Understanding the concepts of arc length and sector area is essential for grasping the wider subject of circle geometry. These calculations allow us to measure the length of a segment of the circle's circumference (arc length) and the area of a 'slice' of the circle (sector area), using the circle's radius and the central angle that subtends the arc or sector. This section delves into the mathematical procedures for calculating these measures, incorporating the constant π and the relationship of the sector's angle to the full 360° of the circle.

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Introduction to Arc Length

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The arc length of a circle is the distance along the edge of the arc. To calculate this, we use the formula:

where:

• $\theta$ is the central angle in degrees,
• $r$ is the radius of the circle, and
• $\pi$ (pi) is a constant approximately equal to 3.14159.

Example 1: Calculating Arc Length

Given a circle with a radius of 8 cm and a central angle of 45°, find the arc length.

$\text{Arc length} = \frac{45}{360} \times 2 \times \pi \times 8$

$\text{Arc length} = \frac{1}{8} \times 16\pi$

$\text{Arc length} = 2\pi \text{ cm}$

Therefore, the arc length is approximately 6.28 cm.

Introduction to Sector Area

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The sector area is the area of the 'slice' of the circle formed by two radii and the arc between them. The formula for calculating the sector area is:

where:

• $\theta$ is the central angle in degrees,
• $r$ is the radius of the circle, and
• $\pi$ is a constant approximately equal to 3.14159.

Example 2: Calculating Sector Area

Consider a circle with a radius of 10 cm and a central angle of 60°. Calculate the sector area.

$\text{Sector area} = \frac{60}{360} \times \pi \times 10^2$

$\text{Sector area} = \frac{1}{6} \times \pi \times 100$

$\text{Sector area} = \frac{100\pi}{6} \text{ cm}^2$

$\text{Sector area} = \frac{50\pi}{3} \text{ cm}^2$

Thus, the sector area is approximately $52.36 \text{ cm}^2$.

Key Concepts and Formulas

• Central Angle $(\theta)$: The angle formed at the circle's centre by two radii. It's crucial in determining the fraction of the circle's circumference or area that the arc or sector represents.
• Radius $(r)$: The distance from the centre of the circle to any point on its circumference. It's essential for calculating both arc length and sector area.
• Pi $(\pi)$: A mathematical constant approximately equal to 3.14159. It represents the ratio of a circle's circumference to its diameter and is pivotal in calculations involving circles.

Practice Problems

Problem 1

A circle has a radius of 7 cm, and the central angle of the sector is 90°. Calculate the arc length and the sector area.

Solution:

• Arc length $= \dfrac{90}{360} \times 2\pi \times 7$

• Sector area $= \dfrac{90}{360} \times \pi \times 7^2$

Problem 2

Find the arc length and sector area of a circle with radius 12 cm and a central angle of 30°.

Solution:

• Arc length $= \dfrac{30}{360} \times 2\pi \times 12$

• Sector area $= \dfrac{30}{360} \times \pi \times 12^2$