Question

Hold a pencil in front of your eyes at a position where its blunt end just blocks out the moon (see the figure below). Make the appropriate measurements to estimate the diameter of the moon, given that the earth-to--moon distance is$3.8\times {10}^5\mathrm{km}$.

Short answer

The diameter of the moon is given by the formula

$D=l\frac{d_{\circ }}{s}$.

Here, s is the distance of the pencil from the eyes, l is the length of the blunt end, $d_{\circ }$is earth-to-moon distance, and D is the diameter of the moon.

Step-by-step solution

Step 1. Properties of similar triangles

Similar triangles have the same ratio of their corresponding sides, and their corresponding angles are equal.

Step 2. Given data and assumption

The distance between the earth and the moon, $d_{\circ }=3.8\times {10}^8\mathrm{km}$.

Assume that you are holding a pencil at a distance of s from your eyes, the length of the blunt end is l, and the diameter of the moon is D.

Step 3. Applying properties of similar triangles

As can be seen in the figure above, both $\Delta \mathrm{ABC}$and $\Delta \mathrm{ADE}$are similar to each other by the AAA rule. (The common lines intersect one common angle and two angles on parallel lines.)

Thus, the ratio of the corresponding sides will be equal. So,

$\frac{s}{d_{\circ }}=\frac{l}{D}$.

Step 4. Method for estimating the diameter of the moon

By the relation derived from the properties of triangles, you can find the diameter as

$D=l\frac{d_{\circ }}{s}$.

Here, you know the value of $d_{\circ },$and you can measure the values of s and l.