PROBLEM
Find the area of the sector with central angle 76∘ in a circle with radius 11. Round to the nearest hundredth.
STEP 1
Assumptions
1. Circle N has a central angle m∠MNP which is 76∘.
2. The radius MN of circle N is 11 units.
3. The area of sector MNP is to be found.
4. The area of the sector is proportional to the central angle of the sector.
5. The formula for the area of a sector with central angle θ in degrees and radius r is given by:
Area of sector=360θ×πr2
STEP 2
First, we need to calculate the area of the entire circle using the formula for the area of a circle, which is πr2.
Area of circle=πr2
STEP 3
Now, plug in the given value for the radius r to calculate the area of the circle.
Area of circle=π(11)2
STEP 4
Calculate the area of the entire circle.
Area of circle=π(121)
STEP 5
Next, we calculate the area of sector MNP using the formula for the area of a sector.
Area of sector MNP=360m∠MNP×πr2
STEP 6
Plug in the given values for the central angle m∠MNP and the radius r to calculate the area of sector MNP.
Area of sector MNP=36076×π(11)2
STEP 7
Calculate the area of sector MNP.
Area of sector MNP=36076×π(121)
STEP 8
Simplify the fraction 36076.
Area of sector MNP=9019×π(121)
STEP 9
Calculate the numerical value of the area of sector MNP.
Area of sector MNP=9019×π×121
STEP 10
Use a calculator to find the numerical value of the area of sector MNP to the nearest hundredth.
Area of sector MNP≈9019×3.14159×121
SOLUTION
Perform the multiplication to find the area of sector MNP.
Area of sector MNP≈9019×3.14159×121≈80.42 The area of sector MNP is approximately 80.42 square units.
Start understanding anything
Get started now for free.