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.

The formula for the area of a sector with angle θ and radius r is (θ)/(360^°) × π r². We substitute the given values into this formula to find the area of the minor sector.

Substitute the values of θ = 30^°, r = 4 cm, and π = 3.14 into the formula. First, simplify the fraction (30)/(360) to (1)/(12). Then, calculate 4² = 16. Finally, multiply all values to get the sector area.

Substitute the values of θ = 30^°, r = 4 cm, and π = 3.14 into the formula. First, simplify the fraction (30)/(360) to (1)/(12). Then, calculate 4² = 16. Finally, multiply all values to get the sector area.

Substitute the values: (330)/(360) × 3.14 × 16. Simplify the fraction (330)/(360) to (11)/(12). Then perform the multiplication.

Multiply (11)/(12) by 3.14 and 16. This gives (552.64)/(12), 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.

Multiply (11)/(12) by 3.14 and 16. This gives (552.64)/(12), 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.

Substitute r=4 cm and π=3.14 into the formula. Calculate 4² = 16. Then, perform the multiplication 3.14 × 16.

The area of the full circle is 50.24 cm 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.

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.

Find the area of the sector of a circle with radius $4$ cm and of angle $30^\circ$ . Also, find the area of the corresponding major sector (Use $\pi = 3.14$ ).

Answer: The area of the minor sector is $4.19\text{ cm}^2$ and the area of the major sector is $46.05\text{ cm}^2$ .

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 $\theta = 30^\circ$ . We are also instructed to use $\pi = 3.14$ for calculations.

Step 2: Calculate Area of Minor Sector

The formula for the area of a sector with angle $\theta$ and radius $r$ is $\frac{\theta}{360^\circ} \times \pi r^2$ . 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 $\theta = 30^\circ$ , $r = 4$ cm, and $\pi = 3.14$ into the formula. First, simplify the fraction $\frac{30}{360}$ to $\frac{1}{12}$ . Then, calculate $4^2 = 16$ . Finally, multiply all values to get the sector area.

Step 4: Perform Calculation

Perform the multiplication: $\frac{1}{12} \times 3.14 \times 16 = \frac{50.24}{12}$ . 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 $360^\circ$ . To find the angle of the major sector, we subtract the angle of the minor sector from $360^\circ$ . This gives us $360^\circ - 30^\circ = 330^\circ$ .

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 $330^\circ$ . Substitute $\theta = 330^\circ$ , $r = 4$ cm, and $\pi = 3.14$ into the formula.

Step 7: Substitute Values and Calculate (Major Sector Method 1)

Substitute the values: $\frac{330}{360} \times 3.14 \times 16$ . Simplify the fraction $\frac{330}{360}$ to $\frac{11}{12}$ . Then perform the multiplication.

Step 8: Perform Calculation (Major Sector Method 1)

Multiply $\frac{11}{12}$ by $3.14$ and $16$ . This gives $\frac{552.64}{12}$ , 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 $\pi r^2$ .

Step 10: Substitute Values and Calculate (Area of Circle)

Substitute $r=4$ cm and $\pi=3.14$ into the formula. Calculate $4^2 = 16$ . Then, perform the multiplication $3.14 \times 16$ .

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.

More from this chapter

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Find the area of the sector of a circle with radius 444 cm and of angle 30∘30^\circ30∘. Also, find the area of the corresponding major sector (Use π=3.14\pi = 3.14π=3.14).