Answer
3500 km.
Work Step by Step
The pencil diameter is approximately 7 mm. Suppose it is held 0.75 m from the eye and just blocks out the Moon.
From the diagram we see that two similar triangles are formed, with the pencil width corresponding to the moon width (the former blocks out the latter), and the pencil distance corresponding to the moon distance.
$$\frac{pencil\;diameter}{ pencil\;distance }=\frac{ moon\;diameter}{moon\;distance }$$
$$moon\;diameter=(moon\;distance)\frac{pencil\;diameter}{ pencil\;distance }$$
Putting in the estimated numbers, we obtain an estimate of 3500 km.
$$moon\;diameter=(3.8\times 10^5 km)\frac{0.007m}{0.75m }\approx 3500 km$$
This estimate is pretty close to the moon's diameter.
