# Area of Circle/Proof 1

## Theorem

The area $A$ of a circle is given by:

$A = \pi r^2$

where $r$ is the radius of the circle.

## Proof

From Equation of Circle:

$x^2 + y^2 = r^2$

Thus $y = \pm \sqrt {r^2 - x^2}$.

It follows that from the geometric interpretation of the definite integral:

 $\ds A$ $=$ $\ds \int_{-r}^r \paren {\sqrt {r^2 - x^2} - \paren {-\sqrt {r^2 - x^2} } } \rd x$ $\ds$ $=$ $\ds \int_{-r}^r 2 \sqrt {r^2 - x^2} \rd x$ $\ds$ $=$ $\ds \int_{-r}^r 2 r \sqrt {1 - \frac {x^2} {r^2} } \rd x$

Let $x = r \sin \theta$ (note that we can do this because $-r \le x \le r$).

Thus $\theta = \map \arcsin {\dfrac x r}$ and $\rd x = r \cos \theta \rd \theta$.

 $\ds A$ $=$ $\ds \int_{\map \arcsin {\frac {-r} r} }^{\map \arcsin {\frac r r} } 2 r^2 \sqrt {1 - \frac {\paren {r \sin \theta}^2} {r^2} } \cos \theta \rd \theta$ Integration by Substitution $\ds$ $=$ $\ds \int_{-\frac \pi 2}^{\frac \pi 2} 2 r^2 \sqrt {1 - \sin^2 \theta} \cos \theta \rd \theta$ $\ds$ $=$ $\ds \int_{-\frac \pi 2}^{\frac \pi 2} 2 r^2 \sqrt {\cos^2 \theta} \cos \theta \rd \theta$ Sum of Squares of Sine and Cosine $\ds$ $=$ $\ds r^2 \int_{-\frac \pi 2}^{\frac \pi 2} 2 \cos^2 \theta \rd \theta$ $\ds$ $=$ $\ds r^2 \int_{-\frac \pi 2}^{\frac \pi 2} \paren {1 + \map \cos {2 \theta} } \rd \theta$ Double Angle Formula for Cosine: Corollary 1 $\ds$ $=$ $\ds r^2 \intlimits {\theta + \frac 1 2 \map \sin {2 \theta} } {-\frac \pi 2} {\frac \pi 2}$ from Definite Integral of Constant and Primitive of Cosine Function $\ds$ $=$ $\ds r^2 \paren {\frac \pi 2 + \frac 1 2 \map \sin {2 \cdot \frac {-\pi} 2} - \frac {-\pi} 2 - \frac 1 2 \map \sin {2 \cdot \frac \pi 2} }$ $\ds$ $=$ $\ds r^2 \paren {2 \cdot \frac \pi 2 + 2 \cdot \frac 1 2 \cdot 0}$ $\ds$ $=$ $\ds \pi r^2$

$\blacksquare$
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://www.proofwiki.org/wiki/Area_of_Circle/Proof_1", "fetch_time": 1686313242000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2023-23/segments/1685224656675.90/warc/CC-MAIN-20230609100535-20230609130535-00384.warc.gz", "warc_record_offset": 1005002589, "warc_record_length": 11864, "token_count": 760, "char_count": 1775, "score": 4.5, "int_score": 4, "crawl": "CC-MAIN-2023-23", "snapshot_type": "latest", "language": "en", "language_score": 0.294101744890213, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00019-of-00064.parquet"}