Count the change a cut makes, not the finished picture
7.G.A.36.G.A.4
Generated variants — 4
The solid shown is a rectangular prism with two of its corners cut off, each cut taking away a triangular pyramid. How many edges does this solid have?
Show solution
1 · Understandwhat's really being asked
A rectangular prism has two of its corners sliced off, each slice taking away a small triangular pyramid. We want the number of edges the remaining solid has.
Givens
- The solid starts as a rectangular prism.
- Two corners are cut off.
- Each cut removes a triangular pyramid and leaves a triangular face.
Unknowns
- The number of edges of the cut solid.
Constraints
- The cuts do not meet each other, so no cut removes another cut's work.
2 · Planchoose the strategy
#16 Count the Complement
Counting the finished picture is unreliable -- some of the cut corners face away from us. Count what one cut changes instead, then apply it as many times as there are cuts.
3 · Execute4 carry out the plan
1Start from the uncut box
2See what one cut destroys
3See what one cut adds
4Apply it two times
4 · Reviewdoes it hold up?
Euler's rule agrees: 8 faces minus 18 edges plus 12 vertices is 2.
Standardsmin grade 7
7.G.A.3Describe the two-dimensional figures that result from slicing three-dimensional figures — Reading the flat face a slice through a solid leaves behind.6.G.A.4Surface area using nets of 3D figures — Counting the prism's own edges before anything is cut.
The solid shown is a rectangular prism with three of its corners cut off, each cut taking away a triangular pyramid. How many edges does this solid have?
Show solution
1 · Understandwhat's really being asked
A rectangular prism has three of its corners sliced off, each slice taking away a small triangular pyramid. We want the number of edges the remaining solid has.
Givens
- The solid starts as a rectangular prism.
- Three corners are cut off.
- Each cut removes a triangular pyramid and leaves a triangular face.
Unknowns
- The number of edges of the cut solid.
Constraints
- The cuts do not meet each other, so no cut removes another cut's work.
2 · Planchoose the strategy
#16 Count the Complement
Counting the finished picture is unreliable -- some of the cut corners face away from us. Count what one cut changes instead, then apply it as many times as there are cuts.
3 · Execute4 carry out the plan
1Start from the uncut box
2See what one cut destroys
3See what one cut adds
4Apply it three times
4 · Reviewdoes it hold up?
Euler's rule agrees: 9 faces minus 21 edges plus 14 vertices is 2.
Standardsmin grade 7
7.G.A.3Describe the two-dimensional figures that result from slicing three-dimensional figures — Reading the flat face a slice through a solid leaves behind.6.G.A.4Surface area using nets of 3D figures — Counting the prism's own edges before anything is cut.
The solid shown is a rectangular prism with four of its corners cut off, each cut taking away a triangular pyramid. How many edges does this solid have?
Show solution
1 · Understandwhat's really being asked
A rectangular prism has four of its corners sliced off, each slice taking away a small triangular pyramid. We want the number of edges the remaining solid has.
Givens
- The solid starts as a rectangular prism.
- Four corners are cut off.
- Each cut removes a triangular pyramid and leaves a triangular face.
Unknowns
- The number of edges of the cut solid.
Constraints
- The cuts do not meet each other, so no cut removes another cut's work.
2 · Planchoose the strategy
#16 Count the Complement
Counting the finished picture is unreliable -- some of the cut corners face away from us. Count what one cut changes instead, then apply it as many times as there are cuts.
3 · Execute4 carry out the plan
1Start from the uncut box
2See what one cut destroys
3See what one cut adds
4Apply it four times
4 · Reviewdoes it hold up?
Euler's rule agrees: 10 faces minus 24 edges plus 16 vertices is 2.
Standardsmin grade 7
7.G.A.3Describe the two-dimensional figures that result from slicing three-dimensional figures — Reading the flat face a slice through a solid leaves behind.6.G.A.4Surface area using nets of 3D figures — Counting the prism's own edges before anything is cut.
The solid shown is a rectangular prism with one of its corners cut off, each cut taking away a triangular pyramid. How many edges does this solid have?
Show solution
1 · Understandwhat's really being asked
A rectangular prism has one of its corners sliced off, each slice taking away a small triangular pyramid. We want the number of edges the remaining solid has.
Givens
- The solid starts as a rectangular prism.
- One corner is cut off.
- Each cut removes a triangular pyramid and leaves a triangular face.
Unknowns
- The number of edges of the cut solid.
Constraints
- The cuts do not meet each other, so no cut removes another cut's work.
2 · Planchoose the strategy
#16 Count the Complement
Counting the finished picture is unreliable -- some of the cut corners face away from us. Count what one cut changes instead, then apply it as many times as there are cuts.
3 · Execute4 carry out the plan
1Start from the uncut box
2See what one cut destroys
3See what one cut adds
4Apply it one time
4 · Reviewdoes it hold up?
Euler's rule agrees: 7 faces minus 15 edges plus 10 vertices is 2.
Standardsmin grade 7
7.G.A.3Describe the two-dimensional figures that result from slicing three-dimensional figures — Reading the flat face a slice through a solid leaves behind.6.G.A.4Surface area using nets of 3D figures — Counting the prism's own edges before anything is cut.
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