Circles and Sectors: Calculating Area and Arc Length

📐 Understanding Circles and Sectors

Let's explore how to calculate the area of circles and sectors, and how to determine arc length. We'll cover the necessary formulas and provide examples to help you master these concepts.

⭕ Circle Area

The area of a circle is the space enclosed within its boundary. The formula to calculate the area of a circle is:

$A = \pi r^2$

Where:

• $A$ is the area
• $\pi$ (pi) is approximately 3.14159
• $r$ is the radius of the circle

Example:

Suppose a circle has a radius of 5 cm. Calculate its area.

import math

radius = 5
area = math.pi * radius**2
print(f"The area of the circle is: {area:.2f} cm²")
# Output: The area of the circle is: 78.54 cm²

🍕 Sector Area

A sector is a portion of a circle enclosed by two radii and an arc. The formula to calculate the area of a sector is:

$A_{sector} = (\frac{\theta}{360}) \pi r^2$

Where:

• $A_{sector}$ is the area of the sector
• $\theta$ is the central angle in degrees
• $r$ is the radius of the circle

Example:

Suppose a sector has a central angle of 60 degrees and the circle's radius is 8 cm. Calculate the sector's area.

import math

angle = 60
radius = 8
sector_area = (angle / 360) * math.pi * radius**2
print(f"The area of the sector is: {sector_area:.2f} cm²")
# Output: The area of the sector is: 33.51 cm²

弧 Arc Length

Arc length is the distance along the curved line making up the arc. The formula to calculate the arc length is:

$L = (\frac{\theta}{360}) 2 \pi r$

Where:

• $L$ is the arc length
• $\theta$ is the central angle in degrees
• $r$ is the radius of the circle

Example:

Suppose a circle has a radius of 10 cm and the central angle is 45 degrees. Calculate the arc length.

import math

angle = 45
radius = 10
arc_length = (angle / 360) * 2 * math.pi * radius
print(f"The arc length is: {arc_length:.2f} cm")
# Output: The arc length is: 7.85 cm

📝 Summary

• Circle Area: $A = \pi r^2$
• Sector Area: $A_{sector} = (\frac{\theta}{360}) \pi r^2$
• Arc Length: $L = (\frac{\theta}{360}) 2 \pi r$

Understanding these formulas and practicing with examples will help you confidently solve problems involving circles, sectors, and arc lengths. 🚀