Circular segment: Difference between revisions

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==Derivation of the area formula==

====INTRODUCTION====

WHAT IS AREA?

The area of the circular sector is&nbsp;<math>\pi R^2 \cdot \frac{\theta}{2\pi} = R^2\left(\frac{\theta}{2}\right)</math>

AREA IS THE AMOUNT OF SURFACE COVERED BY ANY OBJECT.

NOW ABOUT AREA OF A "CIRCLE".

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 CIRCLE DENOTED BY "A"

<math>2\cdot\frac{1}{2}R\sin\frac{\theta}{2} R\cos\frac{\theta}{2}</math>

AREA OF A CIRLE REPRESENTS THE WHOLE SURFACE COVERED

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==