Observe everyday objects like paper cups, cardboard boxes, hourglasses, pyramids, tea containers, diamonds, milk cartons, basketballs, and plumb lines. We notice these objects occupy three-dimensional space. The task of mathematics is to extract their essential characteristics from these intuitive perceptions and systematically study their structural features. We refer to geometric solids formed by plane polygons aspolyhedra, while those generated through rotation are calledrotational solids.
Core Definitions and Classification
According to Chapter 8 of the People's Education Press Selective Compulsory Course 1, we need to master the following fundamental concepts:
- Polyhedron (Polyhedron): a geometric solid enclosed by several planar polygons. The common edge between two adjacent polygons is callededge.
- Prism (Prism): has two faces that are mutually parallel, with all other faces being quadrilaterals, and the common edges between adjacent quadrilaterals are parallel to each other.
- Rotational Surface: a surface formed by rotating a planar curve around a fixed straight line within its own plane.
The study of spatial geometric solids follows the logic of 'point → line → surface → solid,' focusing on using the two core positional relationships—'parallel' and 'perpendicular'—to define different geometric structures.
$$V_{\text{prism}} = Sh, \quad V_{\text{cone}} = \frac{1}{3}Sh, \quad V_{\text{sphere}} = \frac{4}{3}\pi R^3$$
1. Gather polynomial terms: one x² square, three x rectangular strips, and two 1×1 unit squares.
2. Begin geometrically assembling them.
3. They perfectly form a larger continuous rectangle! Width is (x+2), height is (x+1).
QUESTION 1
1. Observe geometric objects around you (such as paper cups, cardboard boxes, hourglasses) and describe their main structural characteristics.
Paper cups are typically frustums, cardboard boxes are rectangular prisms (quadrilateral prisms), and hourglasses are combinations of two cones.
All objects are polyhedra because they all have edges.
The paper cup is a cylinder because it has the same thickness at the top and bottom.
All these objects are generated by rotation.
Correct. According to Section 8.1, cardboard boxes belong to polyhedra (prisms), while paper cups and hourglasses belong to rotational solids. The key to identification lies in how they are generated—whether by enclosing polygons or rotating curves.
Hint: Pay attention to whether the side of the object is a curved surface or a flat plane. The lateral surface of a paper cup unfolds into a sector ring, classifying it as a rotational solid; the lateral surface of a cardboard box is a rectangle, classifying it as a polyhedron.
QUESTION 2
2. Determine whether the following statements are true: (1) A rectangular prism is a quadrilateral prism, and a right quadrilateral prism is a rectangular prism; (2) Quadrilateral prisms, quadrilateral frustums, and pentagonal pyramids are all hexahedra.
(1) False (2) True
(1) True (2) False
(1) True (2) True
(1) False (2) False
Correct. (1) A rectangular prism is indeed a quadrilateral prism. However, a right quadrilateral prism only requires its base to be a parallelogram, not necessarily a rectangle, so it is not necessarily a rectangular prism. (2) A quadrilateral prism has 4 + 2 = 6 faces, a quadrilateral frustum has 4 + 2 = 6 faces, and a pentagonal pyramid has 5 + 1 = 6 faces—all meet the definition of a hexahedron.
Note: The base of a rectangular prism must be a rectangle. In a right quadrilateral prism, lateral edges are perpendicular to the base, but the base only needs to be a parallelogram. When counting faces, do not forget the bases.
QUESTION 3
3. Fill in the blanks: (1) A geometric solid is enclosed by 7 faces, where two faces are mutually parallel and congruent pentagons, and the other faces are all congruent rectangles. Then this solid is ______. (2) A polyhedron has a minimum of ______ faces, and at this point it is ______.
(1) Regular pentagonal prism; (2) 4, triangular pyramid
(1) Pentagonal pyramid; (2) 4, triangular prism
(1) Regular pentagonal prism; (2) 3, triangle
(1) Hexagonal prism; (2) 4, tetrahedron
Correct. (1) The lateral faces are rectangles and perpendicular to the base, and the base is a regular pentagon, so it is a regular pentagonal prism. (2) Three points determine a face. The simplest polyhedron is a triangular pyramid (tetrahedron) formed by four triangles.
Hint: (1) The mention of two parallel faces indicates a prism type. (2) Imagine how many faces are minimally required to enclose a closed space?
QUESTION 4
4. Can a cylinder be obtained by rotating a rectangle, a cone by rotating a right triangle, and can a frustum also be obtained by rotating a planar figure?
Yes, by rotating an isosceles trapezoid around one of its legs
Yes, by rotating a right trapezoid around the leg perpendicular to the base
No, a frustum can only be obtained by truncating a cone
Yes, by rotating a rectangle around its diagonal
Correct. Rotating a right trapezoid around the leg perpendicular to the base forms a frustum, where the remaining three sides generate the surfaces.
Hint: Consider the characteristic that the top and bottom bases of a frustum differ in size but are parallel. The axis of rotation must be perpendicular to both circular faces.
QUESTION 5
5. Regarding Zǔ Gèng’s Principle: 'If the powers and positions are equal, then the volumes cannot differ.' Which of the following understandings is correct?
As long as two geometric solids have the same height, their volumes are equal
As long as two geometric solids have equal base areas, their volumes are equal
If the cross-sectional areas are always equal at the same height, then the volumes are equal
This principle applies only to prisms and not to spheres
Correct. Zǔ Gèng’s Principle emphasizes that for a geometric solid sandwiched between two parallel planes, if any plane parallel to these two cuts produce equal cross-sectional areas, then the volumes are equal. This is the core logic behind deriving the volume of a sphere.
Hint: 'Power' refers to cross-sectional area, 'position' refers to height. Equal total area is both necessary and sufficient for equal volume.
QUESTION 6
6. A polyhedron has one face as a polygon, and all other faces are triangles sharing a common vertex. What is this polyhedron?
Prism
Frustum
Pyramid
Cone
Correct. This is the geometric definition of a pyramid. The common vertex is called the apex of the pyramid, and the polygon is called the base.
Hint: The key phrase is 'triangles sharing a common vertex.' The lateral faces of a prism are parallelograms.
QUESTION 7
7. In the rectangular prism $ABCD-A'B'C'D'$, what is the positional relationship between line $A'B$ and $AC$?
Parallel
Intersecting
Skew
Perpendicular and intersecting
Correct. Line $A'B$ lies in plane $A'B'BA$, while $AC$ intersects this plane at point $A$, and $A$ does not lie on line $A'B$, so the two lines are skew.
Hint: In space, lines that are neither parallel nor intersecting are called skew lines. Try observing in a rectangular prism model whether they lie in the same plane.
QUESTION 8
8. As shown in the diagram, rotate the right trapezoid $ABCD$ around the line containing its lower base $AB$ for one full revolution. What is the structural characteristic of this solid?
A cylinder
A cone
A composite body of a cylinder and a cone
A frustum
Correct. A right trapezoid can be divided into a rectangle and a right triangle. The rectangle generates a cylinder when rotated, and the right triangle generates a cone. Their combination forms a composite body.
Hint: Break down complex figures into basic shapes (rectangle, right triangle) and consider their individual rotation paths.
QUESTION 9
9. How many planes can be determined by four non-coplanar points?
1
2
3
4
Correct. Any three points determine a plane. Choosing any three out of four points gives $C_4^3 = 4$ combinations, forming the four faces of a triangular pyramid (tetrahedron).
Hint: Imagine a triangular pyramid. Its four vertices are the non-coplanar points—how many faces does it have?
QUESTION 10
10. A polyhedron has 6 vertices and 12 edges. What is its number of faces $F$?
6
8
10
12
Correct. According to Euler's formula $V + F - E = 2$, substituting gives $6 + F - 12 = 2$, solving yields $F = 8$. This is a regular octahedron.
Hint: Apply the polyhedron's Euler's formula: number of vertices + number of faces - number of edges = 2.
Challenge: Structural Evolution of Geometric Solids
The Concept of Limits: From Prisms to Cylinders
When studying the volume of geometric solids, we often say 'a cylinder is a regular prism with the number of base edges approaching infinity.' Use the knowledge from this chapter to answer the following logical reasoning questions.
Case Study: Suppose a regular $n$-gonal prism has its base inscribed in a circle of radius $r$. As $n$ increases, how does the relationship between the lateral edges and the base change? How does the volume formula transition?
Q1
If a regular triangular prism, a regular quadrilateral prism, and a regular hexagonal prism all have height $h$ and base area $S$, are their volumes equal? Why?
Answer: Volumes are equal.
Explanation: Based on the prism volume formula $V = Sh$, volume depends only on base area and height. From Zǔ Gèng’s Principle, since they have the same height and equal cross-sectional areas at any horizontal level (both $S$), their volumes must be equal. This embodies the idea of 'if the powers and positions are equal, then the volumes cannot differ'.
Q2
Design a planar figure that, when folded, forms a triangular prism. Explain the relationship between the lateral edges and the base.
Answer: The net should include three adjacent rectangles (lateral faces) and two triangles (bases) connected to the top and bottom ends of one of the rectangles.
Explanation: In a right triangular prism, the creases (lateral edges) must be perpendicular to the sides of the triangle (part of the base perimeter). In an oblique triangular prism, the creases are not perpendicular to the base. This exercise aims to reinforce understanding of the invariance of 'distance' and 'angle' during the unfolding and folding of spatial figures.
Q3
Reasoning: Cutting a pyramid with a plane parallel to its base produces a frustum. If the cross-sectional area is half the base area, what is the ratio of the section height to the original pyramid height?
Answer: $\frac{1}{\sqrt{2}}$ (measured from the apex).
Explanation: According to the properties of similar polyhedra, the ratio of cross-sectional areas equals the square of the ratio of heights. $S_{ ext{section}} : S_{ ext{base}} = h_{ ext{small}}^2 : h_{ ext{large}}^2 = 1 : 2$, thus $h_{ ext{small}} : h_{ ext{large}} = 1 : \sqrt{2}$. This reflects the nonlinear proportional relationship in measuring spatial geometric solids.
✨ Key Points
Polyhedra,are bounded by planes,prisms and pyramids differ in base.Rotational solids,rotate around an axis,cylinders, cones, and spheres are among them.Parallel and perpendicularare central, spatial imagination stands at the core!
💡 Distinguish Polyhedra from Rotational Solids
Polyhedra are 'assembled' from planar polygons (with edges and vertices), while rotational solids are 'swept' from planar figures (typically featuring circular or curved surfaces).
💡 Right Prisms vs. Regular Prisms
In a right prism, lateral edges are perpendicular to the base. A regular prism requires the base to be a regular polygon, in addition to being a right prism. Note: Only a right prism with a rectangular base is a rectangular prism.
💡 The Power of Zǔ Gèng’s Principle
‘If the powers and positions are equal, then the volumes cannot differ.’ As long as the areas of every horizontal cross-section are equal, the volume remains unchanged even if the shape is distorted.
💡 Formula Memorization Tip
The formulas for cylinders, cones, and frustums are unified. When the top base area of a frustum is zero, it becomes a cone (multiply by 1/3); when the top base area equals the bottom base area, it becomes a cylinder.
💡 Determining Skew Lines
The most common method to determine skew lines: a line passing through a point outside a plane and determined by a line inside the plane that does not pass through that point will be skew to the original line in the plane.