Find The Area Of Following Sectors Of A Circle. Take Pie=3.142 .105degrees Inside And 8cm Outside. I Need The Calculation And The Answer Is Very Agently I Need It Now
Introduction
In geometry, a sector of a circle is a region bounded by two radii and an arc. The area of a sector can be calculated using a simple formula, which involves the radius of the circle and the central angle of the sector. In this article, we will discuss how to find the area of a sector of a circle, using a given example.
Understanding the Problem
We are given a circle with a radius of 8cm. There is a sector inside the circle with an angle of 105 degrees. We need to find the area of this sector.
Calculating the Area of a Sector
The formula to calculate the area of a sector is:
Area = (θ/360) × π × r^2
Where:
- θ is the central angle of the sector in degrees
- π is a mathematical constant approximately equal to 3.142
- r is the radius of the circle
Step 1: Convert the Central Angle to Radians
The central angle is given in degrees, but the formula requires the angle to be in radians. We can convert the angle from degrees to radians using the following formula:
θ (in radians) = θ (in degrees) × π/180
In this case, the central angle is 105 degrees. We can convert it to radians as follows:
θ (in radians) = 105 × π/180 = 1.828
Step 2: Calculate the Area of the Sector
Now that we have the central angle in radians, we can calculate the area of the sector using the formula:
Area = (θ/360) × π × r^2
Substituting the values, we get:
Area = (1.828/360) × 3.142 × 8^2
Area = (0.0051) × 3.142 × 64
Area = 1.032
Step 3: Calculate the Area of the Whole Circle
To verify our result, we can calculate the area of the whole circle using the formula:
Area = π × r^2
Substituting the values, we get:
Area = 3.142 × 8^2
Area = 3.142 × 64
Area = 201.06
Step 4: Calculate the Fraction of the Circle
The area of the sector is a fraction of the area of the whole circle. We can calculate this fraction as follows:
Fraction = (Area of sector) / (Area of whole circle)
Substituting the values, we get:
Fraction = 1.032 / 201.06
Fraction = 0.0051
Conclusion
In this article, we discussed how to find the area of a sector of a circle using a given example. We calculated the area of the sector using the formula and verified our result by calculating the area of the whole circle. The final answer is 1.032 square centimeters.
Final Answer
The final answer is: 1.032
Additional Information
- The radius of the circle is 8cm.
- The central angle of the sector is 105 degrees.
- The area of the sector is 1.032 square centimeters.
- The area of the whole circle is 201.06 square centimeters.
- The fraction of the circle occupied by the sector is 0.0051.
Frequently Asked Questions (FAQs) about Finding the Area of a Sector of a Circle ====================================================================================
Q: What is the formula to calculate the area of a sector of a circle?
A: The formula to calculate the area of a sector of a circle is:
Area = (θ/360) × π × r^2
Where:
- θ is the central angle of the sector in degrees
- π is a mathematical constant approximately equal to 3.142
- r is the radius of the circle
Q: What is the central angle of a sector?
A: The central angle of a sector is the angle formed by two radii of the circle. It is measured in degrees.
Q: How do I convert the central angle from degrees to radians?
A: To convert the central angle from degrees to radians, you can use the following formula:
θ (in radians) = θ (in degrees) × π/180
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 fraction of the area of a circle. The area of a sector is calculated using the formula:
Area = (θ/360) × π × r^2
Whereas the area of a circle is calculated using the formula:
Area = π × r^2
Q: How do I calculate the fraction of the circle occupied by the sector?
A: To calculate the fraction of the circle occupied by the sector, you can use the following formula:
Fraction = (Area of sector) / (Area of whole circle)
Q: What is the significance of the radius of the circle in calculating the area of a sector?
A: The radius of the circle is an essential parameter in calculating the area of a sector. It is used in the formula:
Area = (θ/360) × π × r^2
Q: Can I use the formula to calculate the area of a sector if the central angle is not given in degrees?
A: Yes, you can use the formula to calculate the area of a sector even if the central angle is not given in degrees. You can convert the central angle from the given unit to degrees using the following formula:
θ (in degrees) = θ (in radians) × 180/π
Q: What is the relationship between the area of a sector and the area of an arc?
A: The area of a sector is equal to the area of an arc. The area of an arc is calculated using the formula:
Area = (θ/360) × π × r^2
Where:
- θ is the central angle of the sector in degrees
- π is a mathematical constant approximately equal to 3.142
- r is the radius of the circle
Q: Can I use the formula to calculate the area of a sector if the radius of the circle is not given?
A: No, you cannot use the formula to calculate the area of a sector if the radius of the circle is not given. The radius of the circle is an essential parameter in calculating the area of a sector.
Conclusion
In this article, we have discussed frequently asked questions about finding the area of a sector of a circle. We have provided answers to various questions related to the formula, central angle, radius, and fraction of the circle occupied by the sector. We hope that this article has been helpful in clarifying any doubts you may have had about finding the area of a sector of a circle.