonal pyramid has a pentagonal base and triangular faces. Its key properties are:

• It has 6 faces.

• Its 5 side faces are triangular in shape

• Its base is a pentagon.

• It has 6 vertices or corner points.

• It has 10 edges.

Q3: Differentiate Between :

1. Right pyramid and oblique pyramid

2. Regular pyramid and Irregular pyramid

Ans:

1. The position of the apex in a pyramid makes it either a right or an oblique pyramid. In the right pyramid, the apex or top point is directly above the base, while in an oblique pyramid, it is not.