Problem 24 · 2017 Math Kangaroo
Stretch
Geometry & Measurement
area-fractionspatial-reasoning
We look at a regular tetrahedron with volume 1. Its four vertices are cut off by planes that go through the midpoints of the respective edges (see diagram). How big is the volume of the remaining solid?
Show answer
Answer: D — 12
Show hints
Hint 1 of 2
Each cut through edge midpoints slices off a small tetrahedron similar to the whole, at half scale.
Still stuck? Show hint 2 →
Hint 2 of 2
A half-scale tetrahedron has 1/8 the volume; account for all four corners.
Show solution
Approach: subtract four half-scale corner tetrahedra
- Cutting through the midpoints removes a corner tetrahedron with edges half as long, so each has volume (1/2)^3 = 1/8.
- The four corner pieces do not overlap, removing 4 x 1/8 = 1/2 of the volume.
- The remaining solid has volume 1 - 1/2 = 1/2.
Mark:
· log in to save