Angular Size
Apparent angular size of an object (small-angle).
Formula
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Example
Sun and Moon look equal in size
How it works
This formula gives the angular size of an object when it is very small compared with how far away it is. It comes from the small-angle approximation: for tiny angles measured in radians, the arc length is approximately the angle times the radius, so the angle subtended by an object is about its physical size divided by its distance. The result tells you how large the object appears on the sky, not how large it really is.
When to use it
Use this when estimating how wide a planet, moon, spacecraft, building, or distant feature appears in the sky or in an image. It is especially useful in astronomy, where many objects are so far away that their angular size is tiny and can be treated with the small-angle approximation.
Worked example, step by step
A spacecraft is 12 m long and is 3,000 m away from an observer. Use θ = d / D θ = 12 / 3000 θ = 0.004 rad Convert to degrees if needed: 0.004 × 57.3 ≈ 0.23° Final answer: the spacecraft has an angular size of 0.004 radians, or about 0.23°.
Common mistakes to avoid
- Using d and D in different units, such as meters and kilometres, without converting them first
- Forgetting that the formula gives radians unless you convert the answer to degrees
- Using this approximation for very large angles, where the exact tangent relation is needed
Did You Know?
By pure coincidence the Sun and Moon have nearly the same angular size, which is why total eclipses are so perfect.
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