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Expert-verified Found in: Page 479 ### Precalculus Mathematics for Calculus

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 $\mathbf{0}\mathbf{.785}\mathbit{f}{\mathbit{t}}^{\mathbf{2}}$.

See the step by step solution

## Step 1. Given information.

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

## Step 2. Write the concept.

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 $A=\frac{1}{2}{r}^{2}\theta$

## Step 3. Determining the values.

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:

$A=\frac{1}{2}{r}^{2}90\phantom{\rule{0ex}{0ex}}A=0.5×1×\frac{90}{180}\pi \phantom{\rule{0ex}{0ex}}A=0.785f{t}^{2}$ ### Want to see more solutions like these? 