How Do You Find the Arc of a Partial Circle?

To find the arc length of a partial circle, you need to determine the fraction of the entire circle the arc represents and then multiply that fraction by the circle's circumference.

Here's a breakdown of the steps:

Determine the Central Angle: Identify the central angle (θ) that subtends the arc. This angle is usually given in degrees or radians.
Calculate the Fraction of the Circle: Divide the central angle by the total angle of a circle.
- If the angle is in degrees, divide by 360°: Fraction = θ (in degrees) / 360°
- If the angle is in radians, divide by 2π: Fraction = θ (in radians) / 2π
Calculate the Circumference: The circumference (C) of the entire circle is given by the formula: C = 2πr, where r is the radius of the circle.
Calculate the Arc Length: Multiply the fraction of the circle by the circumference: Arc Length = Fraction * C
- Arc Length = (θ/360°) * 2πr (if θ is in degrees)
- Arc Length = (θ/2π) * 2πr = θr (if θ is in radians)

Example:

Let's say you have a circle with a radius of 8, and the arc is subtended by a central angle of 3/4 of the circle. We want to find the arc length.

The radius r = 8
The fraction of the circle is 3/4.
The circumference is C = 2 * pi * 8 = 16pi
The Arc Length is (3/4) * 16pi = 12pi

In Summary:

To find the arc length of a partial circle, determine the central angle, calculate the fraction of the circle it represents, find the circumference of the entire circle, and then multiply the fraction by the circumference. The formula simplifies to Arc Length = θr when the angle θ is in radians.