Depth · one concept from first principles · step 2 of 4
Cutting a piece away removes that much volume
The rectangular prism at the right is cut along the red line. What is the volume of one of the two pieces, in ?
Geometry
Your answer
How to solve
Strategy Visualize Spatial Relationships — The two pieces are triangular prisms, and a diagonal splits a rectangle into two congruent triangles. So the pieces are congruent, and there is no need to work either one out on its own -- halve the whole.
1STEP 1
See that the two pieces match
The diagonal cuts the top into two congruent triangles and the cut goes straight down, so the two pieces are congruent triangular prisms.
Each piece is exactly half the box.
6.G.A.1Visualize Spatial RelationshipsThe two pieces the cut makes are the same size.
Why?
A diagonal cuts the top rectangle into two triangles that lie exactly on each other after a half turn, so they cover the same area.
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
Why?
The cut runs straight down, so each piece is that triangle carried through the same of height, and the two pieces fill the box between them.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
2STEP 2
Find the whole prism's volume
Multiply the three edges.
16 × 10 × 7 = 1120
The box is 1120 cm³.
6.G.A.2Draw A Diagram3STEP 3
Halve it
The two congruent pieces share the volume equally.
1120 ÷ 2 = 560
One piece is 560 cm³.
6.G.A.2Identify SubproblemsAnswer
560 cm³
Work one piece out directly: its triangular face is 16 × 10 ÷ 2 = 80 cm², and 80 × 7 = 560 cm³.
Takeaway
A diagonal cut through a box gives two matching halves -- so halve the whole instead of measuring a piece.
- See that the two pieces match
- Find the whole prism's volume
- Halve it
Where next?
Another one like thissuggested
Step back and solidifysuggested
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