Angular Size Calculator

About the Angular Size Calculator

The Angular Size Calculator is a powerful tool used in trigonometry, astronomy, and various engineering applications. It helps you determine the relationship between an angle, the arc length it subtends, and the radius of the circle or sphere.

What is Angular Size?

Angular size, also known as angular diameter or apparent size, is a measure of how large an object appears from a given point of view. In circular motion or when dealing with circular sectors, it relates the arc length, radius, and central angle of a circle.

Formula

The formula for angular size is:

\[ \theta = \frac{s}{r} \]

Where:

• \( \theta \) is the angle in radians
• \( s \) is the arc length
• \( r \) is the radius

Calculation Steps

1. Identify the two known values among angle, arc length, and radius.
2. If the angle is given in degrees, convert it to radians using the formula: \( \theta_{rad} = \theta_{deg} \times \frac{\pi}{180} \)
3. Apply the appropriate formula based on which value you need to calculate:
   • To find angle (in radians): \( \theta = \frac{s}{r} \)
   • To find arc length: \( s = \theta \times r \)
   • To find radius: \( r = \frac{s}{\theta} \)
4. If you calculated the angle in radians, convert it back to degrees using: \( \theta_{deg} = \theta_{rad} \times \frac{180}{\pi} \)

Example

Let's calculate the angular size of the Sun as seen from Earth:

• The Sun's diameter is approximately 1,392,700 km
• The average distance from Earth to Sun is about 149,600,000 km

Using these values:

\[ \theta = 2 \times \arctan(\frac{1,392,700}{2 \times 149,600,000}) \approx 0.00931 \text{ radians} \]

Converting to degrees:

\[ 0.00931 \times \frac{180}{\pi} \approx 0.533° \]

Therefore, the angular size of the Sun as seen from Earth is approximately 0.533 degrees.