What Is The Area Of A Sector With A Central Angle Of $\frac{\pi}{3}$ Radians And A Radius Of 12.4 M?Use 3.14 For $\pi$ And Round Your Final Answer To The Nearest Hundredth.Enter Your Answer As A Decimal In The Box: $\square \,

What is the Area of a Sector with a Central Angle of $\frac{\pi}{3}$ Radians and a Radius of 12.4 m?

In geometry, a sector is a portion of a circle enclosed by two radii and an arc. The area of a sector can be calculated using the formula: Area = (θ/360) × πr^2, where θ is the central angle in degrees, π is a mathematical constant approximately equal to 3.14, and r is the radius of the circle. In this article, we will calculate the area of a sector with a central angle of $\frac{\pi}{3}$ radians and a radius of 12.4 m.

Understanding the Central Angle

The central angle is the angle formed by two radii that intersect at the center of the circle. In this problem, the central angle is given as $\frac{\pi}{3}$ radians. To convert this angle to degrees, we can use the conversion factor: 1 radian = 180/π degrees. Therefore, the central angle in degrees is:

$\frac{\pi}{3}$ radians × (180/π) degrees/radian = 60 degrees

Calculating the Area of the Sector

Now that we have the central angle in degrees, we can use the formula: Area = (θ/360) × πr^2 to calculate the area of the sector. Plugging in the values, we get:

Area = (60/360) × 3.14 × (12.4)^2

Simplifying the Expression

To simplify the expression, we can first calculate the square of the radius:

(12.4)^2 = 154.76

Now, we can plug this value back into the expression:

Area = (60/360) × 3.14 × 154.76

Evaluating the Expression

To evaluate the expression, we can first calculate the product of 3.14 and 154.76:

3.14 × 154.76 = 485.55

Now, we can plug this value back into the expression:

Area = (60/360) × 485.55

Simplifying the Fraction

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 60:

(60/360) = (1/6)

Now, we can plug this value back into the expression:

Area = (1/6) × 485.55

Evaluating the Expression

To evaluate the expression, we can multiply the fraction by the value:

Area = 0.17 × 485.55

Area = 82.24

Rounding the Answer

Finally, we need to round the answer to the nearest hundredth. Therefore, the final answer is:

82.24

In this article, we calculated the area of a sector with a central angle of $\frac{\pi}{3}$ radians and a radius of 12.4 m. We used the formula: Area = (θ/360) × πr^2 to calculate the area, and then simplified the expression to get the final answer. The final answer is 82.24, rounded to the nearest hundredth.
Frequently Asked Questions (FAQs) About the Area of a Sector

In our previous article, we calculated the area of a sector with a central angle of $\frac{\pi}{3}$ radians and a radius of 12.4 m. In this article, we will answer some frequently asked questions (FAQs) about the area of a sector.

Q: What is the formula for the area of a sector?

A: The formula for the area of a sector is: Area = (θ/360) × πr^2, where θ is the central angle in degrees, π is a mathematical constant approximately equal to 3.14, and r is the radius of the circle.

Q: What is the central angle in degrees?

A: The central angle in degrees is 60 degrees, which is equivalent to $\frac{\pi}{3}$ radians.

Q: How do I convert radians to degrees?

A: To convert radians to degrees, you can use the conversion factor: 1 radian = 180/π degrees.

Q: What is the radius of the circle?

A: The radius of the circle is 12.4 m.

Q: How do I calculate the area of a sector with a central angle of 90 degrees and a radius of 10 m?

A: To calculate the area of a sector with a central angle of 90 degrees and a radius of 10 m, you can use the formula: Area = (90/360) × π(10)^2. Plugging in the values, you get:

Area = (1/4) × 3.14 × 100

Area = 78.5

Q: How do I calculate the area of a sector with a central angle of 120 degrees and a radius of 15 m?

A: To calculate the area of a sector with a central angle of 120 degrees and a radius of 15 m, you can use the formula: Area = (120/360) × π(15)^2. Plugging in the values, you get:

Area = (1/3) × 3.14 × 225

Area = 235.5

Q: What is the relationship between the area of a sector and the area of a circle?

A: The area of a sector is a fraction of the area of a circle. The fraction is equal to the ratio of the central angle to 360 degrees.

Q: How do I calculate the area of a sector with a central angle of x degrees and a radius of r m?

A: To calculate the area of a sector with a central angle of x degrees and a radius of r m, you can use the formula: Area = (x/360) × πr^2.

In this article, we answered some frequently asked questions (FAQs) about the area of a sector. We provided examples and formulas to help you calculate the area of a sector with different central angles and radii. We hope this article has been helpful in understanding the concept of the area of a sector.