 equal to $\frac{1}{3}$ times the base area times the height.
In this case, the base of pyramid $ABCH$ is the triangle $ABC$, which is a right triangle with legs of length 1.
The height of the pyramid is the length of the segment $AH$, which is also 1.
Therefore, the volume of pyramid $ABCH$ is $\frac{1}{3} \cdot \frac{1}{2} \cdot 1 \cdot 1 = \boxed{\frac{1}{6}}$.The answer is: \frac{1}{6}
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
$ABCDEFGH$ shown below is a cube with volume 1.  Find the volume of pyramid $ABCH$.

[asy]
import three;
triple A,B,C,D,EE,F,G,H;
A = (0,0,0);
B = (1,0,0);
C = (1,1,0);
D= (0,1,0);
EE = (0,0,1);
F = B+EE;
G = C + EE;
H = D + EE;
draw(B--C--D);
draw(B--A--D,dashed);
draw(EE--F--G--H--EE);
draw(A--EE,dashed);
draw(B--F);
draw(C--G);
draw(D--H);
label("$A$",A,S);
label("$B$",B,W);
label("$C$",C,S);
label("$D$",D,E);
label("$E$",EE,N);
label("$F$",F,W);
label("$G$",G,SW);
label("$H$",H,E);
[/asy]

### Answer:
