• Pixel-art sticker of Spanno, personifying Angular Size, with the caption “Farther Away, Looks Smaller.”

IndexRelativity & CosmosSticker

Spanno

Angular Size Physics

\[\theta \approx \frac{D}{d}\]
Regular price $4.99 USD
Regular price Sale price $4.99 USD
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Meet Spanno, a crate who looks smaller the farther away he is. For small angles, angular size in radians is real size divided by distance. A 1.8 m person 100 m away covers 0.018 rad, about 1.0°; at 200 m it halves to about 0.52°. Pixel-art kiss-cut sticker in 2, 3, 4 and 6 inches; equation, symbols and sources in the physics panel below.

Size
Surface

Flat $6 shipping worldwideMade to order, ships in 2–5 business days

Details

Size3″ × 3″

3″ × 3″ kiss-cut sticker.

Shipping

Flat $6 shipping on every order, worldwide, whatever's in it.

Delivery times & where we ship

Returns & replacements21 days

Every item is printed to order, so we can’t accept returns or exchanges for change of mind, or if you ordered the wrong size or colour.

Print defect, misprint, damage or the wrong item? We’ll send a free replacement, or a refund if you prefer. Report it within 21 days of delivery with a photo; no need to send it back.

Full refund & replacement policy

The physics

Meet Spanno, a crate who looks smaller the farther away he is.

How big something looks depends on both its real size and its distance. Its angular size is the angle it covers in your view. For small angles, the angular size in radians is the real size divided by the distance, θ ≈ D/d.

A person 1.8 m tall standing 100 m away covers θ ≈ 1.8 / 100 = 0.018 rad, about 1.0°. Move the same person to 200 m and θ ≈ 1.8 / 200 = 0.009 rad, about 0.52°. Doubling the distance halves the angular size.

This is why two identical objects at different distances look different in size, and why artists draw distant things smaller to show depth.

The shortcut θ ≈ D/d only works for small angles. For something very close and wide, the exact relation θ = 2 arctan(D/2d) is needed. At these small angles the shortcut is very accurate: for the person 100 m away, the exact formula also gives 0.018 rad.

Equations

\[\theta \approx \frac{D}{d}\]
Symbols
SymbolMeaningUnit
\(\theta\) angular size \(\mathrm{rad}\)
\(D\) real size of the object \(\mathrm{m}\)
\(d\) distance to the object \(\mathrm{m}\)