Yusuf Abu Muhanna, YusufAqel, Khitam Zuhair2015-06-022015-06-022015-0629.232-2015.03http://hdl.handle.net/11073/7823A Master of Science thesis in Mathematics by Khitam Zuhair Aqel entitled, "Some applications of the Hyperbolic Metric," submitted in June 2015. Thesis advisor is Dr. Yusuf Abumuhanna. Soft and hard copy available.This thesis is an expository of the hyperbolic geometry in general and the hyperbolic metric for several domains in particular. The thesis also focuses on the differences between the Euclidean geometry and hyperbolic geometry. Mainly, it focuses on the difference between the Euclidean area of triangle and the hyperbolic area. At the end, a new application of the hyperbolic metric is given. In specific, a new bound for the second coefficient of harmonic univalent maps is determined. The thesis was written in such a clear way to make the subject accessible to people with average knowledge of mathematics.en-USHyperbolic geometryHyperbolic metricPoincare metricHarmonic mapsUnivalent mapsCoefficient boundGeometry, HyperbolicThesis