[Math] Area of a sector – Inscribed Angle and Central Angle

areacirclesgeometry

I know the formula for the area of a sector of an arc made by central angle is
$$\text{Area}_\text{Sector}= \frac{\text{Arc Angle} \times \text{Area of Circle} }{360}$$
Now my question is , Is this formula also applicable for Arcs formed by inscribed angles rather than Central Angles ? (I know that angle of an intercepted arc of an inscribed angle is twice the measure of the inscribed angle.)

Best Answer

Here are some equal angle inscribed angles (they all subtend half the angle of the red arc). It is clear that the area of the last sector is properly contained in the areas of the others.

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The area of the sector is the sum of the area between the arc and its chord (a constant area) and the area of the triangle, which is $\frac12$ the product of the length of the chord (a constant length) and the altitude of the triangle, which varies as the vertex moves around the circle.

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