AMC 10 · 2015 · #16

Grade 10 geometry-3d

Archetype Net Folding / Face Adjacency └ Sub-archetype Net-Edge Matching for Length, Area, or Volume

volume-pyramidarea-regular-hexagonpythagorean-theoremnet-folding spatial-visualizationidentify-subproblems ↑ Prerequisites: pythagorean-theoremvolume-pyramid

📏 Long solution 💡 3 insights

Problem

Triangles glued to each side of a regular hexagon fold up into a closed pyramid. Find the volume.

Pick an answer.

(A)

(B)

162

(C)

$36\sqrt{21}$

(D)

$18\sqrt{138}$

(E)

$54\sqrt{21}$

How to solve

Strategy Visualize Spatial Relationships

The whole problem is a folding problem, so the first job is to see what the fold guarantees: the six triangle tips must all land on the same point, and each of the six slanted edges keeps its length 8. Once I see that, the volume formula for a pyramid splits the work into two independent subproblems (Identify Subproblems): the area of the hexagon base, and the height of the apex above it. The height is the delicate half, because the formula only works if I use the true vertical height, not the slanted length of a triangle. So I first prove the apex sits exactly over the hexagon's center, then compute the height twice from two different right triangles (Organize Information in More Ways) to make sure I picked the right horizontal leg.

1STEP 1

See what the fold forces

Folding makes every slant edge the same.

each flap: sides 8, 8, 6 → PV₁ = PV₂ = … = PV₆ = 8

2STEP 2

Measure the hexagon from its center

The hexagon is measured from its centre.

OV_i = 6, OM = 6 · √(3)/2 = 3√(3) ≈ 5.196

3STEP 3

Add up the base area

Its area comes to 54√3.

[hexagon] = 6 · √(3)/4 · 6² = 6 · 9√(3) = 54√(3)

4STEP 4

Prove the apex is over the center

Equal edges put the apex over the centre.

FV_i^ 2 = 8² - PF² (same for every i) → F = O

5STEP 5

Get the height two different ways

Two routes give the same height.

h² = 8² - 6² = 28; PM = √(8² - 3²) = √(55), h² = 55 - (3√(3))² = 55 - 27 = 28 → h = 2√(7)

6STEP 6

Multiply into the volume

Multiplying gives 36√21, choice (C).

V = 1/3 · 54√(3) · 2√(7) = 36√(3 · 7) = 36√(21)

Answer

36√(21)

Numerically the height is about 5.29 and the base area is about 93.5, so the volume is about 93.5 times 5.29 divided by 3, roughly 165. And 36 times the square root of 21 is about 164.97, so choice (C) matches. A shape check: the pyramid must fit inside the hexagonal prism of the same base and height, whose volume is about 495, and 165 is exactly one third of that, as a pyramid should be. The wrong choices are all height mistakes with the same base, which is why the double check in step 5 mattered. Using the center-to-edge distance 3 times the square root of 3 as if it were the height gives exactly 162, choice (B). Using half the side, 3, instead of the center-to-edge distance under the slant of the square root of 55 gives a height of the square root of 46 and lands on 18 times the square root of 138, choice (D). Choice (E) is one and a half times the true value. Also worth noting the fold only works because 8 is strictly bigger than the corner-to-center distance 6; if the triangles had legs of exactly 6 the flaps would fold flat and the volume would be zero.

💡Key takeaway

Folding keeps every edge length, so all six tips land on one point above the center, and the pyramid's height is the leg you get when you pair a slanted 8 with the right flat distance on the base.

See what the fold forces
Measure the hexagon from its center
Add up the base area
Prove the apex is over the center
Get the height two different ways
Multiply into the volume