Arc Length of a Circle Formula - Sector Area, Examples, Radians, In Terms of Pi, Trigonometry

This paragraph introduces the concept of arc length in a circle and explains how to calculate it. It starts with a visual description of an arc segment within a circle and defines the arc length 's' as the distance along the circle between two points A and B, based on the angle Theta. The relationship between arc length, the radius of the circle, and the angle in radians is established, with the formula s = Theta * R. Two examples are provided to illustrate the calculation: the first with an angle of 5 radians and a radius of 12T, resulting in an arc length of 60 feet; the second with a radius of 8 cm and an angle of 150°, which requires converting degrees to radians and results in an arc length of approximately 20.944 cm. An alternative method is also mentioned for calculating arc length using the angle in degrees, which involves the formula s = (angle in degrees / 360) * 2πR.

The second paragraph delves into the calculation of the area of a sector of a circle and the arc length as a fraction of the circle's circumference. It explains that the arc length can be thought of as the circumference times a fraction of the circle, represented by Theta over 360. The paragraph provides a step-by-step solution for finding the arc length when the angle is given in degrees, using the formula (angle in degrees / 360) * 2πR, and demonstrates this with an example of a 150° angle and an 8 cm radius, yielding an arc length of approximately 20.944 cm. The area of a sector is then discussed, with the formula A = (1/2) * Theta * R^2 for angles in radians, and its conversion for angles in degrees, A = (Theta / 360) * πR^2. Examples are given to show how to calculate the area of sectors for both radians and degrees, emphasizing the relationship between the sector's area, the circle's area, and the angle's fraction of the circle.

This paragraph continues the discussion on calculating the area of a sector in a circle, focusing on the use of radians and degrees. It provides the formula for sector area when the angle is in radians, A = (1/2) * Theta * R^2, and shows its application with an example of a 90° angle and a 10 cm radius, resulting in an area of 25π square cm. The paragraph also explains how to convert the radians-based formula to one that uses degrees by multiplying by 2π/360, leading to the formula A = (Theta / 360) * πR^2. The similarities between the processes of finding arc length and sector area are highlighted, showing how both can be derived from the original formula by adjusting for the angle's unit. Additional examples are given, including one with a 60° angle and a 5T radius, resulting in an area of approximately 13.1 square feet, and another with a 2 radian angle and an 8 cm radius, yielding an area of 64 square cm.

The final paragraph succinctly wraps up the topic by reiterating the formulas for calculating the area of a sector when the angle is given in radians or degrees. It emphasizes the straightforward calculation of the sector area as 64 square cm for an angle of two radians and a radius of 8 cm. The paragraph serves as a brief conclusion to the detailed explanations and examples provided in the previous sections, reinforcing the understanding of sector area calculation in the context of circle geometry.