Question

An arc of a great circle on a sphere of radius 3959 mi. subtends at the center an angle of 5 degree. Find the length of the arc in degrees, in radians, and in nautical miles.Note: 1 mile = 0.868976 nautical miles

Answer 1

Given:

Central angle: 5°

r = 3959 mi

Converting radius from miles to km;

1 mile = 1.609km; then:

3959 x 1.609 = 6370.031 km

• Arc length in degrees:

`\begin{gathered} \frac{\theta}{360\text{\degree}}\text{ x 2}\pi r \\ arclength\text{ = }\frac{5\text{\degree}}{360\text{\degree}}\text{ x }2\pi(6370.031)\text{ = }555.89\text{ km} \end{gathered}`

• Arc length in radians:

`\begin{gathered} \frac{\theta\text{ x }\pi}{180\text{\degree}}\text{ x r} \\ \text{arc length = }\frac{5\text{ x }\pi}{180\text{ \degree}}\text{ x 6370.031 = }555.89\text{ km} \end{gathered}`

• In Nautical Miles:

1nm = 1.852km

? nm = 555.89 km

`\frac{1\text{ x 555.89}}{1.8252}=\text{ 300.156 nautical miles}`