Calculating the Sector Area

Oct 13, 2025 by miodeya.com 132 views

Introduction

When it comes to designing and maintaining a tennis court, proper lighting is crucial for player safety and optimal gameplay. In this scenario, we have a tennis court area illuminated by four corner lights. The illumination of the sector area close to each light is considered to be good (G), while the remaining area is considered to be lit satisfactorily (S). In this article, we will delve into the mathematics behind calculating the percentage of the area that is lit satisfactorily.

Understanding the Problem

To solve this problem, we need to understand the geometry of the tennis court and the illumination pattern of the corner lights. Let's assume that the tennis court is a square with side length 's'. Each corner light illuminates a sector area of the court, which we will consider as the good (G) area. The remaining area is the satisfactorily (S) lit area.

Calculating the Sector Area

The sector area close to each light can be calculated using the formula for the area of a sector of a circle:

A = (θ/360) * πr^2

where A is the area of the sector, θ is the angle subtended by the sector at the center of the circle, and r is the radius of the circle.

In this case, the radius of the circle is half the side length of the tennis court, i.e., s/2. The angle θ is 90 degrees, as each corner light illuminates a quarter of the court.

Calculating the Good (G) Area

Using the formula for the area of a sector, we can calculate the good (G) area as follows:

AG = (90/360) * π(s/2)^2 AG = (1/4) * π(s/2)^2 AG = (1/4) * (πs^2)/4 AG = (πs^2)/16

Calculating the Satisfactorily (S) Area

The satisfactorily (S) area is the remaining area of the court, which can be calculated as follows:

AS = s^2 - AG AS = s^2 - (πs^2)/16 AS = (16s^2 - πs^2)/16 AS = (16 - π)s^2/16

Calculating the Percentage of Lit Area

To calculate the percentage of the area that is lit satisfactorily, we need to divide the satisfactorily (S) area by the total area of the court and multiply by 100:

Percentage of lit area = (AS/s^2) * 100 Percentage of lit area = ((16 - π)s^2/s2) * 100 Percentage of lit area = (16 - π) * 100

Numerical Calculation

Now, let's calculate the percentage of the area that is lit satisfactorily using the value of π as approximately 3.14:

Percentage of lit area = (16 - 3.14) * 100 Percentage of lit area = 12.86 * 100 Percentage of lit area = 1286%

However, this is not the correct answer. We need to consider that the satisfactorily (S) area is the remaining area of the court, which is 1 - (good (G) area/total area).

Correct Calculation

The correct calculation for the percentage of the area that is lit satisfactorily is as follows:

Percentage of lit area = (1 - (good (G) area/total area)) * 100 Percentage of lit area = (1 - (πs^2/16s2)) * 100 Percentage of lit area = (1 - (π/16)) * 100 Percentage of lit area = (1 - 0.19635) * 100 Percentage of lit area = 0.80365 * 100 Percentage of lit area = 80.365%

Conclusion

In this article, we calculated the percentage of the area that is lit satisfactorily on a tennis court illuminated by four corner lights. We first calculated the good (G) area using the formula for the area of a sector of a circle. Then, we calculated the satisfactorily (S) area as the remaining area of the court. Finally, we calculated the percentage of the area that is lit satisfactorily by dividing the satisfactorily (S) area by the total area of the court and multiplying by 100. The correct answer is approximately 80.365%.

Introduction

In our previous article, we calculated the percentage of the area that is lit satisfactorily on a tennis court illuminated by four corner lights. In this article, we will answer some frequently asked questions related to the illumination of a tennis court.

Q: What is the formula for calculating the good (G) area?

A: The formula for calculating the good (G) area is:

AG = (θ/360) * πr^2

where A is the area of the sector, θ is the angle subtended by the sector at the center of the circle, and r is the radius of the circle.

Q: What is the formula for calculating the satisfactorily (S) area?

A: The formula for calculating the satisfactorily (S) area is:

AS = s^2 - AG AS = s^2 - (πs^2)/16 AS = (16s^2 - πs^2)/16 AS = (16 - π)s^2/16

Q: How do I calculate the percentage of lit area?

A: To calculate the percentage of lit area, you need to divide the satisfactorily (S) area by the total area of the court and multiply by 100:

Percentage of lit area = (AS/s^2) * 100 Percentage of lit area = ((16 - π)s^2/s2) * 100 Percentage of lit area = (16 - π) * 100

Q: What is the correct calculation for the percentage of lit area?

A: The correct calculation for the percentage of lit area is:

Q: What is the significance of the percentage of lit area?

A: The percentage of lit area is an important factor in determining the safety and visibility of a tennis court. A higher percentage of lit area indicates better visibility and safety for players.

Q: How can I apply this calculation to a real-world scenario?

A: To apply this calculation to a real-world scenario, you need to know the side length of the tennis court and the angle subtended by the sector at the center of the circle. You can then use the formulas provided to calculate the good (G) area, satisfactorily (S) area, and percentage of lit area.

Q: What are some common mistakes to avoid when calculating the percentage of lit area?

A: Some common mistakes to avoid when calculating the percentage of lit area include:

Not considering the angle subtended by the sector at the center of the circle
Not using the correct formula for calculating the good (G) area
Not using the correct formula for calculating the satisfactorily (S) area
Not considering the total area of the court when calculating the percentage of lit area

Conclusion

In this article, we answered some frequently asked questions related to the illumination of a tennis court. We provided formulas and calculations for calculating the good (G) area, satisfactorily (S) area, and percentage of lit area. We also discussed the significance of the percentage of lit area and provided tips for applying this calculation to a real-world scenario.