Calculate trapezoid segment

Question

I want to calculate the diameter of a shield which, when placed between the Sun and Earth, would cast a shadow on Earth equal to Earth's diameter essentially totally blocking the Sun. If the shield is to be placed 75% of the way to the Sun and

Base Sun diam = 1 400 000 million Km Base Earth diam = 12 742 Km Legs = 150 000 000 Km

how big would the shield's diameter be?

If the median segment of a trapezoid is ( Base Sun diam + Base Earth diam ) / 2 (50 percent) does this mean that any segment length can be calculated using the percentage of the distance from the segment to any of the 2 bases? if so, what is the exact formula for any "median" segment length in relation to its distance to any of the bases?

thanks

Answer 1

What you have is a conical frustum with diameter of one end as $d_e$ and the other end as $d_s$ and,
$d_e \lt d_s \, $ where $d_e$ is the diameter of the earth and $d_s$ is the diameter of the sun.

Hence the shield will have to be circular so it can block sun in all directions. Yes to solve this, you can look at the trapezoid made by $3$ diameters all in one plane parallel to each another.

Say the ratio $r = \dfrac {l_s} {l_e} \,$ where $l_s = GI, l_e = AI$ are distances from sun and earth to the shield respectively.

$ \frac {EI}{CG} = \frac {AI}{AG} = \frac{l_e}{l_e + l_s}$

$ EI = \frac{l_e}{l_e + l_s} \times CG = \frac{l_e}{l_e + r \times l_e} \times \frac{d_s - d_e}{2}$

$ EF = 2 \times EI + AB = \frac{1}{r + 1} \times (d_s - d_e) + d_e = \frac {d_s + \, r \, d_e}{r+1}$

where $EF$ is the diameter of the circular shield to block sun rays reaching the earth.