Square Pyramid
A square pyramid is a three dimensional figure with a square base and four triangular lateral faces. It has a total of five faces, hence it is also called a pentahedron.
All triangular faces meet at a point called the apex of the pyramid. The most relatable real life example of a square pyramid is the pyramid of Giza.
- If the apex of a pyramid is right above the center of base, it is called the right pyramid.
- The perpendicular distance from the apex to the base of a pyramid is called its altitude.
- The perpendicular distance from apex to the base along the lateral face of the pyramid is called its slant height.
Properties of a Square Pyramid
- It has four triangular faces.
- It has a square base.
- It has five vertices.
- It has eight edges.
Types of Square Pyramids
There are two types of square pyramids:
- Right square pyramids: In a right square pyramid, the apex lies exactly over the center of the square base.
- Oblique square pyramids: In an oblique square pyramid, the apex lies slightly away from the center of the square base.
Formulas used for Pyramids
We can calculate the volume, total surface area and slant height of a pyramid using these formulas:
- Volume of pyramid, \(V=\frac{1}{3}Bh\)
- Surface Area, S = Area of base + Areas of lateral surfaces
- Slant height, \(l=\sqrt{h^2~+~\left(\frac{s}{2}\right)^2}\)
Where,
- B is the area of the base of the pyramid. For a square pyramid the base is square shaped hence the base area is given by side 2
- h is the height of the pyramid
- And, s is the side or edge length of the square base.
[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.]
Net of Square Pyramid
The net of a pyramid can be obtained when it is flattened and opened up on a two-dimensional surface. Here the net of a square pyramid will give us an accurate view of the square base and four triangular lateral faces.
The net of a pyramid gives a better understanding of the total surface area of the pyramid.
Solved Square Pyramid Examples
Example 1: A square pyramid shaped box has an edge length of 10 inch and height of 15 inch. Find the volume of the pyramid.
Solution:
\(V=\frac{1}{3}Bh\) Write the formula for volume of the pyramid
\(V=\frac{1}{3}~\times~10^2~\times~15\) Substitute 102 for B and 15 for h
\(V=\frac{1500}{3}\) Multiply
\(V=500\) Divide
So, the volume of the box is 500 cubic inches.
Example 2: Find the surface area of the wooden roof structure shown in the image below.
Solution:
The wooden roof structure is in the shape of a square pyramid with triangular lateral faces.
So the area of the wooden structure will be the area of the lateral faces.
Area of the wooden roof structure = 4 x area of triangular lateral face [There are 4 triangular lateral faces]
⇒ \(4~\times~\frac{1}{2}~\times~10~\times~12\) [Area of triangle formula]
⇒ 240
So, the area of the wooden roof structure is 240 square feet.
Example 3: Find the slant height of a square pyramid whose height and edge length are 9 cm and 6 cm respectively.
Solution:
\(l=\sqrt{h^2~+~\left(\frac{s}{2}\right)^2}\) Write the formula for slant height
\(l=\sqrt{9^2~+~\left(\frac{6}{2}\right)^2}\) Substitute 6 for s and 9 for h
\(l=\sqrt{81~+~9}\) Simplify
\(l=\sqrt{90}\) Add
\(l=3\sqrt{10}\) Positive square root of 90
So, the slant height is \(3\sqrt{10}\) cm.
A pyramid with a square base is called a square pyramid.
There are five faces in a square pyramid.
The lateral faces are triangular shaped in pyramids.
There is only one base in the pyramid.
A prism has rectangular shaped lateral faces whereas a pyramid has triangular faces.