{"id":4304,"date":"2022-09-12T11:23:22","date_gmt":"2022-09-12T09:23:22","guid":{"rendered":"https:\/\/erlebnisland-mathematik.de\/?page_id=4304"},"modified":"2024-02-22T14:45:42","modified_gmt":"2024-02-22T13:45:42","slug":"advanced-text-how-big-is-the-area-of-a-circle","status":"publish","type":"page","link":"https:\/\/erlebnisland-mathematik.de\/en\/advanced-text-how-big-is-the-area-of-a-circle\/","title":{"rendered":"Advanced text How big is the area of a circle?"},"content":{"rendered":"

[vc_row drowwidth=”sidebar-biest-default sidebar-biest”][vc_column][vc_column_text]<\/p>\n

How big is the area of a circle?<\/h1>\n

This question already preoccupied the Egyptians around 2000 BC. This exhibit should help you find an answer to this question yourself. By “cleverly<\/em>” flipping sectors of a given circle, one can see that its area is exactly the same as that of a parallelogram-like figure (cf. Figure 3). In this way, one obtains a relationship between the circumference and the area of a circle.[\/vc_column_text][vc_column_text]<\/p>\n

And now … the mathematics:<\/h3>\n

[\/vc_column_text][vc_column_text]First, you decompose the given circle into \"n\" congruent circle sectors<\/em>. Here \"n\" should be even (for odd \"n\" an analogous procedure, but with a “<\/em>trapezoid”<\/em> instead of a “parallelogram”<\/em>, would lead to the same result). In our example \"n=12\" was chosen.[\/vc_column_text][vc_single_image image=”1216″ img_size=”large” alignment=”center”]Figure 1: The circle divided into twelve sectors[\/vc_single_image][vc_column_text]Each of these sectors has the following shape:[\/vc_column_text][vc_single_image image=”1220″ img_size=”large” alignment=”center”]Figure 2: A circle sector[\/vc_single_image][vc_column_text]Anyone who takes a closer look at the exhibit quickly realises that simple “trial and error<\/em>” or even “brute force<\/em>” will not get you anywhere with the task of determining the area of the circle, at least approximately. But there is a good approximate solution. This looks like this:[\/vc_column_text][vc_single_image image=”1224″ img_size=”large” alignment=”center”]Figure 3: The solution offered by the exhibit[\/vc_single_image][vc_column_text]One now looks for correlations between the underlying circle and the associated “parallelogram<\/em>” (Figure 3). It applies:<\/p>\n