Track Faces, Edges, and Vertices after Cutting and Folding
切割与折叠后的面、棱和顶点
A cut can remove old parts and create new ones. A net can look flat but hide a complete solid. Learn to keep a structural ledger so nothing is lost or counted twice.
Speak the language of a solid
Before tracking a change, name exactly what is being counted.
Hidden edges are dashed. A drawing is not a measurement tool.
Face
A flat surface of a solid. A cube has six square faces.
Edge
A line segment where two faces meet. A cube has twelve edges.
Vertex
A corner point where edges meet. A cube has eight vertices.
Check the vocabulary
Keep a cut ledger
In Missions 2–5, one plane cuts off exactly one original cube corner. It meets the three edges from that corner, either inside each edge or at its other endpoint. Keep the other original vertices and count only the larger remaining solid; set the removed corner piece aside. The remaining solid gains one cut face. Counting both pieces, as in Lesson 6.2, would give two new faces.
The original vertex is no longer part of the remaining solid.
The plane crosses the interiors of the three edges that met at the removed corner.
They form one new triangular face.
Complete the generic cut ledger
Make the remaining solid have as many vertices as possible
Place all three intersections inside edges. This creates the maximum number of new vertices.
The goal is the greatest possible number of remaining vertices under Mission 2’s single-corner rule. The key calculation is:
8 − 1 + 3 = 10 verticesThe edge and face counts are additional instructional scaffolds. The three old edges at the corner become shorter but remain edges; the triangular cut adds three new edges.
12 + 3 = 15 edges 6 + 1 = 7 facesAudit the maximum case
Make the remaining solid have as few vertices as possible
Move the cutting plane through the three vertices joined by an edge to the removed corner. Keep those three vertices on the remaining solid. Then no new interior edge-points are created.
The cut still creates one triangular face, so the solid has seven faces. Its three removed corner edges are replaced by the three sides of the new triangular face, so the edge count remains twelve.
Audit the minimum case
Compare the two legal corner cuts
The phrase “cut off one corner” does not determine one vertex count. The position of the plane matters.
| Structural count | Plane crosses 3 edge interiors | Plane passes through 3 neighboring vertices |
|---|---|---|
| Old corner removed | 1 | 1 |
| New vertices created | 3 | 0 |
| Vertices remaining | 10 | 7 |
| New cut face | triangle | triangle |
| Total faces | 7 | 7 |
| Total edges | 15 | 12 |
What stays the same?
Both cuts remove one corner and add one triangular face.
What changes?
The number of new intersection vertices depends on where the plane meets the cube’s edges.
Optional convex-solid check
This relationship is an optional verification tool, not required for this lesson:
vertices − edges + faces = 2Maximum case: 10−15+7=2. Minimum case: 7−12+7=2.
Choose the true statements
Fold a composite net into one solid
The net contains one regular pentagon, five matching rectangles, and five equilateral triangles. Edges joined in the displayed net have equal lengths, and folding preserves each face without overlap. Understand the folded solid before counting its edges.
Five rectangles
They wrap into the side faces of a pentagonal prism section.
One pentagon
It closes one end of the prism section.
Five triangles
Their free tips meet at one apex, creating a pentagonal pyramid section.
Read the folded structure
Count the folded solid’s edges without double counting
Count edges by structural families after folding, not by tracing every segment in the flat net.
The total is therefore 20.
Optional check using faces and vertices
The solid has 11 faces and 11 vertices. For this convex solid:
11−20+11=2Complete the edge-family count
A wire frame counts edge lengths—not surface area
A cuboid frame is taken apart and its wire reused to make all twelve edges of a cube frame. No wire is lost, added, or overlapped.
A cuboid has four edges of each dimension:
total wire = 4(length+width+height)A cube has twelve equal edges:
cube edge = total wire ÷ 12Solve the wire-frame problem
Use the fixed 15 × 8 × 7 dm dimensions, regardless of the sliders.
Faces–edges–vertices workshop
Use a structural count or a cut ledger. Do not estimate from appearance.
1 · Cube structure
How many faces does a cube have?
Hint
Top, bottom, front, back, left, right.
2 · Maximum corner cut
A plane crosses the interiors of the three edges at one cube corner. How many vertices remain?
Hint
Use 8−1+3.
3 · Minimum corner cut
The plane passes through the three neighboring existing vertices. How many vertices remain?
Hint
No new interior vertex is created.
4 · Maximum-case edges
How many edges does Mission 3’s solid have, when the plane crosses three edge interiors?
Hint
The triangular cut adds three edges.
5 · Composite solid faces
One pentagon, five rectangles, and five triangles fold into how many faces?
Hint
Add the face pieces in the net.
6 · Composite solid vertices
Two pentagonal rings and one apex give how many vertices?
Hint
5+5+1.
7 · Composite solid edges
Find the total from four edge families of five.
Hint
5+5+5+5.
8 · Wire frame
A 15×8×7 dm cuboid frame becomes a cube frame. Find the cube edge in decimetres.
Hint
4(15+8+7)÷12.
Exit ticket
Complete five objective checks without using the worked solutions above.
Certificate of mastery
Solid-Structure Tracker
This certifies that the learner can track faces, edges, and vertices after cutting and folding.
Teaching notes
The edge and face counts for the two cut solids, the staged fold visualizations, and the optional convex-solid check are added instructional scaffolds.