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Q. 71

Expert-verifiedFound in: Page 479

Book edition
7th Edition

Author(s)
James Stewart, Lothar Redlin, Saleem Watson

Pages
948 pages

ISBN
9781337067508

**Area of a Sector of a Circle Three circles with radii 1, 2, and ****3 ft are externally tangent to one another, as shown in the ****figure. Find the area of the sector of the circle of radius 1 that ****is cut off by the line segments joining the center of that circle ****to the centers of the other two circles.**

The area of sector is $\mathbf{0}\mathbf{.785}\mathit{f}{\mathit{t}}^{\mathbf{2}}$.

The radius of circle $r=1\text{ft}$

Before applying the formula for area of sector we need to find the central angle theta which the sector subtends.

As, shown in the figure

$AB=5ft$, $BC=3ft$ and $AB=4ft$

Applying Pythagoras theorem $A{C}^{2}=A{B}^{2}+B{C}^{2}$

Also, the area of sector is given by $A=\frac{1}{2}{r}^{2}\theta $

Putting the values of Ab, BC and AB in Pythagoras Theorem we find the central angle of the sector as ${90}^{\circ}$.

Therefore the area of sector is:

$A=\frac{1}{2}{r}^{2}90\phantom{\rule{0ex}{0ex}}A=0.5\times 1\times \frac{90}{180}\pi \phantom{\rule{0ex}{0ex}}A=0.785f{t}^{2}$

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