Prisms and Cylinders

Welcome to the World of "3D Shapes"! Let's Master Prisms and Cylinders

Hello! Let's study the unit on "Prisms and Cylinders" together. Up until now, we have mostly dealt with "2D" shapes drawn on paper, but in this chapter, we will learn about "3D" shapes—the kinds of objects that take up space, like boxes.

There are so many 3D shapes hidden all around us: soda cans, pencils, skyscrapers, and more. Once you start spotting them and saying, "Hey, that’s a prism!" or "That’s a cylinder!", math becomes much more fun. It might feel a little tricky at first, but don't worry, I’ll break it down into key points for you. Let's relax and get started!

1. What are Prisms and Cylinders?

First, let's learn two basic terms: "base" and "lateral face" (the side face).

● What is a Base?

These are the two faces located at the top and bottom of the solid that face each other. In prisms and cylinders, these two bases are "congruent" (meaning they have the same shape and size) and are "parallel" to each other.
Example: Think of them like the two pieces of bread holding a sandwich together.

● What is a Lateral Face?

This is the surface around the shape that connects the two bases.

【Prisms】

A solid with "polygons" (like triangles or quadrilaterals) as its bases is called a prism. The name is determined by the shape of the base:
・If the base is a triangle ＝ Triangular prism
・If the base is a quadrilateral ＝ Quadrilateral prism
・If the base is a pentagon ＝ Pentagonal prism

Key Point: All lateral faces of a prism are "rectangles" (or squares).

【Cylinders】

A solid with "circles" as its bases is called a cylinder.
Example: Canned goods and the cardboard core of a toilet paper roll are examples of cylinders.

Key Point: The lateral face of a cylinder is a curved surface.

2. Counting Vertices, Edges, and Faces (Rules for Prisms)

Prisms have interesting rules regarding the number of vertices, edges, and faces. Knowing these will help you avoid counting mistakes on tests!

Let's consider a base shaped like an "\(n\)-gon".
1. Number of Vertices: Twice the number of vertices on the base.
Formula: \(n \times 2\)
2. Number of Edges: Three times the number of edges on the base.
Formula: \(n \times 3\)
3. Number of Faces: The number of edges on the base plus 2 (for the top and bottom bases).
Formula: \(n + 2\)

【Common Mistake】

Don't confuse the number of "lateral faces" with the "total number of faces." For example, in a triangular prism:
・Number of lateral faces is 3
・Total number of faces is 2 bases + 3 lateral faces ＝ 5
Always read carefully to see if the question asks for the "total" or just the "lateral faces."

3. Mastering Nets!

A diagram created by cutting open a 3D shape and laying it flat on a single piece of paper is called a "net". Many people find this difficult, but it's simple once you grasp the trick.

● Nets of Prisms

When you open up a prism, the lateral faces become one large rectangle. Attached to the top and bottom of that rectangle are the two polygonal bases.
Key Point: The lengths of the edges that meet when you fold the shape back together will always be the same.

● Nets of Cylinders

Surprisingly, when you open up a cylinder, the lateral face becomes a "rectangle"!
Here is the most important rule:
"The width of the lateral rectangle" ＝ "The circumference of the base circle"

Knowing this helps you solve calculation problems.
Remember, the formula for circumference is ＝ \(Diameter \times 3.14\).
So, you can calculate the width of the lateral rectangle as \(Diameter \times 3.14\).

【Fun Fact: Why does the lateral face become a rectangle?】

You can see this by making a cylinder out of construction paper. If you cut the circular tube straight down vertically and spread it out, it forms a perfect rectangle—just like when you peel the label off a snack container!

4. Summary and Study Tips

Finally, let's review the important points we learned today.

● Prisms and cylinders have two identical "bases."
● The lateral faces of a prism are "rectangles," and the lateral face of a cylinder unfolds into a "rectangle."
● In the net of a cylinder, the width of the lateral face ＝ the circumference of the base.
● The height is the length of the straight line perpendicular to the two bases.

★ A final word of encouragement ★
You might feel like, "Visualizing these shapes is tough..." When that happens, try taking a box from around the house and actually cutting it open, or just observe a tissue box. By looking at real objects, you will definitely reach a moment where it just "clicks." Let's keep moving forward one step at a time!