生成适合4生5年纪学上的，生动有趣的教学视频。---**Extracted Content:** **Header Information:** * **Top Right Title:** 图形与几何 (Shapes and Geometry) * **Top Right Subtitle:** 专题五 (Topic Five) * **Vertical Text on Right Side:** 学之舟 (Learning Boat) **Main Content - Section Title:** * 第二节 立体图形的认识 (Section Two: Understanding Three-dimensional Shapes) **Main Content - Subsection Title:** * 考点一 长方体的认识 (Key Point One: Understanding Cuboids) --- **Detailed Content Extraction:** **1. 长方体的定义 (Definition of a Cuboid)** * **Text:** 长方体也称为平行六面体，它是由六个面围成的立体图形，相对的面是平行且全等的长方形，相邻的两个面相互垂直。如下图： (A cuboid is also called a rectangular parallelepiped. It is a three-dimensional figure enclosed by six faces. Opposite faces are parallel and congruent rectangles, and adjacent faces are mutually perpendicular. As shown in the figure below:) * **Chart/Diagram Description:** * **Type:** Geometric figure (3D representation of a cuboid). * **Main Elements:** * **Shape:** A rectangular prism (cuboid) drawn in perspective. Solid lines represent visible edges, dashed lines represent hidden edges. * **Labels:** * "面" (face) is labeled on the front-facing rectangular face. * "顶点" (vertex) is labeled at the bottom-left front corner. * "棱" (edge) is labeled along the bottom-front edge. **2. 长方体的特征 (Characteristics of a Cuboid)** * **Annotation:** →长方体线的特征。 (→Characteristics of cuboid lines.) * **(1) Text:** 六个面都是长方形 (有时有两个相对的面是正方形)。 (All six faces are rectangles (sometimes two opposite faces are squares).) * **(2) Text:** 相对的面面积相等，12条棱中相对的4条棱长度相等。 (Opposite faces have equal area, and among the 12 edges, the 4 opposite edges have equal length.) * **(3) Text:** 相交于一个顶点的三条棱分别叫做长、宽、高。 (The three edges that meet at a vertex are called length, width, and height, respectively.) * **(4) Text:** 两个面相交的边叫做棱，三条棱相交的点叫做顶点。有8个顶点。 (The line where two faces intersect is called an edge, and the point where three edges intersect is called a vertex. There are 8 vertices.) **3. 长方体的表面积 (Surface Area of a Cuboid)** * **(1) Text:** 长方体六个面的总面积叫做长方体的表面积。 (The total area of the six faces of a cuboid is called its surface area.) * **(2) Text:** 长方体的表面积 = (长 × 宽 + 长 × 高 + 宽 × 高) × 2 (Surface Area of a cuboid = (length × width + length × height + width × height) × 2) * **Formula/Text:** 如果用 a、b、h 分别表示长方体的长、宽、高，用 S 表示长方体的表面积，那么长方体表面积的字母公式是 S = 2 (ab + bh + ah)。 (If a, b, h represent the length, width, and height of the cuboid respectively, and S represents the surface area of the cuboid, then the formula for the surface area of a cuboid is S = 2(ab + bh + ah).) **4. 长方体的体积 (Volume of a Cuboid)** * **(1) Text:** 长方体所占空间的大小，叫作长方体的体积。 (The amount of space a cuboid occupies is called its volume.) * **(2) Text:** 长方体的体积等于它的长、宽、高的乘积。即：长方体的体积=长 × 宽 × 高。如果用 a、b、h 分别表示长方体的长、宽、高，用 V 表示长方体的体积，那么长方体体积的字母公式是 V = abh。 (The volume of a cuboid is equal to the product of its length, width, and height. That is: Volume of a cuboid = length × width × height. If a, b, h represent the length, width, and height of the cuboid respectively, and V represents the volume of the cuboid, then the formula for the volume of a cuboid is V = abh.) **5. 长方体的棱长之和 (Sum of Edge Lengths of a Cuboid)** * **Formula/Text:** 因为长方体相对的棱的长度相等，所以长方体的棱长之和 = (长 + 宽 + 高) × 4。 (Because the lengths of opposite edges of a cuboid are equal, the sum of the edge lengths of a cuboid = (length + width + height) × 4.) --- **Right Column - Side Notes:** **实例讲解 (Example Explanation)** * **Text:** 实际上长方体的长、宽、高的位置不是固定不变的。 (In fact, the positions of the length, width, and height of a cuboid are not fixed.) * **Text:** 长方体的摆放不同，长、宽、高也不同。 (Different orientations of a cuboid result in different assignments of length, width, and height.) * **Chart/Diagram Description:** * **Type:** Geometric figures showing cuboids in different orientations. * **Main Elements:** Three cuboid diagrams, each with "长" (length), "宽" (width), and "高" (height) labeled. * **First Cuboid (Left):** Depicts a cuboid lying flat on its largest face. "高" (height) is the shortest vertical dimension. "长" (length) is the longest dimension of the base, and "宽" (width) is the shorter dimension of the base. * **Second Cuboid (Middle):** Depicts a cuboid lying on its side (a narrower face). "高" (height) is the vertical dimension, which is now the original "width" or "length". "长" (length) and "宽" (width) are reassigned to the dimensions of the new base. * **Third Cuboid (Right):** Depicts a cuboid standing upright on its smallest face. "高" (height) is the tallest vertical dimension, which is now the original "length" or "width". "长" (length) and "宽" (width) are reassigned to the dimensions of the new base. **重点说明 (Key Explanation)** * **Text:** 表面积相等的两个长方体或一个长方体和一个正方体，棱长和也不一定相等。 (Two cuboids with equal surface areas, or a cuboid and a cube with equal surface areas, do not necessarily have equal sums of edge lengths.) --- **Footer Information:** * **Text:** 学之舟 109 知识窗 (Learning Boat 109 Knowledge Window) **Overall Document Title:** 小学生知识通 | 小学数学 (Primary School Knowledge Pass | Primary Mathematics) --- **Section: 学之舟 (Boat of Learning)** **重点说明 (Key Explanation)** A cuboid (长方体) has at most 6 faces that are rectangles, at least 4 faces that are rectangles, and at most 2 faces that are squares. **重点说明 (Key Explanation)** * **Diagram:** A Venn diagram-like structure showing two nested ovals. The outer oval is labeled "长方体" (Cuboid), and the inner oval is labeled "正方体" (Cube). * **Text:** A cube (正方体) is a cuboid whose length, width, and height are all equal. **实例讲解 (Example Explanation)** * **Diagram 1:** An illustration of a 3x3x3 Rubik's Cube. * **Diagram 2:** An illustration of a cube labeled "棱长" (Edge length), with an arrow pointing to its unfolded net (a cross shape formed by six squares). * **Formula/Calculation Description:** 正方体的表面积 (Surface Area of a Cube) = 一个面的面积 × 6 (Area of one face × 6) = 棱长 × 棱长 × 6 (Edge length × Edge length × 6) --- **Section: 温馨提示 (Warm Reminder)** * **Text:** If a cuboid (not a cube) has square top and bottom faces, then its four side faces are congruent rectangles. --- **Section: 考点二 正方体的认识 (Knowledge Point Two: Understanding Cubes)** **1. 正方体的定义 (Definition of a Cube)** * **Text:** A cube (正方体) is a special kind of cuboid (长方体). * **Text:** A three-dimensional figure enclosed by six identical squares is called a cube (正方体), also known as a “立方体” (cuboid) or “正六面体” (hexahedron). See figure below. **2. 正方体的特征 (Characteristics of a Cube)** * **Text:** A cube does not have the distinct concepts of length, width, and height (as it's all one dimension, the edge length). * **(1) Text:** All six faces are squares. The areas of the six faces are equal. * **(2) Text:** It has 12 edges (棱), and all edge lengths are equal. It has 8 vertices (顶点). * **(3) Text:** A cube can be seen as a special kind of cuboid. * **Diagram:** A wireframe drawing of a cube. Three visible edges are labeled "棱" (Edge). **3. 正方体的棱长总和 (Total Edge Length of a Cube)** * **Text:** Since a cube has 12 edges, and all edges are equal, the total edge length of a cube is 12 times its edge length, with the following relationships: * **(1) Formula:** 正方体的棱长总和 = 12 × 棱长 (Total edge length of a cube = 12 × Edge length) * **(2) Formula:** 正方体的棱长 = 棱长总和 ÷ 12 (Edge length of a cube = Total edge length ÷ 12) * **Text:** If C represents the total edge length and a represents the edge length, then: * **Formulas:** a = C ÷ 12; C = 12a. **4. 正方体的表面积 (Surface Area of a Cube)** * **(1) Definition:** The total area of the six faces of a cube is called its surface area. * **(2) 正方体表面积的计算 (Calculation of the Surface Area of a Cube)** * **Formula:** 正方体的表面积 = 棱长 × 棱长 × 6 (Surface area of a cube = Edge length × Edge length × 6) * **Text:** If a represents the edge length of a cube and S represents the surface area, the formula for calculating the surface area of a cube expressed in letters is: * **Formula:** S = 6a² * **Example Problem:** 求一个棱长为2分米的正方体的表面积。(Find the surface area of a cube with an edge length of 2 decimeters.) * **Solution:** 根据公式，列式为S = 6 × 2 × 2 = 24（平方分米）。(According to the formula, the expression is S = 6 × 2 × 2 = 24 (square decimeters).) **Header:** 图形与几何 (Figures and Geometry) 专题五 (Topic 5) --- **Left Column Content:** **5. 正方体的体积 (Volume of a Cube)** (1) 正方体的体积：正方体所占空间的大小，叫作它的体积。 (2) 正方体的体积计算： 正方体的体积 = 棱长 × 棱长 × 棱长 如果用 a 表示正方体的棱长，用 V 表示体积，则正方体的体积计算公式用字母表示为：V = a·a·a 或 V = a³。 **温馨提示 (Warm Reminder):** 一个长方体的长、宽、高都相等的时候，该长方体就是正方体，所以，正方体是特殊的长方体。 **Image Description (Warm Reminder):** * Type: Illustration * Main Elements: A rolled-up scroll or document. **考点三 圆柱的认识 (Key Point Three: Understanding Cylinders)** **1. 圆柱 (Cylinder)** 把长方体以它的一条边为旋转轴旋转一周，形成的几何体叫圆柱，也叫直圆柱，如图所示就是一个圆柱。生活中有很多圆柱形的物体，如：水桶、笔筒、电线杆、电池等。 **Chart/Diagram Description (Cylinder Illustrations):** * **Type:** Geometric Figures (3D Cylinders) * **Main Elements:** * **Left Diagram (Cylinder with Axis and Points):** * Shows a cylinder with a top circular face and a bottom circular face. * Point O is at the center of the top base. * Point O' is at the center of the bottom base. * A dashed vertical line connects O and O', representing the axis or height. * Point A is on the circumference of the bottom base. * Point B is on the circumference of the top base, vertically aligned with A. * Label: "底面" (Bottom Face) placed below the bottom base. * **Right Diagram (Cylinder with Labeled Sides):** * Shows a cylinder with a top circular face and a bottom circular face. * A vertical line segment inside the cylinder is labeled "高" (Height). * The curved outer surface is labeled "侧面" (Lateral Surface). * Labels: "底面" (Bottom Face) placed above the top base and below the bottom base. **2. 圆柱的认识 (Understanding Cylinders)** (1) 圆柱的上下两个面叫作底面。圆柱的上下两个面是完全相同的圆。 (2) 圆柱有一个曲面叫作侧面。侧面展开是一个长方形。 (3) 圆柱两个底面之间的距离叫作高。圆柱有无数条高。 **Chart/Diagram Description (Cylinder Properties Diagram):** * **Type:** Explanatory Diagram/Flowchart * **Main Elements:** * **Left Section:** * A vertical rectangle representing the side view of a cylinder or a segment. * Labels: "底面周长" (Base Circumference) along the horizontal dimension, "高" (Height) along the vertical dimension. * "底面" (Base Face) labels above and below the rectangle. * **Arrow:** Points from the left section to the right section. * **Right Section:** * A horizontal rectangle. * Labels: "底面周长" (Base Circumference) along the longer horizontal dimension, "高" (Height) along the shorter vertical dimension. * "底面" (Base Face) labels above and below the rectangle. * **Purpose:** Illustrates the relationship between the dimensions of a cylinder's base and height, possibly showing the dimensions of its unfolded lateral surface. **3. 圆柱的侧面积 (Lateral Surface Area of a Cylinder)** 圆柱的侧面沿着一条高展开后是一个长方形，这个长方形的面积就是圆柱的侧面积。 圆柱的侧面积 = 底面的周长 × 高 --- **Right Column Content:** **重点说明 (Key Explanation):** 长方体和正方体底面的面积叫作底面积。长方体(或正方体)的体积 = 底面积 × 高。 如果用 S 表示底面积，上面的公式可以写成：V = Sh。 **真题解析 (Real Exam Question Analysis):** **(安徽小考) 一个圆柱的侧面展开图是正方形，这个圆柱的底面半径和高的比是 ( )。** **Options:** A. 1 : π B. 1 : 2π C. π : 1 D. 2π : 1 **【答案】B** **重点说明 (Key Explanation):** 圆柱中不是平面的面，我们称之为曲面。 --- **Footer:** 学之舟 111 知识通 (Xue Zhi Zhou 111 Knowledge Hub) The image contains educational content about the properties and formulas of cylinders and cones. There are no explicit questions or options in the traditional sense, but rather explanatory text and diagrams. Here's the extracted content: --- **Section: Cylinder (圆柱)** **Formulas and Explanations:** * **Lateral Surface Area:** S侧 = Ch = 2πrh (r is the radius of the cylinder's base) * **4. Surface Area of a Cylinder:** The sum of the lateral surface area and two base areas of a cylinder is called the surface area of the cylinder. S表 = S侧 + 2S底 = 2πrh + 2πr² * **5. Volume of a Cylinder:** Divide the base of the cylinder into many equal sectors (an even number), then cut the cylinder and piece them together as shown in the figure below, resulting in an approximate cuboid. The more sectors divided, the closer the formed solid figure is to a cuboid. The base area of this cuboid is equal to the base area of the cylinder (S=πr²), and the height is the height of the cylinder (h). Because the volume of a cuboid = base area × height, Therefore, the volume of a cylinder = base area × height. If V represents the volume of the cylinder, S represents the base area of the cylinder, r represents the base radius, and h represents the height, then the algebraic formula for the cylinder's volume is: V = Sh = πr²h. **Example Explanation (Cylinder Unfolding):** * If a cylinder is unfolded along its height, the unfolded shape is a rectangle or a square. If not unfolded along its height, the unfolded shape is a parallelogram or an irregular shape. **Diagrams related to Cylinder:** 1. **Cylinder Unfolding Diagram:** * **Type:** Geometric transformation diagram. * **Main Elements:** * On the left: A 3D cylinder. Labels: "底面 (Base)" on the bottom circular face, "高 (Height)" along the side. * An arrow points from the cylinder to an unfolded 2D shape. * On the right: A 2D rectangle with two circles attached to its top and bottom sides. Labels: "底面 (Base)" on both circles, "底面的周长 (Circumference of the base)" as the length of the rectangle, "高 (Height)" as the width of the rectangle. * **Description:** This diagram illustrates how a cylinder's lateral surface can be unfolded into a rectangle, where the rectangle's length is the circumference of the cylinder's base and its width is the cylinder's height. The top and bottom bases are shown as circles. 2. **Cylinder Volume Derivation Diagram:** * **Type:** Geometric transformation diagram. * **Main Elements:** * On the left: A 3D cylinder conceptually sliced into many vertical sectors from the top base. * An arrow points from the cylinder to a reconstructed 3D shape. * On the right: A 3D shape resembling a cuboid, formed by rearranging the slices of the cylinder. The slices are arranged side-by-side, forming a rectangular base. The height of this cuboid is the same as the cylinder's height. * **Description:** This diagram visually explains how the volume of a cylinder can be understood by cutting it into numerous thin wedges and rearranging them to form an approximate cuboid, thus linking the cylinder's volume to the product of its base area and height. 3. **Cylinder Unfolding Types Diagram:** * **Type:** Flowchart/Categorization diagram. * **Main Elements:** * A cylinder shape labeled "圆柱体 (Cylinder)". * Two arrows emanating from the cylinder: * **Path 1:** "侧面展开 (Lateral surface unfolds) → 长方形 (Rectangle)". Annotation: "(当圆柱体的底面周长和高不等时) (When the circumference of the cylinder's base is not equal to its height)". * **Path 2:** "侧面展开 (Lateral surface unfolds) → 正方形 (Square)". Annotation: "(当圆柱体的底面周长和高相等时) (When the circumference of the cylinder's base is equal to its height)". * **Description:** This diagram categorizes the shapes obtained when unfolding the lateral surface of a cylinder based on the relationship between its base circumference and height. --- **Section: Cone (圆锥)** **Key Point Four: Understanding Cones** **1. Understanding Cones:** * **(1) Definition of a Cone:** A geometric solid formed by rotating a right-angled triangle around one of its right-angle sides for one full revolution is called a cone. * **(2) Properties of a Cone:** * ① The base of a cone is a circle, and its lateral surface is a curved surface. * ② The distance from the apex of the cone to the center of the base circle is the height of the cone. (Annotation next to this point: 圆锥只有一条高。 - A cone has only one height.) * ③ Measuring the height of a cone: First, place the cone with its base flat. Then, place a flat board horizontally on top of the cone's apex, and vertically measure the distance between the board and the base. * ④ Unfolding the lateral surface of a cone yields a sector. **2. Properties of a Cone:** * **(1)** The plane containing the base of a cone is perpendicular to its axis, and the base is a circle. * **(2)** The line passing through the center of the base circle and the apex is the axis of the cone; the height of the cone is the line segment with the center of the base circle and the apex as its endpoints. **Example Explanation (Understanding Cones):** * **Understanding Cones** **Diagrams related to Cone:** 1. **Cone Properties Diagram:** * **Type:** Geometric figure diagram. * **Main Elements:** * A 3D cone with dashed lines indicating its height and axis. * **Labels:** "顶点 (Apex)" at the top, "高 (Height)" along the central vertical dashed line, "侧面 (Lateral Surface)" on the curved side, "底面 (Base)" on the bottom circular face. "O" is labeled at the center of the base. * **Annotations (list format):** * 一个侧面 (One lateral surface) * 一个底面 (One base) * 一个顶点 (One apex) * 只能画一条高 (Only one height can be drawn) * **Description:** This diagram illustrates the basic components of a cone and lists their count and a key property regarding its height. 2. **Cone Height Measurement Diagram:** * **Type:** Geometric figure, measurement illustration. * **Main Elements:** * A 3D cone placed on a flat surface. * A horizontal line segment (representing a flat board) placed above the apex of the cone, parallel to the base. * A vertical double-headed arrow connecting the apex (or the board resting on it) to the base. * **Labels:** "顶点 (Apex)" at the top of the cone, "高 (Height)" along the vertical arrow, "h" (symbol for height) next to the vertical arrow, "底面 (Base)" at the bottom. A dashed line from the apex to the center of the base. * **Description:** This diagram visually explains how the height of a cone is defined and how it can be measured using a flat board. --- Here is the extracted content from the image, formatted as requested: **Overall Context/Header Information:** * **Subject:** 图形与几何 (Graphics and Geometry) * **Topic:** 专题五 (Topic Five) **Right Side Panel Information:** * **Section Title:** 重点说明 (Key Explanation) * **Content:** 很多物体从不同的面观察，看到的形状是不同的，圆球除外。 (Many objects, when viewed from different sides, will appear to have different shapes, with the exception of spheres.) * **Side Label:** 学之舟 (Boat of Learning) * **Section Title:** 实例讲解 (Example Explanation) * **Content:** 站在不同的位置观察物体，看到的形状是不同的。 (When observing an object from different positions, the shape seen is different.) * **Chart Description (for Example Explanation):** * **Type:** Diagram illustrating 3D objects and their 2D views from different perspectives. * **Main Elements:** * **Top:** Two 3D geometric solids are shown side-by-side: a blue sphere (left) and a brown cylinder (right). * **Bottom Row (Views):** Three sets of 2D shapes, each depicting views of the sphere and cylinder from a specific direction: * **左面 (Left View):** Shows a blue circle (representing the sphere) and a larger brown circle (representing the cylinder's end face when viewed from the side). * **上面 (Top View):** Shows a blue circle (representing the sphere) and an orange circle (representing the cylinder's top face). * **正面 (Front View):** Shows a blue circle (representing the sphere) and a brown rectangle (representing the cylinder's side). * **Labels:** Each set of 2D views is labeled with the corresponding observation direction: "左面" (Left), "上面" (Top), "正面" (Front). --- **Main Content (Left Side):** **Section Title:** 考点八 观察物体 (Key Point Eight: Observing Objects) **1. 观察物体的意义 (Meaning of Observing Objects)** * **Textual Information:** 生活中，很多时候都需要从不同角度、不同方位去观察一个物体，以获得该物体大小、形状、颜色等各个方面的信息。 同理，在数学中，也需要从不同的角度、不同的方位去观察被观察物体，从而得到不同的结果。如图所示。 (In life, we often need to observe an object from different angles and directions to obtain information about its size, shape, color, and other aspects. Similarly, in mathematics, we also need to observe objects from different angles and directions, thereby obtaining different results. As shown in the figure.) * **Chart Description:** * **Type:** Cartoon illustration. * **Main Elements:** * Two cartoon children are depicted: a boy on the left (wearing a yellow shirt and green shorts) and a girl on the right (wearing an orange shirt and blue shorts). * Between them, on the ground, is a large three-dimensional number "6" (or "9", its orientation is ambiguous from this perspective). * Above the boy's head, a speech bubble contains the number "6". * Above the girl's head, a speech bubble contains the number "9". * **Relative Position:** The children are facing each other, and the number is positioned centrally between them, illustrating how different viewpoints lead to different perceptions of the same object. **2. 从不同方向观察物体 (Observing Objects from Different Directions)** **(1) 从两个方向观察同一物体的形状 (Observing the Shape of the Same Object from Two Directions)** * **Textual Information:** 从正面和侧面这两个方向观察一个物体，会发现从不同的方向观察同一个物体，所看到的形状可能是不同的。 (When observing an object from the front and side, it will be found that the shape seen from different directions may be different when observing the same object.) **(2) 从三个方向观察立体图形 (Observing 3D Figures from Three Directions)** * **Textual Information:** 从上面、左面、正面三个方向观察一个立体图形，可以确定立体图形的形状。它是随着观察角度的变化而变化。 (Observing a 3D figure from the top, left, and front three directions can determine the shape of the 3D figure. It changes with the change of observation angle.) * **Chart Description:** * **Type:** Collection of cartoon illustrations of 3D objects and their 2D views. * **Main Elements:** * **Left Box:** A horizontal rectangular box containing three distinct 3D cartoon objects: a giraffe (left), a house (middle), and a pig (right). * **Right Side (outside the box):** * **Top Row:** Three 2D shapes, depicted in magenta: a solid square, a house-like outline (representing a front view of a house), and another house-like outline with a sloped roof (possibly a side view). * **Bottom Row:** Three smaller cartoon 3D figures: a giraffe, a pig, and a dog. **3. 从不同方位观察物体 (Observing Objects from Different Positions)** **(1) 不同位置观察物体的范围变化 (Change in the Scope of Objects Observed from Different Positions)** * **Textual Information:** 观察者站得位置不同，观察者看到的观察对象的范围是不同的。 例如：观察者站得位置低，看到的对象的范围就小；观察者站得位置高，看到的对象的范围就大；观察者站得近，看到的对象的范围就小；观察者站得位置远，看到的对象的范围就大。 (If the observer stands in different positions, the scope of objects seen by the observer is different. For example: If the observer stands low, the scope of objects seen is small; if the observer stands high, the scope of objects seen is large; if the observer stands close, the scope of objects seen is small; if the observer stands far, the scope of objects seen is large.) **(2) 不同位置观察同一个对象其形状的变化 (Change in Shape of the Same Object Observed from Different Positions)** * **Textual Information:** 从不同的观测点观察物体，观察到物体的形状也会发生变化。 经常应用的是，用平行光束从不同方向照射物体，在物体后面的墙 (When observing an object from different observation points, the observed shape of the object will also change. It is often applied by illuminating an object with parallel light beams from different directions, on the wall behind the object...) - *Note: The sentence is incomplete as the text is cut off at the bottom of the page.* **Footer Information:** * **Page Number:** 117 * **Text:** 知识通 (Knowledge Pass) The following is a complete and accurate extraction of the content from the provided image: **Header:** 小学生知识通 / 小学数学 (Primary School Knowledge Navigator / Primary School Mathematics) **Sidebar:** 学之舟 (Boat of Learning) **Main Content:** Different shadows will appear, and these shadows can, to some extent, be approximately viewed as the shapes of objects in our eyes. **图解知识 (Illustrated Knowledge)** * **Description of Illustration:** The illustration shows three cartoon children observing a brown rectangular box placed on two green rectangular blocks. Each child is stating how many faces of the box they can see from their respective viewpoints: * A child standing on the top left says: "我一共看到了三个面。" (I saw three faces in total.) * A child kneeling on the top right says: "我看到了两个面。" (I saw two faces.) * A child crouching in the bottom center says: "我只看到了一个面。" (I only saw one face.) * **温馨提示 (Warm Reminder) 1:** (1) 从不同位置观察同一个物体，所看到的图形有可能一样，也有可能不一样。 (Observing the same object from different positions, the shape seen may or may not be the same.) (2) 从同一个位置观察不同的物体，所看到的图形有可能一样，也有可能不一样。 (Observing different objects from the same position, the shape seen may or may not be the same.) **温馨提示 (Warm Reminder) 2:** (1) 从物体的上方和正面观察物体，可以观察到物体的长是一样的。 (Observing an object from above and from the front, the length of the object can be observed to be the same.) (2) 从物体的上方和侧面观察物体，可以观察到物体的宽是一样的。 (Observing an object from above and from the side, the width of the object can be observed to be the same.) (3) 从物体的正面和侧面观察物体，可以观察到物体的高是一样的。 (Observing an object from the front and from the side, the height of the object can be observed to be the same.) **重点说明 (Key Explanation)** 观察物体有诀窍，先看到几个面，再看它的排列法，画图形时要注意，只分上下画数量。 (There's a trick to observing objects: first, see how many faces are visible, then look at their arrangement. When drawing shapes, pay attention to only distinguish between the number of faces drawn above and below.) **考点九 (Knowledge Point Nine)** 立体图形展开图的判断方法 (Methods for Judging Nets of Three-Dimensional Figures) 1. **立体图形的展开图 (Nets of Three-Dimensional Figures)** **Table Content:** | 立体图形 (Three-Dimensional Figure) | 沿着一条线剪开的平面图形 (Plane Figure Cut Along a Line) | | :---------------------------------- | :------------------------------------------------------- | | **Description:** A cuboid (rectangular prism) with visible edges and dashed lines for hidden edges. Length 'a' and width 'b' are indicated on the base. | **Description:** A cross-shaped net consisting of 6 rectangles. It has a central row of 4 rectangles, with one rectangle attached above the second one in the row, and one rectangle attached below the second one in the row. | | **Description:** A cube with visible edges and dashed lines for hidden edges. All visible edges are labeled 'a', indicating equal side lengths. | **Description:** A cross-shaped net consisting of 6 squares. It has a central row of 4 squares, with one square attached above the second one in the row, and one square attached below the second one in the row. | | **Description:** A cylinder with a circular base (labeled with radius 'r' via a dashed line) and height 'h' (labeled via a dashed line). | **Description:** A net consisting of a rectangle (representing the lateral surface) with a circle attached to its top edge and another circle attached to its bottom edge (representing the bases). | | **Description:** A cone with a circular base (dashed outline) and a dashed line indicating its height. | **Description:** A net consisting of a sector of a circle (representing the lateral surface) with a full circle attached to its arc (representing the base). | **实例讲解 (Example Explanation)** **正方体展开图 (Cube Net)** * **Description:** A cross-shaped net of a cube. * The net has a central vertical column of four squares. * The topmost square in the column is labeled "上面" (Top). * The second square from the top in the column is labeled "后面" (Back). * The third square from the top in the column is labeled "下面" (Bottom). * The fourth (bottommost) square in the column is labeled "前面" (Front). * Two additional squares are attached horizontally: * One square is to the left of the "下面" square, labeled "左面" (Left). * One square is to the right of the "下面" square, labeled "右面" (Right). **Footer:** 学之舟 118 知识通 (Boat of Learning 118 Knowledge Navigator) Here is the extracted content from the image: **General Information:** * **Title:** 图形与几何 (Geometry) * **Topic:** 专题五 (Topic 5) * **Source/Brand:** 学之舟 (Xue Zhi Zhou) * **Page/Section:** 学之舟 119 知识窗 (Xue Zhi Zhou 119 Knowledge Window) --- **Section 2: 正方体和长方体的展开图 (Nets of Cubes and Cuboids)** **(1) 正方体的展开图可以分为以下四类。 (Nets of cubes can be divided into the following four categories.)** **① 第一类 ("141" 形展开图):** * **Description:** 中间一排四个连续的小正方形，上、下各一个小的正方形，如图所示，一共有六种情况。 (One row of four consecutive small squares in the middle, one small square each on the top and bottom, as shown in the figure, there are a total of six cases.) * **Annotation:** → 由于少于或多于6个正方形组成的图形不是正方体的平面展开图。 (→ Diagrams composed of fewer or more than 6 squares are not nets of a cube.) * **Summary:** 总结：中间四个一连串，两边各随便放。 (Summary: Four in a row in the middle, two on the sides placed freely.) * **Chart Type:** Grid patterns composed of squares. * **Main Elements (6 patterns for "141" type):** All patterns consist of 6 squares. There is a central horizontal row of 4 squares. The remaining two squares are attached to either the top or bottom of this central row, one above and one below. 1. Central row (squares 1-2-3-4 from left to right). Top square attached above square 2. Bottom square attached below square 3. (Classic cross shape) 2. Central row (squares 1-2-3-4). Top square attached above square 2. Bottom square attached below square 4. 3. Central row (squares 1-2-3-4). Top square attached above square 1. Bottom square attached below square 3. 4. Central row (squares 1-2-3-4). Top square attached above square 1. Bottom square attached below square 4. 5. Central row (squares 1-2-3-4). Top square attached above square 3. Bottom square attached below square 1. 6. Central row (squares 1-2-3-4). Top square attached above square 4. Bottom square attached below square 1. **② 第二类 ("231" 形展开图):** * **Description:** 中间一排三个连续的小正方形，上、下两侧一侧两个小正方形，一侧一个小正方形。如图所示，一共有三种基本图形。 (One row of three consecutive small squares in the middle, two small squares on one side of the top/bottom, and one small square on the other side. As shown in the figure, there are a total of three basic patterns.) * **Annotation:** → 中间为三个正方形，上为两个正方形，下为下一个正方形可以从任意位置。 (→ The middle consists of three squares, the top has two squares, and the bottom has one square that can be in any position.) * **Summary:** 总结：二三竖连错一个，三一相连随便。 (Summary: Two, three vertically connected, one wrong; three, one connected freely.) * **Chart Type:** Grid patterns composed of squares. * **Main Elements (3 patterns for "231" type):** All patterns consist of 6 squares. There is a central horizontal row of 3 squares. The remaining three squares are attached: two on one side (top or bottom) of the central row, and one on the other side. 1. Central row (squares 1-2-3). Two squares attached above squares 1 and 2 (forming an L-shape above). One square attached below square 3. 2. Central row (squares 1-2-3). Two squares attached above squares 2 and 3. One square attached below square 1. 3. Central row (squares 1-2-3). One square attached above square 1, and another square attached above square 3 (separated). One square attached below square 2. **③ 第三类 ("222" 形展开图):** * **Description:** 每一排都是两个连续的小正方形，如图所示，仅有一种基本图形。 (Each row consists of two consecutive small squares, as shown in the figure, there is only one basic pattern.) * **Summary:** 总结：两两相连各错一。 (Summary: Two by two connected, each off by one.) * **Annotation:** → 当分两行的时候只有一种情况。 (→ When divided into two rows, there is only one case.) * **Chart Type:** Grid pattern composed of squares. * **Main Elements (1 pattern for "222" type):** This pattern consists of 6 squares arranged in three rows, each row having two squares. The pattern forms a staircase-like shape where the rightmost square of an upper row is connected to the leftmost square of the row directly below it. **④ 第四类 ("33" 型展开图):** * **Description:** 只有两排图形，每排有三个连续的小正方形。如图所示，仅有一种基本图形。 (There are only two rows of shapes, each row has three consecutive small squares. As shown in the figure, there is only one basic pattern.) * **Summary:** 总结：三个两排一对齐。 (Summary: Three, two rows aligned.) * **Chart Type:** Grid pattern composed of squares. * **Main Elements (1 pattern for "33" type):** This pattern consists of 6 squares arranged in two horizontal rows, with each row having three consecutive squares. The two rows are stacked vertically, with the squares directly aligned. --- **实例讲解 (Example Explanation)** **长方体展开图 (Cuboid Net)** **(2) 长方体的展开图类似正方体的展开图，只是每一个小图形不是全等的正方形，观察下面长方体的展开图，你能发现什么规律？ (The nets of cuboids are similar to the nets of cubes, except that each small shape is not an identical square. Observe the nets of the cuboid below, what pattern can you find?)** * **Chart Type:** Grid patterns composed of rectangles (representing faces of a cuboid). * **Main Elements (4 patterns for cuboid nets):** These patterns are structurally identical to some of the cube nets, but composed of rectangles instead of squares. 1. A central horizontal row of 4 rectangles. One rectangle attached above the second from the left. One rectangle attached below the third from the left. (Corresponds to "141" type, the classic cross shape). 2. A central horizontal row of 3 rectangles. Two rectangles attached above the leftmost two. One rectangle attached below the rightmost. (Corresponds to "231" type). 3. Three rows, each with 2 rectangles, arranged in a staircase fashion. (Corresponds to "222" type). 4. Two horizontal rows, each with 3 rectangles, stacked vertically and aligned. (Corresponds to "33" type). * **Table Content (Example of a cuboid net with face labels):** | | 上面 (Top) | | | | :--- | :--------- | :--- | :--- | | 左面 (Left) | 前面 (Front) | 右面 (Right) | 后面 (Back) | | | 下面 (Bottom) | | |