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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

Reference

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

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A Geometric Limit Problem