Chapter 11.3, Problem 46HP

To determine

The area of the sector

Answer to Problem 46HP

If the central angle and the chord both intercept a semicircle, the area of the sector and the area of the segment (as designated by the brown region) are equal.

Explanation of Solution

Given information:

Is the area of a sector of a circle sometimes, always, or never greater than the area of its corresponding segment?

Formula used:

The area of the segment is calculated on subtracting the area of the triangle determined by the central angle and the chord from the area of the sector.

Calculation:

In most cases, the area of the sector (as designated by the blue region) is greater than the area of the segment (as designated by the red region) for the same central angle. The area of the segment is contained within the area of the sector. In fact, to calculate the area of the segment is to subtract the area of the triangle determined by the central angle and the chord from the area of the sector.

  

However, if the central angle and the chord both intercept a semicircle, the area of the sector and the area of the segment (as designated by the brown region) are equal.

  

Therefore, the statement is sometimes true.

Conclusion:

If the central angle and the chord both intercept a semicircle, the area of the sector and the area of the segment (as designated by the brown region) are equal.