In Exercises 91–92, find the measure of the central angle on a circle of radius r that forms a sector with the given area. Radius, r: 10 feet Area of the Sector, A: 25 square feet

Key Concepts

Central Angle

The central angle of a circle is the angle formed at the center of the circle by two radii. It is measured in degrees or radians and is directly related to the arc length and area of the sector it subtends. Understanding how to calculate the central angle is crucial for solving problems involving sectors.

Area of a Sector

The area of a sector is a portion of the circle defined by a central angle. It can be calculated using the formula A = (θ/360) * πr² for degrees or A = (1/2) * r²θ for radians, where A is the area, r is the radius, and θ is the central angle. This concept is essential for determining the relationship between the area of the sector and the central angle.

Radians and Degrees

Radians and degrees are two units for measuring angles. One complete revolution is 360 degrees or 2π radians. In trigonometry, radians are often preferred for calculations involving circles, as they provide a direct relationship between the angle and the arc length. Understanding how to convert between these units is important for solving problems related to angles in circles.