Calculating Sector Area OAB With Radius 23 Cm And Angle 96 Degrees

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In the realm of geometry, understanding the properties and calculations related to circles and their parts is crucial. One such part is a sector, which is a region bounded by two radii and an arc of a circle. This article delves into the process of calculating the area of a sector, specifically focusing on sector OAB. We will explore the formula, the steps involved, and provide a detailed calculation to arrive at the solution, rounded to one decimal place. This comprehensive guide aims to enhance your understanding of sector area calculations and their applications in various mathematical and real-world scenarios.

Understanding the Fundamentals of Sector Area

Before diving into the specific calculation for sector OAB, it's essential to grasp the fundamental concepts. A sector, as mentioned earlier, is a pie-shaped portion of a circle defined by two radii and the intercepted arc. The area of a sector represents the space enclosed within these boundaries. To calculate this area, we utilize a formula that incorporates the angle subtended by the arc at the center of the circle and the radius of the circle. This formula is derived from the relationship between the sector's area and the total area of the circle. The angle, typically denoted as θ (theta), is measured in degrees, and the radius is denoted as r. The understanding of these fundamentals is the cornerstone for accurately calculating the area of any sector.

The Formula for Sector Area

The cornerstone of calculating the area of a sector lies in the formula:

A = (θ/360) × π × r²

Where:

  • A represents the area of the sector.
  • θ (theta) is the angle in degrees subtended by the arc at the center of the circle.
  • Ï€ (pi) is a mathematical constant approximately equal to 3.14159.
  • r is the radius of the circle.

This formula is derived from the proportion of the sector's angle to the total angle of a circle (360 degrees). It essentially calculates the fraction of the circle's total area that the sector occupies. By understanding this formula, one can easily compute the area of any sector given the angle and the radius. This formula is not just a mathematical tool but also a bridge connecting the geometry of circles to practical applications in various fields.

Given Information for Sector OAB

In our specific case, we are tasked with finding the area of sector OAB. We are provided with the following crucial information:

  • Radius (r) = 23 cm: This is the distance from the center of the circle to any point on the circumference, defining the size of the circle and, consequently, the sector.
  • Angle (θ) = 96°: This is the angle subtended by the arc AB at the center O of the circle, defining the proportion of the circle that the sector occupies.

With these two key pieces of information, we have all the necessary components to calculate the area of sector OAB using the formula. The radius provides the scale of the circle, while the angle determines the fraction of the circle that the sector represents. Together, they allow us to accurately determine the area enclosed within the sector's boundaries.

Step-by-Step Calculation of the Area of Sector OAB

Now that we have the formula and the given information, let's proceed with the step-by-step calculation of the area of sector OAB. This process involves substituting the given values into the formula and performing the necessary arithmetic operations. By following these steps carefully, we can ensure an accurate result, rounded to one decimal place as required.

1. Substitute the Values into the Formula

The first step is to substitute the given values of the radius (r) and the angle (θ) into the formula for the area of a sector:

A = (θ/360) × π × r²

Substituting r = 23 cm and θ = 96°:

A = (96/360) × π × (23)²

This substitution sets the stage for the calculation by placing the specific values for our sector OAB into the general formula. It transforms the abstract formula into a concrete expression that we can evaluate to find the area. This step is crucial as it ensures that we are working with the correct parameters for the given sector.

2. Calculate the Square of the Radius

Next, we need to calculate the square of the radius:

(23)² = 23 × 23 = 529

This calculation determines the area of a square with sides equal to the radius, which is a component in determining the total area of the circle. Squaring the radius is a fundamental step in calculating the area of any circle or sector, as it reflects the two-dimensional nature of area. The result, 529, represents the square centimeters that would fit within a square with sides of 23 centimeters. This value will be used in the subsequent steps to determine the sector's area.

3. Multiply by π (pi)

Now, we multiply the result from the previous step by π (pi), which is approximately 3.14159:

529 × π ≈ 529 × 3.14159 ≈ 1661.90111

This multiplication incorporates the circular constant π into the calculation, accounting for the circular shape of the sector. Pi is the ratio of a circle's circumference to its diameter, and it plays a crucial role in all circle-related calculations. Multiplying by π transforms the square of the radius into an area that reflects the circular geometry. The result, approximately 1661.90111, represents the area of a full circle with a radius of 23 cm.

4. Multiply by the Angle Ratio

The next step is to multiply the result by the ratio of the sector's angle to the total angle of a circle (360 degrees):

(96/360) × 1661.90111

First, calculate the angle ratio:

96/360 ≈ 0.266666667

Then, multiply this ratio by the result from the previous step:

  1. 266666667 × 1661.90111 ≈ 443.173629

This multiplication scales the full circle's area down to the proportion represented by the sector. The angle ratio, 96/360, represents the fraction of the circle that the sector occupies. Multiplying this ratio by the full circle's area gives us the area of the sector. The result, approximately 443.173629, is the unrounded area of sector OAB.

5. Round to One Decimal Place

Finally, we round the result to one decimal place as required:

  1. 173629 ≈ 443.2

This rounding provides a practical and easily understandable value for the area. Rounding to one decimal place is a common practice in many applications, as it provides a balance between precision and simplicity. The final result, 443.2, represents the area of sector OAB in square centimeters, rounded to one decimal place. This is the answer we were seeking, and it completes the calculation process.

Final Answer and Conclusion

Therefore, the area of sector OAB, rounded to one decimal place, is approximately 443.2 cm². This calculation demonstrates the application of the sector area formula and the importance of each step in arriving at the correct answer. Understanding these calculations is vital for various applications in mathematics, engineering, and other fields. The ability to accurately calculate sector areas allows for precise measurements and designs in real-world scenarios.

In conclusion, the process of finding the area of a sector involves understanding the formula, substituting the given values, performing the calculations step by step, and rounding the result to the required precision. This detailed guide has provided a clear and comprehensive explanation of this process, specifically for sector OAB. By mastering these concepts, you can confidently tackle similar problems and apply this knowledge in various practical situations. The area of a sector is not just a mathematical concept; it's a tool that enables us to understand and quantify portions of circles, which are fundamental shapes in our world.