Question

The Sun subtends an angle of about 0.5° to us on Earth, 150 million km away. Estimate the radius of the Sun.

Short answer

The radius of the Sun is \(6.54 \times {10^5}\;{\rm{km}}\).

Step-by-step solution

Determination of the angular position of an object

The angular position of an object can be calculated by dividing the arc length by the distance. Its value is altered linearly to the value of the arc length.

Given information

Given data:

The subtended angle is\(\theta = 0.5^\circ \).

The distance between the Earth and the Sun is \(d = 150\;{\rm{million}}\;{\rm{km}}\).

Calculate the radius of the Sun

The diameter of the Sun can be calculated as:

\(\begin{aligned}{l}D &= d\theta \\D &= \left( {\left( {150\;{\rm{million}}\;{\rm{km}}} \right)\left( {\frac{{{{10}^6}\;{\rm{km}}}}{{{\rm{1}}\;{\rm{million}}\;{\rm{km}}}}} \right)} \right)\left( {\left( {0.5^\circ } \right)\left( {\frac{{2\pi \;{\rm{rad}}}}{{360^\circ }}} \right)} \right)\\D = 1.3089 \times {10^6}\;{\rm{km}}\end{aligned}\)

The radius of the Sun can be calculated as:

\(\begin{aligned}{l}R &= \frac{D}{2}\\R &= \frac{{\left( {1.3089 \times {{10}^6}\;{\rm{km}}} \right)}}{2}\\R = 6.54 \times {10^5}\;{\rm{km}}\end{aligned}\)

Thus, the radius of the Sun is \(6.54 \times {10^5}\;{\rm{km}}\).