geometry
Find the Area of a segment of a circle if the central angle of the segment is $105^\circ$ degrees and the radius is $70$.
Formulas I have:
Please, could you explain it step by step so I can understand, thanks
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.
The length of the edge of the sector is $2\pi r$ because that is the circumference of the base of the cone.
The radius of the sector is $l$ because that is the distance from the tip of the cone to the base, which becomes the distance from the center of the circle to the edge.
For any arc of length $L$ on a circle of radius $R$, the angle that it subtends is $\frac{L}{R}$ radians, essentially by definition.
Thus, the angle of the sector is $\theta=\frac{2\pi r}{l}$.
Best Answer
You can work out the area of the sector then subtract the area of the triangle.
The area of a sector is given by $\frac{1}{2}r^2\theta$ if $\theta$ is in radians or $\frac{1}{2}r^2\pi\frac{\theta}{180^\circ}$ if $\theta$ is in degrees.
The area of the triangle is give by $\frac{1}{2}r^2\sin\theta$.
Combining these two gives: $\frac{1}{2}\times70^2\times\pi\times\frac{105^\circ}{180^\circ}-\frac{1}{2}\times70^2\times\sin105^\circ \approx 2123.34$