An arc length 3 feet is cut off by a central angle of pi/4 radians, find the area of the sector formed
Sector area= 0.5 r^2 theta (in radians)
arc length = rtheta
3=r(0.7854)
r=3.82
sector area= 0.5 (3.82)^2 * 0.7854 = 5.73 sq ft
Answer:
Area of the sector = 5.73 feet²
Step-by-step explanation:
An arc length is 3 feet is cut off by a central angle of \(\frac{\pi }{4}\)
We have to find the area of the sector formed.
Area of the sector formed = \(\frac{1}{2}r^{2}\theta\)
If the arc length = 3 feet
Central angle = \(\frac{\pi }{4}\)
Since arc length = \(r\theta\)
3 = r(\frac{\pi }{4})
r = \(\frac{3}{\frac{\pi }{4}}=\frac{(3)(4)}{\pi }=\frac{12}{\pi }\)
Now area of the sector = \(\frac{1}{2}(\frac{12}{\pi })^{2}\frac{\pi }{4}\)
= \(\frac{1}{2}(\frac{144}{\pi ^{2}})(\frac{\pi }{4})\)
= \(\frac{144}{\pi ^{2}}(\frac{\pi }{8})\)
= \(\frac{18}{\pi }\)
= \(\frac{18}{3.14}=5.73\) square feet
Therefore, area of the sector formed is 5.73 feet²