Square Prism
A square prism is a three dimensional figure. It has two square bases and four parallelogram shaped faces. A Rubik’s cube and a bar magnet are some real life examples of objects that are shaped like a square prism.
Properties of a Square Prism
- A square prism has 6 faces, 12 edges and 8 vertices.
- Opposite sides or edges of a square prism are congruent and parallel to one another.
- Each angle of a right square prism is right angle.
- Diagonals of a right square prism are equal in length.
Types of Square Prisms
- Right square prism
- Oblique prism
1. Right square prism:
If the lateral faces of a prism are perpendicular to the square bases, it is called a right square prism. In other words, if the lateral surfaces are rectangular in shape, the prism is a right square prism.
2. Oblique prism:
If the lateral faces and square bases are not perpendicular, it is called an oblique prism. In other words, the faces of an oblique prism are parallelograms in shape.
Surface Area and Volume of a Square Prism
1. Surface area of square prism:
The surface area of a prism is the sum of the area of its bases and lateral faces. Here the surface area S of a square prism is the sum of the area of its two square bases and four rectangular faces.
Surface area is measured in square units.
\(S=2~\times~area~of~square~base~+~4~\times~area~of~rectangular~face\)
\(S=2a^2+~4ah\) square units.
Where a is the edge length of square base and h is the height of the square prism.
2. Volume of square prism:
Volume is the amount of space occupied by a solid. Here the volume of a square prism is equal to the number of unit cubes that can be filled in it. To calculate the volume of the square prism, multiply the area of base by the height of the prism.
Volume of square prism, \(V=area~of~square~base~\times~height~of~prism\)
\(V=a^2h\) cubic units.
Note : The volume of oblique prisms is calculated in the same way as the volume of the right prism.
Net of a Square Prism
When all the faces of a prism are separated along the edge and flattened to make a two dimensional figure, the figure is called the net of the prism. Here the net of a square prism will give us an accurate view of square and rectangular surfaces.
Solved Square Prism Examples
Example 1: A square prism shaped box has an edge length as 15 cm and height as 50 cm. Find the total surface area of the box.
Solution:
\(S=2a^2~+~4ah\) Write the formula for surface area of a square prism
\(S=2~\times~15^2~+~4~\times~15~\times~50\) Substitute 15 for a and 50 for h
\(S=450~+~3000\) Multiply
\(S=3450\) Add
So, the total surface area of the box is 3450 square centimeters.
Example 2: A fish tank is in the shape of a square prism with base edge length as 1.5 ft. and height as 5 ft. Find the volume of the fish tank.
Solution:
\(V=a^2h\) Write the formula for the volume of a square prism
\(V=1.5^2~\times~5\) Substitute 1.5 for a and 5 for h
\(V=11.25\) Multiply
So, the volume of the fish tank is 11.25 cubic feet.
Example 3: Find the height of the square prism if the edge length is 5 inch and volume is 250 cubic inches.
Solution:
\(V=a^2h\) Write the formula for the volume of a square prism
\(250=5^2~\times~h\) Substitute 5 for a and 250 for V
\(250=25~\times~h\) Square of 5 is 25
\(\frac{250}{25}=h\) Divide both sides by 25
\(10=h\)
So, the height of the square prism is 10 inches.
There are six faces in a square prism.
There are two types of square prisms:
- Right square prisms
- Oblique square prisms
The volume of the prism is equal to the product of area of its base and height.
Yes, all cubes are square prisms but not vice versa.
There are two square bases in a square prism.