Find the area of a sector of a circle having radius r and central angle θ. Express answers to the nearest tenth. See Example 5. r = 40.0 mi, θ = 135°

Key Concepts

Area of a Sector

The area of a sector of a circle is a portion of the circle defined by a central angle θ. It can be calculated using the formula A = (θ/360) * πr², where A is the area, θ is the angle in degrees, and r is the radius. This formula derives from the fact that the area of a full circle is πr², and the sector's area is a fraction of that based on the angle.

Central Angle

The central angle θ is the angle formed at the center of the circle by two radii. It is measured in degrees or radians and determines the size of the sector. In this problem, θ is given as 135°, which indicates that the sector represents a significant portion of the circle, specifically 135 out of 360 degrees.

Units of Measurement

When calculating the area of a sector, it is essential to ensure that all measurements are in compatible units. In this case, the radius is given in miles, so the area will be expressed in square miles. Additionally, rounding the final answer to the nearest tenth is important for clarity and precision in reporting the result.