Observing everyday objects like paper cups, cardboard boxes, hourglasses, pyramids, tea containers, diamonds, milk cartons, basketballs, and plumb lines, we notice they occupy three-dimensional space. The task of mathematics is to extract their essential properties from these intuitive perceptions and systematically study their structural characteristics. We refer to geometric solids formed by plane polygons aspolyhedron, while those generated by rotation are calledrotational solid.
Core Definitions and Classification
According to Chapter 8 of the People's Education Press' Compulsory Course 2, we need to master the following fundamental concepts:
- Polyhedron (Polyhedron): a geometric solid formed by several plane polygons. The common edge between two adjacent polygons is callededge.
- Prism (Prism): two faces are mutually parallel, all other faces are quadrilaterals, and the common edges between adjacent quadrilaterals are parallel to each other.
- Rotational Surface: a curved surface formed by rotating a planar curve around a fixed straight line within its plane.
The study of spatial geometric solids follows the logic of 'point → line → surface → solid,' with emphasis on using the two core positional relationships—parallelism and perpendicularity—to define different geometric structures.
$$V_{\text{prism}} = Sh, \quad V_{\text{pyramid}} = \frac{1}{3}Sh, \quad V_{\text{sphere}} = \frac{4}{3}\pi R^3$$
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 features.
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 have edges.
The paper cup is a cylinder because it has the same width at the top and bottom.
All these objects are obtained through rotation.
Correct. According to Section 8.1, cardboard boxes belong to polyhedra (prisms), while paper cups and hourglasses are rotational solids. The key to identification lies in how they are generated—whether formed by enclosing polygons or by rotating curves.
Hint: Pay attention to whether the side of the object is a curved surface or a flat plane. The side of a paper cup unfolds into a sector ring, making it a rotational solid; the side of a cardboard box is a rectangle, making it 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) A quadrilateral prism, a quadrilateral frustum, and a pentagonal pyramid are all hexahedra.
(1) Incorrect (2) Correct
(1) Correct (2) Incorrect
(1) Correct (2) Correct
(1) Incorrect (2) Incorrect
Correct. (1) A rectangular prism is indeed a quadrilateral prism. However, the base of a right quadrilateral prism only needs 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. The lateral edges of a right quadrilateral prism 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 all other faces are congruent rectangles. Then this solid is ______. (2) A polyhedron has a minimum of ______ faces, and at that 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 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 needed to enclose a closed space?
QUESTION 4
4. A cylinder can be obtained by rotating a rectangle, and a cone by rotating a right triangle. 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 generates a frustum, with the remaining three sides forming the surface.
Hint: Consider the characteristic that the top and bottom bases of a frustum are unequal in size but parallel. The axis of rotation must be perpendicular to both circular faces.
QUESTION 5
5. Regarding Zu Geng'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 equal heights, their volumes are equal
As long as two geometric solids have equal base areas, their volumes are equal
If the cross-sectional areas at equal heights are always equal, then the volumes are equal
This principle applies only to prisms and not to spheres
Correct. Zu Geng'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 used to derive 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 term 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 line $AC$?
parallel
intersecting
skew
perpendicular and intersecting
Correct. Line $A'B$ lies in plane $A'B'BA$, and line $AC$ intersects this plane at point $A$, but point $A$ does not lie on line $A'B$. Therefore, 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 combination of a cylinder and a cone
a frustum
Correct. A right trapezoid can be divided into a rectangle and a right triangle. The rectangle forms a cylinder when rotated, and the right triangle forms a cone. Together, they form a composite solid.
Hint: Decompose complex shapes into basic ones (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. Selecting 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 four 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 $6 + F - 12 = 2$, so $F = 8$. This is a regular octahedron.
Hint: Apply Euler's formula for polyhedra: vertices + faces - edges = 2.
Challenge: The 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: Assume a regular $n$-gonal prism whose base is inscribed in a circle of radius $r$. How does the relationship between the lateral edges and the base change as $n$ increases? 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 Zu Geng's Principle, since they have equal heights and identical cross-sectional areas ($S$) at every horizontal level, their volumes must be equal. This embodies the idea of 'if powers and positions are equal, then 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) attached to the top and bottom 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). If it is 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 unfolding and folding of spatial figures.
Q3
Reasoning: Cutting a pyramid with a plane parallel to the base produces a frustum. If the cross-sectional area is half the base area, what is the ratio of the cut 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. $S_{cut} : S_{base} = h_{small}^2 : h_{large}^2 = 1 : 2$, so $h_{small} : h_{large} = 1 : \sqrt{2}$. This illustrates the nonlinear proportional relationship in measuring spatial geometric solids.
✨ Key Points
Polyhedron,formed by planes,prisms and pyramids differ in their bases.Rotational solid,rotated around an axis,cylinders, cones, and spheres are central.Parallel and perpendicularare key, spatial imagination stands at the core!
💡 Distinguish Polyhedra from Rotational Solids
Polyhedra are 'assembled' from plane polygons (with edges and corners), while rotational solids are 'swept' out by planar figures (typically having circular or curved surfaces).
💡 Right Prism vs. Regular Prism
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 Zu Geng's Principle
‘If powers and positions are equal, then volumes cannot differ.’ As long as the area of each horizontal cross-section is equal, the volume remains unchanged even if the shape is distorted.
💡 Formula Memory Tip
The formulas for cylinders, pyramids, and frustums are unified. When the top base area of a frustum is zero, it becomes a pyramid (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 another line within the plane that does not pass through that point will be skew to the original line within the plane.