Understanding Angles 1

The idea of an angle is one of the earliest geometrical concepts developed by humans along with that of lines. But even today it remains a concept which is not clearly understood in schools. This is mostly because of inappropriate pedagogy which focuses more on formal definitions, construction rules & measurements rather than on “intuitive” understanding.

Linear Vs Rotational Distance

The idea of “distance” between 2 points actually can be seen in 2 perspectives. The idea of a “linear distance” is easier and more familiar to children. It can be understood as the number steps required to reach one of the points from the other point. In Geometry it can also be seen as the length of a tightly held string which connects 2 points. This “linear” distance between 2 points does not change and can be expressed as a number measured in cm or m, as convenient.

Astronomers have been studying stars & planets for more than 3000 years. Until recently they did not have the science to measure the linear distance between a star & Earth or between 2 stars. So, they studied the distance between 2 stars or planets by a concept called “rotational distance”. If an astronomer is looking at star A and has to rotate his head (or his instrument), without physically changing the location, by a certain magnitude X so that he can look at star B, then X is called the “rotational distance” between stars A & B. This is how the science of Trigonometry started.

Children are introduced formally to the idea of “rotational distance” only in school through Geometry. But they come with enough life experiences which can be related to this idea.

Daily Experiences of Rotational Distances

One of the earliest experiences that children have is about opening & closing books & doors. They also understand the concept of “less open” or “more open” in relation to the above experiences. To hand over a packet of milk, the door can be “less” open. But to bring in a chair, the door needs to be “more” open. To read a book comfortably it has to be “more” open. But to insert a book mark it can be “less” open.

But the idea of “more” or “less” in these situations cannot easily be related to the idea of linear distance that they are familiar with. When a book is open, the distance between the pages (whichever way you measure them) increases as we go from the spine of the book to the edges of the pages. At the spine both the pages are joined or the distance between them is 0. At the 2 edges, furthest from the spine, the distance between the same 2 pages is the greatest. These distances also increase as the book is opened “more”. Hence there is no single number which can indicate this linear distance.

The same issues can be seen in the case of a door also. Here the door hinges are the spine and the handle is the edge.

Rotational Distance and the Angle

Humans realized that the space between the pages of the book, as it is opened or closed, can be related to the idea of rotational “distance”. In any particular position of an “open” book, the amount or “rotational distance” the reader has to “rotate” or “turn” the pages in order to close the book remains same. Lesser the “rotational distance”, less is the book or door open & vice versa.

Geometers applied this idea in the study of spaces formed when lines intersect.

Geometers called the space formed by 2 lines as “angle”. The measure of the angle formed by 2 lines was conceptualized as the amount of rotation, one of the lines had to undergo (about the point of intersection) so that it coincides with the other line. Which of the lines rotated to coincide with the other lines was not relevant since both indicated the same amount of rotation? Please note that the term “angle” denotes both the shape and its measure. We say “this is an acute angle” as well as "the angle is 60°."