Area and Arc Length of a Sector of a Circle

Area and Arc Length of a Sector of a Circle

Area of a Sector

The area of a sector is a fundamental concept in precalculus, relating the portion of a circle defined by a central angle to the total area of the circle.

• Definition: A sector of a circle is the region bounded by two radii and the arc between them.

• Formula: For a circle of radius r and central angle θ (in radians), the area A of the sector is given by:

$A = \frac{1}{2} r^2 \theta$

• Important Note: This formula is valid only when the angle θ is measured in radians. If the angle is given in degrees, it must be converted to radians first.

• Conversion: To convert degrees to radians, use the formula:

$\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180}$

• Example: Find the area of a sector with radius 5 units and central angle 60°.

First, convert 60° to radians: $60 \times \frac{\pi}{180} = \frac{\pi}{3}$ radians. Then, apply the formula: $A = \frac{1}{2} \times 5^2 \times \frac{\pi}{3} = \frac{25\pi}{6}$ units2.

Arc Length of a Sector

The arc length is the distance along the curved edge of the sector, determined by the central angle and the radius.

• Definition: The arc length is the length of the portion of the circle's circumference intercepted by the central angle.

• Formula: For a circle of radius r and central angle θ (in radians), the arc length s is given by:

$s = r\theta$

• Important Note: This formula is valid only when the angle θ is measured in radians. If the angle is given in degrees, it must be converted to radians first.

• Example: Find the arc length of a sector with radius 4 units and central angle 45°.

First, convert 45° to radians: $45 \times \frac{\pi}{180} = \frac{\pi}{4}$ radians. Then, apply the formula: $s = 4 \times \frac{\pi}{4} = \pi$ units.

Comparison Table: Area vs. Arc Length Formulas

Quantity	Formula (θ in radians)	Depends on
Area of Sector	$A = \frac{1}{2} r^2 \theta$	Radius squared, central angle
Arc Length	$s = r\theta$	Radius, central angle

Summary: Both the area and arc length formulas for a sector of a circle require the central angle to be in radians. Always convert degrees to radians before applying these formulas.