What Is The Area Of A Sector With A Central Angle Of $\frac{4 \pi}{3}$ Radians And A Radius Of $12.5 \text{ Cm}$?Use $3.14$ For $\pi$ And Round Your Final Answer To The Nearest Hundredth. Enter Your Answer As A

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Introduction

When dealing with circular shapes, it's essential to understand the concept of sectors and their areas. 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: $A = \frac{\theta}{360} \pi r^2$, where $\theta$ is the central angle in degrees, $\pi$ is a mathematical constant approximately equal to $3.14$, and $r$ is the radius of the circle. In this article, we will explore how to calculate the area of a sector with a central angle of $\frac{4 \pi}{3}$ radians and a radius of $12.5 \text{ cm}$.

Understanding the Central Angle

The central angle is a crucial component in calculating the area of a sector. In this case, the central angle is given as $\frac{4 \pi}{3}$ radians. To use the formula $A = \frac{\theta}{360} \pi r^2$, we need to convert the central angle from radians to degrees. We can do this by multiplying the central angle in radians by $\frac{180}{\pi}$.

Converting Radians to Degrees

To convert the central angle from radians to degrees, we multiply $\frac{4 \pi}{3}$ by $\frac{180}{\pi}$.

$$\frac{4 \pi}{3} \times \frac{180}{\pi} = 240$$

Calculating the Area of the Sector

Now that we have the central angle in degrees, we can use the formula $A = \frac{\theta}{360} \pi r^2$ to calculate the area of the sector.

$$A = \frac{240}{360} \times 3.14 \times (12.5)^2$$

Simplifying the Expression

To simplify the expression, we can start by evaluating the expression inside the parentheses.

$$(12.5)^2 = 156.25$$

Continuing the Simplification

Now we can substitute this value back into the expression.

$$A = \frac{240}{360} \times 3.14 \times 156.25$$

Evaluating the Expression

To evaluate the expression, we can start by multiplying the numbers together.

$$\frac{240}{360} = \frac{2}{3}$$

Continuing the Evaluation

Now we can substitute this value back into the expression.

$$A = \frac{2}{3} \times 3.14 \times 156.25$$

Final Evaluation

To get the final answer, we can multiply the numbers together.

$$A = \frac{2}{3} \times 3.14 \times 156.25 = 415.51$$

Conclusion

In conclusion, the area of a sector with a central angle of $\frac{4 \pi}{3}$ radians and a radius of $12.5 \text{ cm}$ is approximately $415.51 \text{ cm}^2$. This can be calculated using the formula $A = \frac{\theta}{360} \pi r^2$, where $\theta$ is the central angle in degrees, $\pi$ is a mathematical constant approximately equal to $3.14$, and $r$ is the radius of the circle.

Formula for Calculating the Area of a Sector

The formula for calculating the area of a sector is:

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

Where:

• $A$ is the area of the sector
• $\theta$ is the central angle in degrees
• $\pi$ is a mathematical constant approximately equal to $3.14$
• $r$ is the radius of the circle

Example of Calculating the Area of a Sector

To calculate the area of a sector, we can use the formula:

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

For example, if we have a sector with a central angle of $120^\circ$ and a radius of $10 \text{ cm}$, we can calculate the area of the sector as follows:

$$A = \frac{120}{360} \times 3.14 \times (10)^2$$

Simplifying the Expression

To simplify the expression, we can start by evaluating the expression inside the parentheses.

$$(10)^2 = 100$$

Continuing the Simplification

Now we can substitute this value back into the expression.

$$A = \frac{120}{360} \times 3.14 \times 100$$

Evaluating the Expression

To evaluate the expression, we can start by multiplying the numbers together.

$$\frac{120}{360} = \frac{1}{3}$$

Continuing the Evaluation

Now we can substitute this value back into the expression.

$$A = \frac{1}{3} \times 3.14 \times 100$$

Final Evaluation

To get the final answer, we can multiply the numbers together.

$$A = \frac{1}{3} \times 3.14 \times 100 = 104.67$$

Conclusion

In conclusion, the area of a sector with a central angle of $120^\circ$ and a radius of $10 \text{ cm}$ is approximately $104.67 \text{ cm}^2$. This can be calculated using the formula $A = \frac{\theta}{360} \pi r^2$, where $\theta$ is the central angle in degrees, $\pi$ is a mathematical constant approximately equal to $3.14$, and $r$ is the radius of the circle.

Formula for Calculating the Area of a Sector in Radians

The formula for calculating the area of a sector in radians is:

$$A = \frac{\theta}{2\pi} \pi r^2$$

Where:

• $A$ is the area of the sector
• $\theta$ is the central angle in radians
• $\pi$ is a mathematical constant approximately equal to $3.14$
• $r$ is the radius of the circle

Example of Calculating the Area of a Sector in Radians

To calculate the area of a sector in radians, we can use the formula:

$$A = \frac{\theta}{2\pi} \pi r^2$$

For example, if we have a sector with a central angle of $\frac{4 \pi}{3}$ radians and a radius of $12.5 \text{ cm}$, we can calculate the area of the sector as follows:

$$A = \frac{\frac{4 \pi}{3}}{2\pi} \times 3.14 \times (12.5)^2$$

Simplifying the Expression

To simplify the expression, we can start by evaluating the expression inside the parentheses.

$$(12.5)^2 = 156.25$$

Continuing the Simplification

Now we can substitute this value back into the expression.

$$A = \frac{\frac{4 \pi}{3}}{2\pi} \times 3.14 \times 156.25$$

Evaluating the Expression

To evaluate the expression, we can start by multiplying the numbers together.

$$\frac{\frac{4 \pi}{3}}{2\pi} = \frac{2}{3}$$

Continuing the Evaluation

Now we can substitute this value back into the expression.

$$A = \frac{2}{3} \times 3.14 \times 156.25$$

Final Evaluation

To get the final answer, we can multiply the numbers together.

$$A = \frac{2}{3} \times 3.14 \times 156.25 = 415.51$$

Conclusion

In conclusion, the area of a sector with a central angle of $\frac{4 \pi}{3}$ radians and a radius of $12.5 \text{ cm}$ is approximately $415.51 \text{ cm}^2$. This can be calculated using the formula $A = \frac{\theta}{2\pi} \pi r^2$, where $\theta$ is the central angle in radians, $\pi$ is a mathematical constant approximately equal to $3.14$, and $r$ is the radius of the circle.

Introduction

Calculating the area of a sector is a crucial concept in mathematics, particularly in geometry and trigonometry. In our previous article, we explored how to calculate the area of a sector with a central angle of $\frac{4 \pi}{3}$ radians and a radius of $12.5 \text{ cm}$. In this article, we will answer some frequently asked questions about calculating the area of a sector.

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

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

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

Where:

• $A$ is the area of the sector
• $\theta$ is the central angle in degrees
• $\pi$ is a mathematical constant approximately equal to $3.14$
• $r$ is the radius of the circle

Q: How do I convert radians to degrees?

A: To convert radians to degrees, you can multiply the central angle in radians by $\frac{180}{\pi}$.

Q: What is the difference between the formula for calculating the area of a sector in degrees and radians?

A: The formula for calculating the area of a sector in degrees is:

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

Where:

• $A$ is the area of the sector
• $\theta$ is the central angle in degrees
• $\pi$ is a mathematical constant approximately equal to $3.14$
• $r$ is the radius of the circle

The formula for calculating the area of a sector in radians is:

$$A = \frac{\theta}{2\pi} \pi r^2$$

Where:

• $A$ is the area of the sector
• $\theta$ is the central angle in radians
• $\pi$ is a mathematical constant approximately equal to $3.14$
• $r$ is the radius of the circle

Q: How do I calculate the area of a sector with a central angle of $120^\circ$ and a radius of $10 \text{ cm}$?

A: To calculate the area of a sector with a central angle of $120^\circ$ and a radius of $10 \text{ cm}$, you can use the formula:

$$A = \frac{120}{360} \times 3.14 \times (10)^2$$

Q: How do I calculate the area of a sector with a central angle of $\frac{4 \pi}{3}$ radians and a radius of $12.5 \text{ cm}$?

A: To calculate the area of a sector with a central angle of $\frac{4 \pi}{3}$ radians and a radius of $12.5 \text{ cm}$, you can use the formula:

$$A = \frac{\frac{4 \pi}{3}}{2\pi} \times 3.14 \times (12.5)^2$$

Q: What is the area of a sector with a central angle of $\frac{4 \pi}{3}$ radians and a radius of $12.5 \text{ cm}$?

A: The area of a sector with a central angle of $\frac{4 \pi}{3}$ radians and a radius of $12.5 \text{ cm}$ is approximately $415.51 \text{ cm}^2$.

Q: Can I use the formula for calculating the area of a sector to calculate the area of a circle?

A: Yes, you can use the formula for calculating the area of a sector to calculate the area of a circle. If the central angle is $360^\circ$ or $2\pi$ radians, the area of the sector is equal to the area of the circle.

Q: What is the area of a circle with a radius of $12.5 \text{ cm}$?

A: The area of a circle with a radius of $12.5 \text{ cm}$ is approximately $491.12 \text{ cm}^2$.

Conclusion

In conclusion, calculating the area of a sector is a crucial concept in mathematics, particularly in geometry and trigonometry. By understanding the formula for calculating the area of a sector and how to convert radians to degrees, you can calculate the area of a sector with ease. We hope this Q&A article has been helpful in answering your questions about calculating the area of a sector.