Skinny triangle

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

${\displaystyle b\simeq r\theta \,}$

${\displaystyle {\text{area}}\simeq {\frac {1}{2}}\theta r^{2}\,}$

This is based on the small-angle approximations;

${\displaystyle \sin \theta \simeq \theta ,\quad \theta \ll 1\,}$

and,

${\displaystyle \cos \theta =\sin \left({\frac {\pi }{2}}-\theta \right)\simeq 1,\quad \theta \ll 1}$

when ${\displaystyle \scriptstyle \theta }$ is in radians.

The proof of the skinny triangle solution follows from the small-angle approximation by applying the law of sines. Again referring to figure 1;

${\displaystyle {\frac {b}{\sin \theta }}={\frac {r}{\sin \left({\frac {\pi -\theta }{2}}\right)}}}$

The term ${\displaystyle \scriptstyle {\frac {\pi -\theta }{2}}}$ 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;

${\displaystyle {\frac {b}{\theta }}\simeq {\frac {r}{1}}}$

the desired result.

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. The error is 10% or less for angles less than about 43°,[2][3] and improves quadratically: when the angle decreases by a factor of k, the error decreases by k².

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

${\displaystyle {\text{area}}={\frac {\sin \theta }{2}}r^{2}}$

Applying the small angle approximations results in;

${\displaystyle {\text{area}}\simeq {\frac {1}{2}}\theta r^{2}\,}$

Fig.3 The right skinny triangle

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

${\displaystyle b\simeq h\theta }$

This is based on the small-angle approximation;

${\displaystyle \tan \theta \simeq \theta ,\quad \theta \ll 1}$

which when substituted into the exact solution;

${\displaystyle b=h\tan \theta \ }$

yields the desired result.

The error of this approximation is less than 10% for angles 31° or less.[4]

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.

• Infinitesimal oscillations of a pendulum

• George Ogden Abell, David Morrison, Sidney C. Wolff, Exploration of the Universe, Saunders College Pub., 1987 ISBN 0-03-005143-6.
• Jim Breithaupt, Physics for Advanced Level, Nelson Thornes, 2000 ISBN 0-7487-4315-4.
• Charles H. Holbrow, James N. Lloyd, Joseph C. Amato, Enrique Galvez, Beth Parks, Modern Introductory Physics, Springer, 2010 ISBN 0-387-79079-9.
• Srini Vasan, Basics of Photonics and Optics, Trafford Publishing, 2004 ISBN 1-4120-4138-4.
• Tom A. Warlow, Firearms, the law and forensic ballistics, Taylor & Francis, 1996 ISBN 0-7484-0432-5.