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CAHSEE Math Exam: Tutoring Solution21 chapters | 211 lessons

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Lesson Transcript

Instructor:
*Joseph Vigil*

In this lesson, you will learn what prisms are and what makes rectangular prisms unique. There are several rules that identify a three-dimensional shape as a prism. You will also consider a few everyday items as examples of rectangular prisms.

Prisms are a certain kind of shape, but what makes them stand out? A **prism** is two polygons, or multiple-sided closed shapes, joined together to form an enclosed three-dimensional shape. For example, a box is a prism because it consists of two squares or rectangles joined together to make the enclosed three-dimensional figure seen here:

Two-dimensional shapes make the flat top and bottom sides, or **faces**, of a prism.

So any box, stacks of Post-It Notes, and even legal pads are all prisms because they're three-dimensional objects with congruent polygons as faces.

There are five rules for identifying a prism:

__Rule 1:__ is that the faces of a prism must be polygons. So even though it may seem that cylinders would be prisms, they don't fit the definition. This is because their faces are circles or ovals, which don't have sides and are therefore not polygons. So a can isn't a prism. No sides, no polygons, no prism.

A prism can have triangles as faces.

__Rule 2:__ states that the sides of a prism must be parallelograms. Parallelograms are quadrilaterals whose opposite sides are parallel. Consider a box and a triangular prism, all of their sides are parallelograms.

__Rule 3:__ The third rule says a prism's faces must be parallel. If one face lies at an angle to the other, they will at some point intersect, and the figure is not a prism. Also, if the faces aren't parallel, the sides won't be parallelograms.

In this figure, for example, the rear face lies at an angle compared to the front face. As a result, two of the sides are not parallelograms. Non-parallel faces create sides that are not parallelograms.

__Rule 4:__ states that the bases must be congruent. **Congruent** means the exact same shape and size.

A figure that has differently shaped faces creates sides that are not parallelograms. Since the faces are not congruent, this figure isn't a prism.

__Rule 5:__ Lastly, rule five says if a cross-section is taken of a prism parallel to its faces, the cross-section would be congruent with the prism's faces. This happens because the faces are congruent and parallel. This is true no matter where you slice it.

Think of it like a loaf of bread. If it's sliced parallel to the ends, no matter where you cut, the slices will have the same shape.

With all these rules, it's important to realize that there is something that isn't a rule. The angles between the sides and the faces do not have to be right angles. An **oblique prism** has slanted sides, but the faces are still congruent and parallel.

Although this type of prism's sides slant, its pentagonal faces are congruent and parallel and its sides are still parallelograms.

So where does all this get us with regard to rectangular prisms? Since there are many different shapes that can serve as faces, there are many different types of prisms. In fact, prisms are named for the shape of their faces. So a **rectangular prism** is simply a prism that has rectangles as its faces. It's an enclosed three-dimensional shape, but it's based on two rectangles. There are many common items that are in fact rectangular prisms.

This jewelry box is a three-dimensional object with a rectangular top and bottom. That makes it a rectangular prism!

In the same way, a mattress is a rectangular prism.

A mattress has a rectangular top and bottom. The congruent and parallel rectangular faces make it a rectangular prism.

And one more example:

This book stands on narrow rectangles. But it's still a three-dimensional figure based on two rectangles, so it is a rectangular prism.

A **prism** is two polygons joined together to form an enclosed three-dimensional shape. The rules that define a prism state that the faces must be polygons, the sides must be parallelograms, and the faces must be parallel. A three-dimensional shape that is a prism also must have faces that are congruent and cross-sections parallel to the faces must be congruent to the faces. Cylinders are not prisms because of a circular top and bottom.

**Oblique prisms** have slanted sides but are prisms because they have parallel and congruent faces. A **rectangular prism** is simply a prism with rectangular faces. Because a rectangular prisms faces must be rectangles, at least two of its sides will always be rectangles, even if the rest of its sides are squares.

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CAHSEE Math Exam: Tutoring Solution21 chapters | 211 lessons

- Properties of Shapes: Rectangles, Squares and Rhombuses 5:46
- Properties of Shapes: Quadrilaterals, Parallelograms, Trapezoids, Polygons 6:42
- Properties of Shapes: Triangles 5:09
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- Orthographic Projection: Definition & Examples 4:01
- The Golden Rectangle: Definition, Formula & Examples 6:12
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