Hello! Welcome to the World of "Area, Rectangular Prisms, and Cubes"!
Hello, Math team! In this chapter, we are going to explore area, which measures the amount of surface around us, as well as 3D shapes like dice and tissue boxes called rectangular prisms and cubes.
You might think, "Geometric shapes and big unit conversions sound a little tricky...", but don't worry! If we look at the basic rules step by step, solving these problems becomes as fun as doing a puzzle. Let's master them together!
Part 1: Expressing Size with Numbers! "Area"
1. What is Area?
When comparing the size of your desk to your notebook, it is very convenient to show which one is bigger—and by how much—using numbers. The measure of this flat surface size is called area.
To measure area, we choose a small square as our basic unit. In arithmetic, we use the area of a square with sides of \(1\text{ cm}\) as our standard, written as \(1\text{ cm}^2\) (one square centimeter).
By counting "how many \(1\text{ cm}^2\) squares can fit inside," we can express any flat surface as a number.
2. Area Formulas for Rectangles and Squares
Instead of counting grid squares one by one, you can calculate the area in a flash using multiplication! These are the area formulas.
[Key Point: Formulas for the Area of Rectangles and Squares]
・Area of a rectangle ＝ \((\text{length}) \times (\text{width})\) or \((\text{width}) \times (\text{length})\)
・Area of a square ＝ \((\text{side}) \times (\text{side})\)
(Example 1) What is the area of a rectangle with a length of \(4\text{ cm}\) and a width of \(6\text{ cm}\)?
Equation: \(4 \times 6 = 24\)
Answer: \(24\text{ cm}^2\)
(Example 2) What is the area of a square with sides of \(5\text{ cm}\)?
Equation: \(5 \times 5 = 25\)
Answer: \(25\text{ cm}^2\)
3. Smart Ways to Find Area (L-shaped and Combined Figures)
For shapes like stairs or an "L-shape," you cannot use the formula right away. However, you can easily solve them using one of these two methods!
Method 1: Split into two rectangles
Cut the shape vertically or horizontally to split it into two rectangles (or squares). Calculate each area separately, and then add them together.
Method 2: Subtract the missing piece from a larger rectangle
Fill in the missing corner to make one large rectangle. Find the total area, and then subtract the area of the extra piece you added.
4. Units for Larger Areas (\(\text{m}^2\), \(\text{a}\), \(\text{ha}\), \(\text{km}^2\))
Measuring the size of a classroom, a park, or a town in \(\text{cm}^2\) would make the numbers way too huge! That is why we use larger units.
・\(1\text{ m}^2\) (one square meter)
The area of a square with sides of \(1\text{ m}\).
Since \(1\text{ m} = 100\text{ cm}\), \(100 \times 100 = 10000\text{ cm}^2\).
・\(1\text{ a}\) (one are)
The area of a square with sides of \(10\text{ m}\).
\(10 \times 10 = 100\text{ m}^2\).
・\(1\text{ ha}\) (one hectare)
The area of a square with sides of \(100\text{ m}\).
\(100 \times 100 = 10000\text{ m}^2\), which means \(1\text{ ha} = 100\text{ a}\).
・\(1\text{ km}^2\) (one square kilometer)
The area of a square with sides of \(1\text{ km}\) (\(1000\text{ m}\)).
\(1000 \times 1000 = 1000000\text{ m}^2\), which means \(1\text{ km}^2 = 100\text{ ha}\).
[Helpful Tip]
It is super easy to remember these units if you imagine the "side length" of each square!
\(1\text{ m}^2\) (side is \(1\text{ m}\)) \(\rightarrow\) make the side 10 times longer (\(10\text{ m}\)) to get \(1\text{ a}\) (\(100\text{ m}^2\)) \(\rightarrow\) make the side 10 times longer again (\(100\text{ m}\)) to get \(1\text{ ha}\) (\(10000\text{ m}^2\)) \(\rightarrow\) make the side 10 times longer once more (\(1000\text{ m} = 1\text{ km}\)) to get \(1\text{ km}^2\)!
Every time the side length is multiplied by \(10\), the area becomes \(100\) times bigger (adding two \(0\)s).
[Common Mistake]
If the length is \(3\text{ m}\) and the width is \(40\text{ cm}\), you cannot multiply them right away because their units are different!
Always convert the measurements to the same unit first (for example, \(300\text{ cm}\) and \(40\text{ cm}\)) before calculating.
Part 2: Exploring Box Shapes! "Rectangular Prisms and Cubes"
1. What are Rectangular Prisms and Cubes?
Let's look at 3D shapes (solids) enclosed by flat surfaces, just like snack boxes or dice.
・Rectangular Prism: A solid shape enclosed only by rectangular faces (which may sometimes include squares).
・Cube: A solid shape enclosed only by identical, congruent square faces (like a standard die).
Both of these shapes have the exact same number of faces, edges, and vertices!
[Key Point: Features of Rectangular Prisms and Cubes]
・Number of faces: \(6\)
・Number of edges: \(12\)
・Number of vertices: \(8\)
*A special feature of a cube is that all \(12\) of its edges are equal in length!
2. Parallel and Perpendicular Lines and Planes
A flat, level surface extending in all directions is called a plane. When observing a rectangular prism, you will find special relationships between faces, between faces and edges, and between edges.
・Relationships Between Faces
Faces that are opposite each other will never meet, no matter how far they are extended. We say these faces are parallel (there are \(3\) pairs of opposite faces).
Faces that meet each other form right angles (\(90^\circ\)). We say these faces are perpendicular.
・Relationships Between Faces and Edges
Edges standing upright are perpendicular to the bottom base face.
Edges along the top that do not touch the bottom base face are parallel to the bottom base face.
3. Perspective Drawings (Sketches)
A drawing sketched from an angle to help you see the entire 3D shape clearly is called a perspective drawing (or sketch).
[Key Points for Drawing a Perspective Sketch]
1. Draw edges that can actually be seen using solid lines.
2. Draw hidden edges that cannot be seen using dashed lines.
3. Edges that are parallel in real life should also be drawn parallel on paper.
4. Nets (Unfolded Views)
A flat drawing made by cutting along the edges of a box and flattening it onto a single surface is called a net.
Folding the net back up will recreate the original rectangular prism or cube.
[Tips for Checking a Net]
・Does it have exactly \(6\) faces? (If it has \(5\) or fewer, or \(7\) or more, it cannot make a box).
・Do any of the faces overlap when folded?
・Do opposite faces have the exact same shape and size? (For a cube, all faces must be identical squares).
[Common Mistake]
In the net of a cube, opposite faces never touch side-by-side; they are always separated by one square (or around a corner fold). Imagine folding the net in your head to check that faces don't overlap and end up missing a lid!
Part 3: Describing Positions of Objects
1. Describing Positions on a Flat Surface (Plane)
When identifying a position on a flat surface, such as a classroom seating chart or grid paper, choose a reference starting point (corner) and describe the location using a pair of numbers in two directions.
(Example) "Move \(3\) across and \(2\) up"
Using two numbers like this lets anyone find the exact location without getting confused.
2. Describing Positions in Space (3D)
How do we describe the position of something inside a 3D box or a set of lockers?
In addition to the "across" and "length" of a flat surface, we need the height (or depth). In other words, we describe the position using a set of numbers in three directions (across, length, and height).
(Example) "Move \(2\) across, \(3\) deep, and \(1\) high"
Wrap-Up Review!
Finally, let's review the most important points we learned today!
1. Area of a rectangle ＝ \((\text{length}) \times (\text{width})\), Area of a square ＝ \((\text{side}) \times (\text{side})\)
2. Area unit conversions: \(1\text{ m}^2 = 10000\text{ cm}^2\), \(1\text{ a} = 100\text{ m}^2\), \(1\text{ ha} = 100\text{ a} = 10000\text{ m}^2\), \(1\text{ km}^2 = 100\text{ ha}\)
3. Rectangular prisms and cubes: Both have \(6\) faces, \(12\) edges, and \(8\) vertices.
4. In a rectangular prism, opposite faces are parallel, and adjacent faces are perpendicular.
5. For a perspective sketch, draw hidden edges with dashed lines; for a net, pay attention to which faces will end up opposite each other.
6. To describe a position, use two numbers on a flat surface and three numbers in 3D space.
Instead of just memorizing the formulas, try to picture them in your head: "Why does this formula work?" or "Where will these faces meet when I fold this die?" Have fun practicing!