Linear Size Calculator
Enter an object's angular size in degrees and its distance from the observer, and this Linear Size Calculator returns the object's actual linear size D using D = 2·L·tan(α/2). Useful for astronomy, optics, photography, and surveying.
Find Size From Distance and Angle
Linear Size Formula
Where:
- \(D\) — linear size of the object (the dimension perpendicular to the line of sight)
- \(L\) — distance from the observer to the object
- \(\alpha\) — angular size subtended by the object, in degrees or radians
The formula comes from simple isosceles triangle geometry: the observer, one edge of the object, and the opposite edge form a triangle that the distance line bisects, so half the linear size equals \(L \tan(\alpha/2)\). Note that \(\alpha\) must be converted to radians before applying standard tangent functions.
Linear Size – Calculation Example
Example: a distant object is 10 units away and subtends an angular size of 30°. What is its linear size?
\[ D = 2 \cdot 10 \cdot \tan\left(\frac{30^\circ}{2}\right) = 20 \cdot \tan(15^\circ) \approx 20 \cdot 0.26795 \approx 5.359 \]
Result: the linear size is \(D \approx 5.3590\) units — for example, an object 10 m away that spans 30° of your view is about 5.36 m wide.
How to Use the Linear Size Calculator
- Enter the Angular size (°) — the angle the object occupies in your field of view, measured in degrees.
- Enter the Distance to the object (L) — the straight-line distance from the observer to the object, in any unit of length (meters, kilometers, feet, miles).
- Read the Linear size (D) — the result appears in the same unit of length you used for the distance.
Use decimal degrees for the angular size; for instance, enter 0.5° instead of 30 arcminutes. The result is rounded to four decimal places.
What Is Linear Size?
Linear size is the actual physical dimension of an object — its true width, height, or diameter — as opposed to its angular size, which is how large it appears from a particular viewpoint. The same object looks smaller as it moves farther away: the Sun and the Moon both subtend roughly 0.5° in our sky, yet the Sun is about 400 times wider than the Moon because it is about 400 times farther away.
The relationship between the two is governed by triangle geometry. If you draw lines from your eye to the two edges of an object, you form an isosceles triangle whose apex angle is the angular size \(\alpha\) and whose equal sides have length \(L\). Bisecting the apex angle splits the object's size into two right triangles, each with a far side of \(L \tan(\alpha/2)\), which gives the formula \(D = 2L\tan(\alpha/2)\).
This calculation appears everywhere angular measurements are made: astronomers use it to compute the diameters of planets, nebulae, and galaxies from their apparent sizes; surveyors use it to measure inaccessible objects such as cliffs or towers; photographers use it to check whether a subject will fill the frame; and optometrists and display engineers rely on it when working with field of view and screen viewing angles.
To interpret the result, keep in mind that \(D\) is the dimension perpendicular to the line of sight. An object that is tilted or extended along the viewing direction will have other dimensions this formula cannot recover — it gives one linear measurement from one angular measurement and one distance.
Assumptions, Limitations, and Tips
- Flat geometry: the formula assumes straight-line (Euclidean) geometry, which is excellent for everyday distances but must be treated carefully for very large astronomical scales where curvature matters.
- Small-angle approximation: for small angles (below about 10°), \(D \approx L \alpha\) with \(\alpha\) in radians is a good shortcut. This calculator uses the exact tangent formula, so it stays accurate even for large angles.
- Accurate inputs matter: the distance \(L\) should be measured to the object's center along the line of sight; the result inherits any error in both the angle and the distance.
- Maximum angle: the formula is meaningful for angular sizes between 0° and 180°. At \(\alpha = 180^\circ\) the tangent is undefined, and angles above that have no physical meaning in this context.
- Units: the output has the same length unit as the distance input — there is no built-in unit conversion.
If you need to work in the opposite direction, our Angular Size Calculator finds the apparent angle from a known size and distance, and the Distance Perspective Projection Calculator handles related projection problems.
Quick Reference Table
Linear size for a distance of \(L = 100\) at various angular sizes:
| Angular size α | Linear size D |
|---|---|
| 0.5° | 0.873 |
| 1° | 1.745 |
| 5° | 8.749 |
| 10° | 17.633 |
| 30° | 53.590 |
| 45° | 82.843 |
| 90° | 200.000 |
Notice how the linear size grows faster than the angle: doubling the angular size more than doubles \(D\) once the angle becomes large, because \(\tan\) is nonlinear.
Frequently Asked Questions
How do I convert arcminutes or arcseconds to degrees for the angular size input?
Divide by 60 for each step: 1 degree = 60 arcminutes and 1 arcminute = 60 arcseconds. So an angular size of 30 arcminutes equals \(30/60 = 0.5^\circ\), and 30 arcseconds equals \(30/3600 \approx 0.0083^\circ\). Many astronomy sources quote sizes in arcminutes or arcseconds, so convert them to decimal degrees before entering the value — the calculator only accepts degrees.
How do I calculate linear size by hand without a calculator?
Convert the angle to radians (multiply degrees by \(\pi/180\)), halve it, take the tangent, and multiply by twice the distance: \(D = 2L\tan(\alpha/2)\). For a quick mental estimate with angles under about 10°, use the small-angle shortcut \(D \approx L\alpha\) with \(\alpha\) in radians. Example: 2° at 50 m gives \(D \approx 50 \times 0.035 \approx 1.74\) m.
How can I estimate angular size in the field without instruments?
Use your hand at arm's length: your pinky finger covers about 1°, three fingers together about 5°, a fist about 10°, and the span from thumb to pinky about 20–25°. Compare the object's apparent width to these references to get a rough angle, then enter it with your distance estimate. This works well for hiking, surveying, and quick astronomy checks.
What are the most common mistakes when using this formula?
The biggest one is forgetting to halve the angle — the formula uses \(\tan(\alpha/2)\), not \(\tan(\alpha)\), because the distance line bisects the viewing triangle. The second is mixing units: entering degrees into a tangent function that expects radians, or using different length units for distance and expected output. Also make sure the distance is measured along the line of sight to the object, not along the ground.
Can the angular size be larger than 90 degrees?
Yes. Any value between 0° and 180° is physically valid. An object spanning exactly 90° at distance \(L\) has a linear size of exactly \(2L\) — for example, a wall 3 m away that fills a quarter of your view around you is 6 m wide. As the angle approaches 180°, the tangent grows without bound, meaning the object would have to extend infinitely far to the sides, so 180° itself is undefined.
Does the formula work if the object is tilted or not facing me directly?
Only partially. The result is the dimension perpendicular to your line of sight — the apparent projected width. If a rectangular object is rotated, its true depth or the other dimension can't be recovered from a single angle and distance. To fully describe a 3D object you need additional angular measurements from other directions or independent information about its orientation.
How do astronomers use this to measure the size of the Moon or a planet?
They take the object's measured angular diameter and its distance, then apply the same formula. The Moon subtends about 0.5° and lies roughly 384,400 km away, giving \(D \approx 384{,}400 \times 0.0087 \approx 3{,}475\) km — close to the accepted 3,474 km. For very distant or cosmological objects, corrections for expansion and curvature of space are needed, so this simple formula is best for nearby astronomy.