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2020 AMC 10A Problem 10

2020 AMC 10A Problem 10AMC 10 Step by Step

Name: Date: September 13, 2026

Solid Geometry · difficulty 2 of 5 · about 3 min · position P1-10

Seven cubes, whose volumes are $1$ , $8$ , $27$ , $64$ , $125$ , $216$ , and $343$ cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except for the bottom cube, the bottom face of each cube lies completely on top of the cube below it. What is the total surface area of the tower (including the bottom) in square units?

A)
$644$
B)
$658$
C)
$664$
D)
$720$
E)
$749$

Answer

Key insight

Seen from above the tower shows exactly a 7-by-7 square of exposed top faces, and likewise from below; only the side faces need summing.

Solution

The cubes have side lengths $1, 2, 3, 4, 5, 6, 7$ .

Side faces: every cube shows all four of its vertical faces, contributing
$4(1^2 + 2^2 + \cdots + 7^2) = 4 \cdot 140 = 560.$

Top faces: looking straight down, every point of the bottom cube's $7 \times 7$ footprint is covered by exactly one exposed top surface (part of the $7$ -cube's top, or the top of some smaller cube resting above it). So the exposed top area is $49$ .

Bottom face: the base of the $7$ -cube, area $49$ .

Total: $560 + 49 + 49 = 658$ .

The answer is $\boxed{\textbf{(B)}\ 658}$ .

Why this works

Stacked-solid surface areas split into vertical and horizontal parts. Vertical faces are never hidden in a straight stack, while the horizontal exposed area is simply the footprint seen from above (and from below), regardless of how the smaller cubes sit. Projection replaces a messy overlap subtraction.

Alternative approach

Sum all surface areas, $6 \cdot 140 = 840$ , then remove each contact region twice (the hidden top of the lower cube and the hidden bottom of the upper cube): $2(1 + 4 + 9 + 16 + 25 + 36) = 182$ . Then $840 - 182 = 658$ .

The trap

Subtracting the hidden overlap area only once (top of a cube hidden but not the bottom of the cube above), which gives 749.

Common mistakes

Subtracting the hidden overlap area only once (top of a cube hidden but not the bottom of the cube above), which gives 749.
Forgetting to include the bottom of the tower (giving $609$ ) or using volumes instead of side lengths for face areas.

Techniques

Cut the figure into known shapes (triangles, rectangles, sectors) · Exploit symmetry to reduce work or pair up objects