A Geometric Limit Problem

The following constitutes a popular problem in calculus courses. Consider the sector of a unit circle with angle with the measurements as shown above. Let be the area of triangle , while is the area of the curved shape . What is the limit of the ratio as the angle approaches 0? The answer is 3/4. The ratio takes the value when and when .

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Reference
[1] D. N. Adler, "Some Extensions of a Ubiquitous Geometric Limit Problem," The College Mathematics Journal, 27(4), 1996 pp. 290–291. maa.org/sites/default/files/0746834242060.di020773.02p0249o.pdf.
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