Find the area of the sector of a circle with radius 4 cm and of angle 30∘. Also, find the area of the corresponding major sector (Use π=3.14).
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Step-by-Step Solution
Step 1: Identify Given Values
We are given the radius of the circle r=4 cm and the angle of the minor sector θ=30∘. We are also instructed to use π=3.14 for calculations.
Step 2: Calculate Area of Minor Sector
The formula for the area of a sector with angle θ and radius r is 360∘θ×πr2. We substitute the given values into this formula to find the area of the minor sector.
Step 3: Substitute Values and Calculate
Substitute the values of θ=30∘, r=4 cm, and π=3.14 into the formula. First, simplify the fraction 36030 to 121. Then, calculate 42=16. Finally, multiply all values to get the sector area.
Step 4: Perform Calculation
Perform the multiplication: 121×3.14×16=1250.24. Dividing 50.24 by 12 yields approximately 4.1866... cm 2. Rounding to two decimal places, this is 4.19 cm 2.
Step 5: Calculate Angle of Major Sector
The total angle in a circle is 360∘. To find the angle of the major sector, we subtract the angle of the minor sector from 360∘. This gives us 360∘−30∘=330∘.
Step 6: Calculate Area of Major Sector (Method 1: Using angle)
We can calculate the area of the major sector using the same sector area formula, but with the major sector angle of 330∘. Substitute θ=330∘, r=4 cm, and π=3.14 into the formula.
Step 7: Substitute Values and Calculate (Major Sector Method 1)
Substitute the values: 360330×3.14×16. Simplify the fraction 360330 to 1211. Then perform the multiplication.
Step 8: Perform Calculation (Major Sector Method 1)
Multiply 1211 by 3.14 and 16. This gives 12552.64, which is approximately 46.0533... cm 2. Rounding to two decimal places, this is 46.05 cm 2. Note that this method sometimes leads to minor rounding differences compared to Method 2 due to multiple calculations.
Step 9: Calculate Area of Circle
Alternatively, we can find the area of the major sector by subtracting the area of the minor sector from the total area of the circle. First, calculate the area of the full circle using the formula πr2.
Step 10: Substitute Values and Calculate (Area of Circle)
Substitute r=4 cm and π=3.14 into the formula. Calculate 42=16. Then, perform the multiplication 3.14×16.
Step 11: Perform Calculation (Area of Circle)
The area of the full circle is 50.24 cm 2.
Step 12: Calculate Area of Major Sector (Method 2: Using total area)
Subtract the previously calculated area of the minor sector from the total area of the circle to find the area of the major sector.
Step 13: Substitute Values and Calculate (Major Sector Method 2)
Substitute the area of the circle (50.24 cm 2) and the area of the minor sector (4.19 cm 2) into the equation. Perform the subtraction.
Step 14: Perform Calculation (Major Sector Method 2)
Subtracting 4.19 from 50.24 gives 46.05. So, the area of the major sector is 46.05 cm 2. Both methods yield approximately the same result, confirming the calculation.