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====INTRODUCTION====
==Derivation of the area formula==
WHAT IS AREA?
AREA IS THE AMOUNT OF SURFACE COVERED BY ANY OBJECT.
The area of the circular sector is <math>\pi R^2 \cdot \frac{\theta}{2\pi} = R^2\left(\frac{\theta}{2}\right)</math>
NOW ABOUT AREA OF A "CIRCLE"
AREA OF CIRCLE DENOTED BY "A"
If we bisect angle <math>\theta</math>, and thus the triangular portion, we will get two triangles with the area <math>\frac{1}{2} R\sin \frac{\theta}{2} R\cos \frac{\theta}{2}</math> or
AREA OF A CIRLE REPRESENTS THE WHOLE SURFACE COVERED
<math>2\cdot\frac{1}{2}R\sin\frac{\theta}{2} R\cos\frac{\theta}{2}</math>
BY THE CIRCLE.
<math>= R^2\sin\frac{\theta}{2}\cos\frac{\theta}{2}</math>
Since the area of the segment is the area of the sector decreased by the area of the triangular portion, we have
<math>R^2\left(\frac{\theta}{2}-\sin\frac{\theta}{2}\cos\frac{\theta}{2}\right)</math>
According to trigonometry, '''<big><math>2\sin x\cos x = \sin 2x</math></big>''', therefore
<math>R\sin\frac{\theta}{2}R\cos\frac{\theta}{2} = \frac{R^2}{2}\sin\theta</math>
The area is therefore:
<math>R^2\left(\frac{\theta}{2}-\frac{1}{2}\sin\theta\right)</math>
<math>= \frac{R^2}{2}\left(\theta-\sin\theta\right)</math>
==See also==
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