Question

The sun's angular diameter is measured to be $1920\text{seconds}$. The distance of the sun from the earth is $1.496\times {10}^{11}\text{m}$. Find the diameter of the sun.

Answer 1

Step 1: Given

The sun's angular diameter, $\theta =1920\text{seconds}$
$$\begin{gathered} =\frac{1920}{3600} \\ =0.53° \\ =0.53\times \frac{\mathrm{\pi }}{180} \\ =0.009\text{rad} \end{gathered}$$.

The distance from the sun to the earth is, $D=1.496\times {10}^{11}\text{m}$.

Let $d$ be the diameter of the sun

Step 2: Formulas used

From trigonometry, we know that
$\tan \theta =\frac{\mathrm{Perpendicular}}{\mathrm{Base}}$

For very small values of $\theta$,
$\tan \theta =\theta$

Step 3: Solution

From the figure, we can say that,
$\tan \left(\frac{\theta }{2}\right)=\frac{{\displaystyle \frac{d}{2}}}{D}$

Since, $\frac{\theta }{2}$ is very close to $0$,
$\begin{array}{cc}& \frac{\theta }{2}=\frac{d}{2·D}\\ \Rightarrow & \theta =\frac{d}{D}\\ \Rightarrow & d=D·\theta \\ \Rightarrow & d=1.496\times {10}^{11}\times 0.009\\ \Rightarrow & d=1.346\times {10}^9\text{m}\end{array}$

Therefore, the diameter of the sun is $1.346\times {10}^9\text{m}$.