# Skinny triangle

File:Isosceles skinny triangle.svg
Fig. 1 Isosceles skinny triangle

A skinny triangle in trigonometry is a triangle whose height is much greater than its base. The solution of such triangles can be greatly simplified by using the approximation that the sine of a small angle is equal to the angle in radians. The solution is particularly simple for skinny triangles that are also isosceles or right triangles: in these cases the need for trigonometric functions or tables can be entirely dispensed with.

The skinny triangle finds uses in surveying, astronomy and shooting.

## Isosceles triangle

Table of sine small-angle approximation errors
Large angles Small angles
angle error
1 0.017 < 0.01
5 0.087 0.13
10 0.175 0.51
15 0.262 1.15
20 0.349 2.06
25 0.436 3.25
30 0.524 4.72
35 0.611 6.50
40 0.698 8.61
45 0.785 11.07
50 0.873 13.92
55 0.960 17.19
60 1.047 20.92
angle error
1 0.0003 0.01
5 0.0015 0.35
10 0.0029 1.41
15 0.0044 3.17
20 0.0058 5.64
25 0.0073 8.81
30 0.0087 12.69
35 0.0102 17.28
40 0.0116 22.56
45 0.0131 28.56
50 0.0145 35.26
55 0.0160 42.66
60 0.0175 50.77

The approximated solution to the skinny isosceles triangle, referring to figure 1, is;

$b\simeq r\theta \,$ $area\simeq {\frac {1}{2}}\theta r^{2}\,$ This is based on the small-angle approximations;

$\sin \theta \simeq \theta ,\quad \theta \ll 1\,$ and,

$\cos \theta =\sin \left({\frac {\pi }{2}}-\theta \right)\simeq 1,\quad \theta \ll 1$ The proof of the skinny triangle solution follows from the small-angle approximation by applying the law of sines. Again referring to figure 1;

${\frac {b}{\sin \theta }}={\frac {r}{\sin \left({\frac {\pi -\theta }{2}}\right)}}$ The term ${\frac {\pi -\theta }{2}}$ represents the base angle of the triangle and is this value because the sum of the internal angles of any triangle (in this case the two base angles plus θ) are equal to π. Applying the small angle approximations to the law of sines above results in;

${\frac {b}{\theta }}\simeq {\frac {r}{1}}$ the desired result.

File:Length of arc.svg
Fig.2 Length of arc l approaches length of chord b as angle θ decreases

This result is equivalent to assuming that the length of the base of the triangle is equal to the length of the arc of circle of radius r subtended by angle θ. This approximation becomes ever more accurate for smaller and smaller θ. The error is 10% or less for angles less than about 43°.

The side-angle-side formula for the area of the triangle is;

$area={\frac {\sin \theta }{2}}r^{2}$ Applying the small angle approximations results in;

$area\simeq {\frac {1}{2}}\theta r^{2}\,$ ## Right triangle

File:Right skinny triangle.svg
Fig.3 The right skinny triangle
Table of tangent small-angle approximation errors
Large angles Small angles
angle error
1 0.017 −0.01
5 0.087 −0.25
10 0.175 −1.02
15 0.262 −2.30
20 0.349 −4.09
25 0.436 −6.43
30 0.524 −9.31
35 0.611 −12.76
40 0.698 −16.80
45 0.785 −21.46
50 0.873 −26.77
55 0.960 −32.78
60 1.047 −39.54
angle error
1 0.0003 −0.03
5 0.0015 −0.71
10 0.0029 −2.82
15 0.0044 −6.35
20 0.0058 −11.28
25 0.0073 −17.63
30 0.0087 −25.38
35 0.0102 −34.55
40 0.0116 −45.13
45 0.0131 −57.12
50 0.0145 −70.51
55 0.0160 −85.32
60 0.0175 −101.54

The approximated solution to the right skinny triangle, referring to figure 3, is;

$b\simeq h\theta$ This is based on the small-angle approximation;

$\tan \theta \simeq \theta ,\quad \theta \ll 1$ which when substituted into the exact solution;

$b=h\tan \theta \$ yields the desired result.

The error of this approximation is less than 10% for angles 31° or less.

## Applications

Applications of the skinny triangle occur in any situation where the distance to a far object is to be determined. This can occur in surveying, astronomy, and also has military applications.

### Astronomy

The skinny triangle is frequently used in astronomy to measure the distance to solar system objects. The base of the triangle is formed by the distance between two measuring stations and the angle θ is the parallax angle formed by the object as seen by the two stations. This baseline is usually very long for best accuracy; in principle the stations could be on opposite sides of the Earth. However, this distance is still short compared to the distance to the object being measured (the height of the triangle) and the skinny triangle solution can be applied and still achieve great accuracy. The alternative method of measuring the base angles is theoretically possible but not so accurate. The base angles are very nearly right angles and would need to be measured with much greater precision than the parallax angle in order to get the same accuracy.

The same method of measuring parallax angles and applying the skinny triangle can be used to measure the distances to stars; at least the nearer ones. In the case of stars however, a longer baseline than the diameter of the Earth is usually required. Instead of using two stations on the baseline, two measurements are made from the same station at different times of year. During the intervening period, the orbit of the Earth around the Sun moves the measuring station a great distance, so providing a very long baseline. This baseline can be as long as the major axis of the Earth's orbit or, equivalently, two Astronomical units (AU). The distance to a star with a parallax angle of only one arcsecond measured on a baseline of one AU is a unit known as the parsec (pc) in astronomy and is equal to about 3.26 light years. There is an inverse relationship between the distance in parsecs and the angle in arcseconds. For instance, two arcseconds corresponds to a distance of 0.5 pc and 0.5 arcseconds corresponds to a distance of two parsecs.