Trigonometry and Geometry Compared

Mathematics is a subject that traditionally has been divided into various branches, some of which are interrelated: algebra, trigonometry, geometry, statistics, calculus, etc. Geometry and trigonometry are but two of the branches that we study probably in junior high school and beyond.


Geometry is one of the earliest math topics covered by school-children, other than arithmetic. It is one of the ‘oldest’ concepts in math history too. Early Greek and European mathematicians had discovered many useful formulae concerning the area of circle, triangle and other geometric shapes, among them Ptolemy, Apollonius, Pythagaros and Euclid. In elementary schools, children are introduced to various geometrical shapes e.g. square, circle, rectangle, triangle, and other polygons. At this stage, the idea is merely to introduce the concept of shapes and sizes to these early school-goers. So early studies of geometry do not go much beyond recognizing shapes and sizes and learning the names of the various geometrical figures. It is only in middle high school and beyond that the study of geometry covers topics like co-ordinate or Cartesian geometry and other theorems of Euclidean and non-Euclidean geometry. Here the various theorems concerning triangles, trapeziums, rhombuses, and the circle and ways of calculating their areas are studied. In the study of co-ordinate or Cartesian geometry (first invented by Rene Descartes), points and lines in a 2-dimensional space are given ‘co-ordinates’ and equations respectively, thus bringing algebra into the picture. At this level too students learn some of the properties of triangle, parallel and perpendicular lines, cyclic quadrilaterals and so on. So broadly speaking, geometry is the study of shapes of figures, lines; the areas of the various geometrical shapes (finding area of a square, rectangle, circle, etc.) and the associated formula and theorems.


Trigonometry is a subject that is not normally introduced early in high school because it is perceived to be more difficult or abstract than geometry. Elementary trigonometry is closely related to geometry in that the fundamental ratios of trigonometry (sine, cosine, tangent and later secant, cosecant and cotangent) are defined using right-angled triangles when the angles are acute or less than 90 degrees. Of course when the relevant angles are larger than 90 degrees to cover angles of any magnitude, then other geometric shapes e.g. the circle – are then used to lend meaning to the fundamental trigonometric ratios. Elementary trigonometry is useful in determining the areas of triangles, quadrilaterals etc. But in advanced trigonometry (studied in senior high school and college) trigonometry can be a very complex and difficult topic: areas like solving trigonometric equations, trigonometric identities and the calculus of trigonometric functions can be quite daunting to the uninitiated and the mentally indolent.


From the aforementioned account, it is evident that trigonometry and geometry are somewhat related but they are of course not one and the same thing: one could safely state that generally trigonometry is more ‘difficult’ at least at the pre-college levels but this is not to say that geometry is easy. In the study of advanced math courses in college, some sub-branches of geometry can be rather complex e.g. in a branch called spherical geometry and also the geometry of surfaces or topology – these are areas of math that even a PhD student in math could study and do research on. The core of trigonometry consists of studies of the basic trigonometric ratios and their applications in areas as varied as engineering, physics and astronomy. And a grasp of geometry and geometric concepts is essential for success in trigonometry too.

As a concluding remark, it is perhaps not too far off the mark to describe the relationship between trigonometry and geometry as complementing each other. While there are differences between them and one normally studies geometry before proceeding to studying trigonometry, in the final analysis, they are more similar than distinctly different and they need to be studied together though not exactly contemporaneously.