## Table of Contents

- What Is a Square Prism?
- Properties of a Square Prism
- Square Prism Examples
- Square Prism Formulas
- Lesson Summary

- What Is a Square Prism?
- Properties of a Square Prism
- Square Prism Examples
- Square Prism Formulas
- Lesson Summary

What is a prism? A **prism** is a three-dimensional solid with two congruent, parallel bases joined together by flat sides. A prism is named by the shape of its parallel bases, so a **square prism** is a prism with two congruent, parallel, square bases. A rectangular prism, on the other hand, has two congruent, parallel, *rectangular* bases. In the example, the square prism has two square bases (in red) and four rectangular sides (in yellow and blue). All square prisms are *not* cubes, although all cubes are square prisms.

What are the properties of a square prism? A square prism has:

**6 faces**: A face is a flat side on a three-dimensional solid shape. Opposite faces are congruent and parallel.**12 edges**: An edge is the line segment where two faces meet.**8 vertices**: The singular form of*vertices*is*vertex*. A vertex is a point, somewhat like a corner, where three or more edges meet.- 1 set of congruent, parallel, square bases: A square prism
*can*have more than two square bases, but there must be*at least two*.

The diagram shows several three-dimensional solids with edges and vertices labeled. There are two prisms, and one of them is a square prism. The 12 edges and 8 vertices on the square prism are marked.

The other two solids are a triangular prism, which has triangle bases, and a square pyramid. Notice how the pyramid has only one square base.

There are two kinds of square prisms:

- A
**right square prism**has sides that are perpendicular to its bases. This means the two bases will line up, one directly above the other. - In an
**oblique square prism**, the sides are not perpendicular to the prism's bases. Although the two bases will be congruent and parallel, one is not directly over the other.

The prisms shown in the diagram are hexagonal, not square, prisms, but they demonstrate the difference between a right prism and an oblique prism. The yellow prism on the left is a square prism, and the blue prism on the right is an oblique prism.

What are some real-world examples of square prisms?

A cube, such as these puzzle cubes, is an example of a square prism because it is a three-dimensional shape with at least two square sides.

This number cube is also a square prism.

This box of tissue is another example of a cube, which is a square prism.

These school lockers are square prisms because the bases of the lockers are squares.

A stack of several square crackers, one on top of the other, would make a square prism.

A closed pizza box is a square prism.

In fact, any kind of box might be a square prism, whether used for moving, holding jewelry, or storing compact discs.

Even a building can be a square prism, as long as the ground floor base is shaped like a square.

A square prism is a three-dimensional shape. Three-dimensional shapes can be constructed from flat, two-dimensional nets.

The net of a right square-based prism will consist of at least two squares. The other sides may be either square or rectangles.

The net of an oblique square-based prism will also consist of two squares, The other sides will be rhombuses or parallelograms.

What is the area of a square prism? Since a square prism is a three-dimensional shape, it has a surface area rather than an area. The **surface area** of a square pyramid is the area of the *outside* of the prism. To find the surface area of a prism, find the sum of the areas of the prism's faces.

Look at the nets of the square prisms. Each net consists of two square bases, and either a large rectangle composed of four smaller rectangles or a large parallelogram composed of four smaller parallelograms. The large rectangle or parallelogram composed of four smaller shapes represents the sides of the prism. To find the surface area, find the area of the two square bases, and add to that total the area of the large shape composed of the lateral sides. That larger shape has a width that is equal to the perimeter of the base of the square prism, and a length that is equal to the height of the prism. So, the formula for the surface area of a square prism is:

{eq}Surface\ area\ of\ square\ prism\ =\ 2B\ +\ Ph \\ B\ =\ area\ of\ square\ base,\ P\ =\ perimeter\ of\ square\ base,\ and\ h\ =\ height\ of\ prism {/eq}

Use this formula to find the surface area of a square prism with a base that is a square 10 inches by 10 inches, and a height of 25 inches.

{eq}B\ =\ area\ of\ square\ base,\ or\ 10\ \times\ 10\ =\ 100\ in^2 \\ P\ =\ perimeter\ of\ square\ base,\ or\ 10\ +\ 10\ +\ 10\ +\ 10\ =\ 40\ in \\ h\ =\ height\ of\ prism.\ or\ 25\ in \\ Surface\ area\ of\ square\ prism\ =\ 2B\ +\ Ph \\ SA\ =\ 2\ \times\ 100\ +\ 40\ \times\ 25 \\ SA\ =\ 200\ +\ 1,000 \\ SA\ =\ 1,200\ in^2 {/eq}

The prism has a surface area of 1,200 square inches.

What is the volume of a square prism? The **volume** of a square prism is the amount of space on the *inside* of the prism. To find the volume of a square prism, multiply the area of its square base times the prism's height. *Note:* This is the perpendicular height of the prism, rather than the slant height in the case of an oblique square prism. Use the formula:

{eq}Volume\ of\ square\ prism\ =\ Bh \\ B\ =\ area\ of\ square\ base\ and\ h\ =\ height\ of\ prism {/eq}

Use this formula to find the volume, or space inside, the square prism with a 10-inch by 10-inch square base and a height of 25 inches.

{eq}B\ =\ area\ of\ square\ base,\ or\ 10\ \times\ 10\ =\ 100\ in^2 \\ h\ =\ height\ of\ prism,\ or\ 25\ in \\ Volume\ of\ square\ prism\ =\ Bh \\ V\ =\ 100\ \times\ 25 \\ V\ =\ 2,500\ in^3 {/eq}

The prism has a volume of 2,500 cubic inches.

A **prism** is a three-dimensional solid with two congruent, parallel bases joined together by flat sides. A prism is named by the shape of its parallel bases, so a **square prism** is a prism with two congruent, parallel, square bases. A rectangular prism, on the other hand, has two congruent, parallel, *rectangular* bases. A square prism has **6 faces**, **12 edges**, **8 vertices**, and 1 set of congruent, parallel, square bases. There are two kinds of square prisms. A **right square prism** has sides that are perpendicular to its bases. This means the two bases will line up, one directly above the other. In an **oblique square prism**, the sides are not perpendicular to the prism's bases. Although the two bases will be congruent and parallel. one is not directly over the other.

The **surface area** of a square pyramid is the area of the *outside* of the prism. To find the surface area of a prism, find the sum of the areas of the prism's faces.

{eq}Surface\ area\ of\ square\ prism\ =\ 2B\ +\ Ph \\ B\ =\ area\ of\ square\ base,\ P\ =\ perimeter\ of\ square\ base,\ and\ h\ =\ height\ of\ prism {/eq}

The **volume** of a square prism is the amount of space on the *inside* of the prism. To find the volume of a square prism, multiply the area of its square base times the prism's height. *Note:* This is the perpendicular height of the prism, rather than the slant height in the case of an oblique square prism.

{eq}Volume\ of\ square\ prism\ =\ Bh \\ B\ =\ area\ of\ square\ base\ and\ h\ =\ height\ of\ prism {/eq}

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Frequently Asked Questions

A square prism has two congruent, parallel *square* bases, while a rectangular prism has two congruent, parallel *rectangle* bases. Every square prism is also a rectangular prism, since all squares are also rectangles.

To find the surface area of a square prism, find the sum of the areas of the prism's faces. The formula is Surface area = *2B + Ph*, where *B* = the area of a square base, *P* = the perimeter of the square base, and *h* = the height of the prism.

A cube is an example of a square prism. A cube has two congruent, parallel, square bases, which makes it a square prism.

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