What Is The Approximate Area Of A Sector Given Θ = 212 ∘ \Theta = 212^\circ Θ = 21 2 ∘ With A Radius Of 45 M?A. 2613.59 M 2 2613.59 \, \text{m}^2 2613.59 M 2 B. 3744.45 M 2 3744.45 \, \text{m}^2 3744.45 M 2 C. 3371.26 M 2 3371.26 \, \text{m}^2 3371.26 M 2 D. 2928.36 M 2 2928.36 \, \text{m}^2 2928.36 M 2

Understanding the Problem

When dealing with circular sectors, it's essential to understand the relationship between the angle, radius, and area. In this problem, we're given a sector with an angle of $\Theta = 212^\circ$ and a radius of 45 m. Our goal is to find the approximate area of this sector.

The Formula for the Area of a Sector

The area of a sector can be calculated using the formula:

$$A = \frac{\Theta}{360} \pi r^2$$

where $A$ is the area of the sector, $\Theta$ is the angle in degrees, $\pi$ is a mathematical constant approximately equal to 3.14159, and $r$ is the radius of the sector.

Plugging in the Values

Now that we have the formula, let's plug in the given values:

$$A = \frac{212}{360} \pi (45)^2$$

Simplifying the Expression

To simplify the expression, we can start by evaluating the fraction:

$$\frac{212}{360} \approx 0.5889$$

Next, we can calculate the square of the radius:

$$(45)^2 = 2025$$

Now, we can substitute these values back into the formula:

$$A \approx 0.5889 \pi (2025)$$

Evaluating the Expression

To evaluate the expression, we can start by multiplying the fraction by the square of the radius:

$$0.5889 \times 2025 \approx 1191.45$$

Next, we can multiply this result by $\pi$:

$$1191.45 \times \pi \approx 3744.45$$

Rounding the Result

Since we're asked to find the approximate area, we can round our result to two decimal places:

$$A \approx 3744.45 \, \text{m}^2$$

Conclusion

In this problem, we used the formula for the area of a sector to find the approximate area of a sector with an angle of $\Theta = 212^\circ$ and a radius of 45 m. Our result is $A \approx 3744.45 \, \text{m}^2$. This value is closest to option B, which is $3744.45 \, \text{m}^2$.

Key Takeaways

• The area of a sector can be calculated using the formula $A = \frac{\Theta}{360} \pi r^2$.
• To find the area of a sector, you need to know the angle in degrees and the radius of the sector.
• The formula for the area of a sector involves multiplying the fraction of the circle by the square of the radius and then multiplying the result by $\pi$.

Practice Problems

• Find the area of a sector with an angle of $\Theta = 180^\circ$ and a radius of 30 m.
• Find the area of a sector with an angle of $\Theta = 270^\circ$ and a radius of 20 m.

Real-World Applications

• The area of a sector can be used to calculate the area of a circular region in a variety of real-world applications, such as architecture, engineering, and design.
• The formula for the area of a sector can be used to calculate the area of a circular region in a variety of fields, such as physics, mathematics, and computer science.

Common Mistakes

• Failing to convert the angle from degrees to radians.
• Failing to square the radius.
• Failing to multiply the fraction by the square of the radius and then by $\pi$.

Tips and Tricks

• Make sure to convert the angle from degrees to radians before plugging it into the formula.
• Use a calculator to evaluate the expression and find the approximate area.
• Check your work by plugging in different values and verifying that the result is correct.
  

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

A: The formula for the area of a sector is:

$$A = \frac{\Theta}{360} \pi r^2$$

where $A$ is the area of the sector, $\Theta$ is the angle in degrees, $\pi$ is a mathematical constant approximately equal to 3.14159, and $r$ is the radius of the sector.

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

A: The area of a sector is a portion of the area of a circle. The area of a circle is given by the formula:

$$A = \pi r^2$$

where $A$ is the area of the circle, $\pi$ is a mathematical constant approximately equal to 3.14159, and $r$ is the radius of the circle.

Q: How do I convert an angle from degrees to radians?

A: To convert an angle from degrees to radians, you can use the following formula:

$$\text{radians} = \frac{\pi}{180} \times \text{degrees}$$

Q: What is the relationship between the angle, radius, and area of a sector?

A: The area of a sector is directly proportional to the angle and the square of the radius. This means that if you increase the angle or the radius, the area of the sector will also increase.

Q: Can I use the formula for the area of a sector to calculate the area of a circular region in a real-world application?

A: Yes, the formula for the area of a sector can be used to calculate the area of a circular region in a variety of real-world applications, such as architecture, engineering, and design.

Q: What are some common mistakes to avoid when calculating the area of a sector?

A: Some common mistakes to avoid when calculating the area of a sector include:

• Failing to convert the angle from degrees to radians.
• Failing to square the radius.
• Failing to multiply the fraction by the square of the radius and then by $\pi$.

Q: How can I verify that my calculation for the area of a sector is correct?

A: You can verify that your calculation for the area of a sector is correct by plugging in different values and verifying that the result is correct.

Q: Can I use a calculator to evaluate the expression and find the approximate area of a sector?

A: Yes, you can use a calculator to evaluate the expression and find the approximate area of a sector.

Q: What is the significance of the value of $\pi$ in the formula for the area of a sector?

A: The value of $\pi$ is a mathematical constant that is approximately equal to 3.14159. It is used in the formula for the area of a sector to calculate the area of a circular region.

Q: Can I use the formula for the area of a sector to calculate the area of a circular region in a field other than mathematics?

A: Yes, the formula for the area of a sector can be used to calculate the area of a circular region in a variety of fields, such as physics, engineering, and computer science.

Q: What are some real-world applications of the formula for the area of a sector?

A: Some real-world applications of the formula for the area of a sector include:

• Calculating the area of a circular region in a building or a bridge.
• Calculating the area of a circular region in a machine or a device.
• Calculating the area of a circular region in a design or a model.

Q: Can I use the formula for the area of a sector to calculate the area of a circular region in a three-dimensional object?

A: Yes, the formula for the area of a sector can be used to calculate the area of a circular region in a three-dimensional object.

Q: What are some tips and tricks for calculating the area of a sector?

A: Some tips and tricks for calculating the area of a sector include:

• Make sure to convert the angle from degrees to radians before plugging it into the formula.
• Use a calculator to evaluate the expression and find the approximate area.
• Check your work by plugging in different values and verifying that the result is correct.