1997 AMC 12 Problem 23

Question

In the figure, polygons

A,

E,

and

F

are isosceles right triangles;

B,

C,

and

D

are squares with sides of length

1;

and

G

is an equilateral triangle. The figure can be folded along its edges to form a polyhedron having the polygons as faces. The volume of this polyhedron is

\frac12

\frac23

\frac34

\frac56

\frac43

Answer: D

Concepts:nets polyhedron volume

Difficulty rating: 2120

Small hint:

Recognize the three unit-square faces as faces meeting at a corner of a unit cube

Big hint:

The triangular faces cap the solid obtained by slicing one corner from that cube

Solution:

The net forms a unit cube with one corner cut off. The removed corner is a triangular pyramid with three mutually perpendicular unit edges, so its volume is

\frac13\cdot\frac12\cdot1=\frac{1}{6}.

The remaining polyhedron has volume

1-\frac{1}{6}=\frac{5}{6},

and the correct answer is D.