Converting the Central Angle to Degrees

Oct 13, 2025 by windailys.com 115 views

Introduction

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: A = (θ/360) × πr^2, where A is the area, θ is the central angle in degrees, and r is the radius of the circle. In this problem, we are given a central angle of $$\frac{5 \pi}{7}$$ radians and a diameter of 5.6 inches. We need to calculate the area of the sector using the given information.

Converting the Central Angle to Degrees

The central angle is given in radians, but the formula for the area of a sector requires the angle to be in degrees. To convert the central angle from radians to degrees, we can use the following conversion factor: 1 radian = (180/π) degrees.

import math
central_angle_radians = 5 * math.pi / 7
central_angle_degrees = (central_angle_radians * 180) / math.pi

Calculating the Radius

The diameter of the circle is given as 5.6 inches. To calculate the radius, we can simply divide the diameter by 2.

# Given diameter
diameter = 5.6
radius = diameter / 2

Calculating the Area of the Sector

Now that we have the central angle in degrees and the radius, we can calculate the area of the sector using the formula: A = (θ/360) × πr^2.

# Calculate area of sector
area_sector = (central_angle_degrees / 360) * math.pi * (radius ** 2)

Rounding the Final Answer

The final answer should be rounded to the nearest hundredth.

# Round final answer to nearest hundredth
final_answer = round(area_sector, 2)

Conclusion

In this problem, we calculated the area of a sector with a central angle of $$\frac{5 \pi}{7}$$ radians and a diameter of 5.6 inches. We first converted the central angle from radians to degrees, then calculated the radius, and finally used the formula for the area of a sector to calculate the final answer.

Final Answer

The final answer is: final_answer

Calculating the Area of a Sector: Step-by-Step Solution

Step 1: Convert the Central Angle to Degrees

To convert the central angle from radians to degrees, we can use the following conversion factor: 1 radian = (180/π) degrees.

import math
central_angle_radians = 5 * math.pi / 7
central_angle_degrees = (central_angle_radians * 180) / math.pi

Step 2: Calculate the Radius

The diameter of the circle is given as 5.6 inches. To calculate the radius, we can simply divide the diameter by 2.

# Given diameter
diameter = 5.6
radius = diameter / 2

Step 3: Calculate the Area of the Sector

Now that we have the central angle in degrees and the radius, we can calculate the area of the sector using the formula: A = (θ/360) × πr^2.

# Calculate area of sector
area_sector = (central_angle_degrees / 360) * math.pi * (radius ** 2)

Step 4: Round the Final Answer

The final answer should be rounded to the nearest hundredth.

# Round final answer to nearest hundredth
final_answer = round(area_sector, 2)

Calculating the Area of a Sector: Example Use Case

Suppose we want to calculate the area of a sector with a central angle of $$\frac{5 \pi}{7}$$ radians and a diameter of 5.6 inches. We can use the following Python code to calculate the final answer:

import math
central_angle_radians = 5 * math.pi / 7
central_angle_degrees = (central_angle_radians * 180) / math.pi
diameter = 5.6
radius = diameter / 2
area_sector = (central_angle_degrees / 360) * math.pi * (radius ** 2)
final_answer = round(area_sector, 2)
print("The final answer is:", final_answer)

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

A: The formula for calculating the area of a sector is: A = (θ/360) × πr^2, where A is the area, θ is the central angle in degrees, and r is the radius of the circle.

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

A: To convert a central angle from radians to degrees, you can use the following conversion factor: 1 radian = (180/π) degrees. You can also use the following formula: θ (in degrees) = θ (in radians) × (180/π).

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

A: A sector is a portion of a circle enclosed by two radii and an arc. A circle is a complete shape with no beginning or end.

Q: Can I use the formula for the area of a sector with a central angle in radians?

A: Yes, you can use the formula for the area of a sector with a central angle in radians. However, you will need to convert the central angle from radians to degrees using the conversion factor: 1 radian = (180/π) degrees.

Q: How do I calculate the radius of a circle when given the diameter?

A: To calculate the radius of a circle when given the diameter, you can simply divide the diameter by 2.

Q: Can I use a calculator to calculate the area of a sector?

A: Yes, you can use a calculator to calculate the area of a sector. Simply enter the values for the central angle and the radius, and the calculator will give you the area.

Q: What is the significance of the value π in the formula for the area of a sector?

A: The value π is a mathematical constant that represents the ratio of a circle's circumference to its diameter. It is approximately equal to 3.14.

Q: Can I use the formula for the area of a sector with a central angle in degrees and a radius in inches?

A: Yes, you can use the formula for the area of a sector with a central angle in degrees and a radius in inches. However, you will need to make sure that the units are consistent (e.g. inches for both the radius and the diameter).

Q: How do I round the final answer to the nearest hundredth?

A: To round the final answer to the nearest hundredth, you can use the round() function in most calculators or programming languages.

Q: Can I use the formula for the area of a sector with a central angle in radians and a radius in centimeters?

A: Yes, you can use the formula for the area of a sector with a central angle in radians and a radius in centimeters. However, you will need to make sure that the units are consistent (e.g. centimeters for both the radius and the diameter).

Calculating the Area of a Sector: Common Mistakes to Avoid

Not converting the central angle from radians to degrees: Make sure to convert the central angle from radians to degrees using the conversion factor: 1 radian = (180/π) degrees.
Not using the correct formula: Use the correct formula for the area of a sector: A = (θ/360) × πr^2.
Not rounding the final answer to the nearest hundredth: Make sure to round the final answer to the nearest hundredth using the round() function.
Not using consistent units: Make sure to use consistent units (e.g. inches for both the radius and the diameter).

Calculating the Area of a Sector: Example Use Cases

Calculating the area of a sector with a central angle of $$\frac{5 \pi}{7}$$ radians and a diameter of 5.6 inches: Use the formula for the area of a sector: A = (θ/360) × πr^2, where θ is the central angle in degrees and r is the radius of the circle.
Calculating the area of a sector with a central angle of 30 degrees and a radius of 2 inches: Use the formula for the area of a sector: A = (θ/360) × πr^2, where θ is the central angle in degrees and r is the radius of the circle.

Calculating the Area of a Sector: Tips and Tricks

Use a calculator to calculate the area of a sector: Use a calculator to calculate the area of a sector, especially if you are not familiar with the formula.
Check your units: Make sure to use consistent units (e.g. inches for both the radius and the diameter).
Round the final answer to the nearest hundredth: Use the round() function to round the final answer to the nearest hundredth.
Use the correct formula: Use the correct formula for the area of a sector: A = (θ/360) × πr^2.