AMC 10 · 2007 · #23

Grade 8 geometry-3d
Archetype Similar-Triangle Length / Area Ratio └ Sub-archetype Synthetic Similar-Triangle Length or Area
similar-figuressurface-areaquadratic-equations convert-to-algebra ↑ Prerequisites: similar-figures
📏 Medium solution 💡 3 insights
Problem
A square-based pyramid is sliced by a plane parallel to the base, 2 units above the base. The top piece is a smaller pyramid whose total surface area is exactly half the total surface area of the original pyramid. Find the height of the original pyramid.

Pick an answer.

(A)
2
(B)
$2+\sqrt{2}$
(C)
$1+2\sqrt{2}$
(D)
4
(E)
$4+2\sqrt{2}$

AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Visualize Spatial Relationships

A plane parallel to the base cuts off a top pyramid that is a shrunk copy of the whole pyramid, so the two solids are similar. Seeing that similarity is the whole game: it links the surface areas to the heights. Then naming the height with a variable turns the area condition into a single equation to solve.

1STEP 1

See the two similar pyramids

A cut parallel to the base leaves a top pyramid that is a scaled copy of the whole one, so the two are similar.

small pyramid ∼ original pyramid
2STEP 2

Areas scale as height squared

For similar solids the surface area scales as the square of the height ratio kk, and here that area ratio is 12\frac{1}{2}.

((small height)/(original height))² = 1/2
3STEP 3

Name the height and build the equation

Let hh be the original height. The cut sits 2 above the base, so the top pyramid's height is h−2h-2.

((h-2)/h)² = 1/2
4STEP 4

Take the square root

Both h−2h-2 and hh are positive lengths, so keep only the positive root: the height ratio is 12\frac{1}{\sqrt{2}}.

(h-2)/h = 1/√(2)
5STEP 5

Solve for the height

Collect the hh terms and rationalize the denominator to get h=4+22h = 4+2\sqrt{2}, which is choice (E).

h - 2 = h/√(2) → h(1 - 1/√(2)) = 2 → h = 2√(2)/(√(2)-1) = 2√(2)(√(2)+1) = 4 + 2√(2)
Answer
4+2√(2)
The height 4 + 2√(2) ≈ 6.83, so the small pyramid's height is about 4.83. Their ratio is about 0.707, which is 1/√(2), and squaring gives 0.5 exactly. The area of the top piece is indeed half the original, so the answer checks out.
💡Key takeaway

When one solid is a shrunk copy of another, its areas shrink by the square of the height ratio, so a half-area copy has height 1 over root 2 of the original.

  • See the two similar pyramids
  • Areas scale as height squared
  • Name the height and build the equation
  • Take the square root
  • Solve for the height

More like this

Same archetype — closest grade level first.