Problem
More faces means more vertices and edges
Two corners of a rectangular prism are sliced off, each cut removing a small triangular pyramid. We need the number of edges the new solid has.
Geometry
Your answer
How to solve
Strategy Visualize Spatial Relationships — Do not recount the whole solid. Start from the prism, which is known, and ask only what one cut changes -- then do that change twice.
1STEP 1
Start from the prism you know
A rectangular prism has 6 faces, 12 edges and 8 vertices.
6, 12, 8
That is the starting point.
6.G.A.4Draw A Diagram2STEP 2
See what one cut changes
Slicing a corner off replaces that one vertex with a triangle: the vertex goes, three new ones appear where the cut meets the three edges, and the triangle's three sides are new edges.
+1 face, +3 edges, -1+3 vertices
One cut adds three edges.
7.G.A.3Visualize Spatial RelationshipsOne corner cut adds three edges.
Why?
Slicing a corner off replaces that single vertex with a triangular face, and each of the triangle's three sides is an edge that was not there before.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
The three edges that met at that vertex are only shortened, not removed, so the prism's own twelve edges all survive the cut.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
3STEP 3
Apply the change twice
Two corners are cut, so the solid gains two triangular faces, six edges, and four vertices overall.
(faces) = 8, (vertices) = 12
8 faces and 12 vertices now.
6.G.A.4Identify Subproblems4STEP 4
Count the edges
The 12 original edges are all still there, and each cut added three more.
12 + 3 + 3 = 18
18 edges.
6.G.A.4Identify SubproblemsAnswer
18
Euler's rule checks it: 12 - 18 + 8 = 2, as it must be for any solid of this kind.
Takeaway
Work out what one change does, then repeat it -- much safer than recounting a complicated picture.
- Start from the prism you know
- See what one cut changes
- Apply the change twice
- Count the edges