In geometry, you will come across with various three-dimensional shapes which are also known as solid shapes or simply solids. Few most common examples are cube, cuboid, cone, pyramid, cylinder, prism, sphere, hemisphere. Lets we talk further about the prisms. In this lesson, we will focus on square prisms only.

A square prism is a three dimensional cuboid figure whose bases are squares. In a square prism, the opposite sides and angles are congruent.

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They are the solid figures having two identical bases that are joined together through flat lateral faces and having same cross section allover. There are variety of prisms found in geometry. They include square prism, rectangular prism, triangular prism and other polygonal prisms.

Below is a picture of square prism.

A prism whose lateral faces are perpendicular to its base is referred as right square prism otherwise oblique. It can also be described as a right prism with square bases.

A right square prism which has square lateral surfaces is a cube.

### Volume of a Square Prism Formula:

Let us find the lateral surface area of the oblique square prism whose base square side is 7 m and the height of the oblique square is 11 m.

Lateral surface area of the oblique square prism = Ph square units

where, P is the perimeter a base of the prism and h is the height of the oblique square prism

Perimeter of the square = 4 Side units = 4 $\times$ 7 = 28

Height of the oblique square prism = 11 m

Lateral surface area of the oblique square prism = 28 $ \times$ 11 = 308

Lateral surface area of the oblique square prism is 308 m$^2$

Given below are some examples on square prism.### Solved Examples

**Question 1: **Find the surface area and volume of a square prism with the side 4 and height 8.

** Solution: **
**Question 2: **Find the base edge of a square prism with height of 3 m and lateral area of 144 m.

** Solution: **

Let s be the base edge of a prism.

Perimeter of square base = 4s

Lateral Area of a square prism = Height of prism x (Perimeter of base)

144 = 3 x (4s)

12 = side

Ans: The base edge of a square prism is 12 m.

Below is a picture of square prism.

A prism whose lateral faces are perpendicular to its base is referred as right square prism otherwise oblique. It can also be described as a right prism with square bases.

**Prisms are of two types:**

- Right Prism
- Oblique Prism

Generally, we consider a prism as a right prism.

A right square prism which has square lateral surfaces is a cube.

There are a number of properties of a square prism. Some of them are as follows,

In a square prism,

- Opposite sides are parallel.
- Opposite sides are congruent.
- Opposite angles are congruent.
- Consecutive angles are supplementary.
- Diagonals bisect each other.

**Surface Area of a Square Prism** is the measure of how much exposed area the solid has. Surface Area of a square prism is expressed in square units. Surface area of prism is the sum of the lateral surface area and twice the base area of the prism.

The surface area of a square prism formula is as follows,

** Area of a Square Prism = 2A + Ph**

where, A = (side)$^2$ = Base area, P = 4 $\times$ side = Perimeter of a base of the prism and h - Height of prism.

Volume of a square prism is a measurement of the occupied units of the solid. The volume of a square prism is the number of units used to fill a cube and is represented by cubic units like cubic centimeter, cubic millimeter and so on.

The volume of a square prism formula is as follows,

**Volume of a square Prism = Area of base $\times$ Height of prism **

If s is the side of the base and h is the height of the prism, then

Area of square base = $s^2$

and volume = $s^2$ h

**Formula for Volume of a square Prism = $s^2$ h**

where, s - side of base and h - height of prism

Oblique prism is a closed geometrical figure having two or three polygons. The prism is an oblique square prism if its lateral edges are oblique to the base edges of the square base.Let us find the lateral surface area of the oblique square prism whose base square side is 7 m and the height of the oblique square is 11 m.

Lateral surface area of the oblique square prism = Ph square units

where, P is the perimeter a base of the prism and h is the height of the oblique square prism

Perimeter of the square = 4 Side units = 4 $\times$ 7 = 28

Height of the oblique square prism = 11 m

Lateral surface area of the oblique square prism = 28 $ \times$ 11 = 308

Lateral surface area of the oblique square prism is 308 m$^2$

Given below are some examples on square prism.

Surface Area of a Prism, SA = 2A + Ph

Base Area, A = side $\times$ side = 4 $\times$ 4 = 16

Perimeter of base = 4 $\times$ side = 4 $\times$ 4 = 16

SA = (2 $\times$ 16) + (16 $\times$ 4)= 32 + 64 = 96

Surface area of square prism is 96 square units.

Volume of Prism = Ah= 16 * 4= 64

Volume of square prism is 64 cubic units.

Base Area, A = side $\times$ side = 4 $\times$ 4 = 16

Perimeter of base = 4 $\times$ side = 4 $\times$ 4 = 16

SA = (2 $\times$ 16) + (16 $\times$ 4)= 32 + 64 = 96

Surface area of square prism is 96 square units.

Volume of Prism = Ah= 16 * 4= 64

Volume of square prism is 64 cubic units.

Let s be the base edge of a prism.

Perimeter of square base = 4s

Lateral Area of a square prism = Height of prism x (Perimeter of base)

144 = 3 x (4s)

12 = side

Ans: The base edge of a square prism is 12 m.

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