Triangular cupola volume
Source: 2014 SDMO Middle School Problem 5
August 28, 2016
Problem Statement
Below is a net consisting of squares, equilateral triangles, and regular hexagon. Each polygon has side length . When we fold this net to form a polyhedron, what is the volume of the polyhedron? (This figure is called a "triangular cupola".)Net:[asy]
pair A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P;
A = origin;
B = (0,3);
C = 3*dir(150);
D = (0,1);
E = (0,2);
F = C+2*dir(30);
G = C+dir(30);
H = 2*dir(150);
I = dir(150);
J = (1,1);
K = J+dir(30);
L = (1,2);
M = F+dir(120);
N = G+dir(120);
O = H+dir(240);
P = I+dir(240);
draw(A--B--C--cycle);
draw(D--E--F--G--H--I--cycle);
draw(D--E--L--J--cycle);
draw(F--G--N--M--cycle);
draw(H--I--P--O--cycle);
draw(J--K--L--cycle);
[/asy]Resulting polyhedron:https://upload.wikimedia.org/wikipedia/commons/9/93/Triangular_cupola.png