AMC 12 2025a Problem 20

The base of the pentahedron shown below is a $13 \times 8$ rectangle, and its lateral faces are two isosceles triangles with base of length $8$ and congruent sides of length $13$, and two isosceles trapezoids with bases of length $7$ and $13$ and nonparallel sides of length $13$.

/* Refined by Mathemagician108 */
import graph3;
size(200);
real l = 13;
real w = 8;
real offset = (l - 7)/2; // 3
real midy = w/2; // 4
real h = 12;
triple O1 = (0,0,0);
triple O2 = (l,0,0);
triple O3 = (l,w,0);
triple O4 = (0,w,0);
triple T1 = (offset, midy, h);
triple T2 = (l - offset, midy, h);
currentprojection=orthographic((-4,-6,3));
draw(O4--O1--O2, linewidth(1));
draw(O2--O3--O4, dashed + linewidth(1));
draw(O3--T2, dashed + linewidth(1));
draw(O1--T1, linewidth(1));
draw(O4--T1, linewidth(1));
draw(O2--T2, linewidth(1));
draw(T1--T2, linewidth(1));
label("13", (O1+O2)/2, 3*-Y); // Bottom length
label("13", (O2+T2)/2, 1.5*X);
label("13", (O4+T1)/2, 2*-X);
label("8", (O1+O4)/2, 2*-X); // Width
label("7", (T1+T2)/2, 1.5*Z); // Top length

What is the volume of the pentahedron?

\textbf{(A)}~416 \textbf{(B)}~520 \textbf{(C)}~528 \textbf{(D)}~676 \textbf{(E)}~832