Area of Segment

A segment is a part of a circle formed by an arc and the chord connecting the arc's endpoints. The area of a segment is the space enclosed within these boundaries.

To find the area of a segment, you'll need to understand both the sector of the circle formed by the arc and the triangular area formed by the chord.

To find the area of a segment, we use the formula: Segment Area = Sector Area - Triangle Area. The sector area is given by (θ/360) × π × r², where θ is the central angle in degrees and r is the radius. The triangle area can be calculated using various methods, such as trigonometry or coordinate geometry.

Derivation of the formula: 1. Calculate the sector area using the formula: (θ/360) × π × r². 2. Find the area of the triangle formed by the chord and the radii. This can be done using trigonometric identities if the angle is known. 3. Subtract the triangle area from the sector area to get the segment area.

We can find the area of a segment using different methods, depending on the available information. Here are some methods: Method using the central angle and radius Method using the chord length and radius Method using the arc length Now let’s discuss these methods.

Method Using the Central Angle and Radius

If the central angle (θ) and radius (r) are given, find the sector area and then subtract the triangle area. For example, if θ is 60 degrees and r is 10 cm: 

Sector Area = (θ/360) × π × r² = (60/360) × π × 10² = (1/6) × π × 100 = 52.36 cm² 

Triangle Area = 0.5 × r² × sin(θ) = 0.5 × 10² × sin(60) = 43.30 cm²

Segment Area = Sector Area - Triangle Area = 52.36 - 43.30 = 9.06 cm²

Method Using the Chord Length and Radius

If the chord length (c) and radius (r) are given, use trigonometry to find θ and proceed as before. For instance, if c is 12 cm and r is 10 cm: 1. Find θ using cos(θ/2) = c/(2r) 2. Calculate the sector and triangle areas as above.

Method Using the Arc Length

If the arc length (l) and radius (r) are given, find θ using θ = (l/r), then proceed with the previous methods.

The area of a segment is measured in square units. The measurement depends on the system used: In the metric system, the area is measured in square meters (m²), square centimeters (cm²), and square millimeters (mm²).

In the imperial system, the area is measured in square inches (in²), square feet (ft²), and square yards (yd²).

Since a segment is part of a circle, its area can be computed using different methods based on available data. Consider these special cases:

Case 1: Use of Central Angle and Radius If the central angle and radius are known, compute the segment area by finding the sector and triangle areas.

Case 2: Use of Chord Length and Radius If the chord length and radius are provided, use trigonometry to find the central angle, then compute the segment area.

Case 3: Using Arc Length If the arc length is known, derive the central angle and apply the standard segment area formula.

To ensure accurate results when calculating the area of a segment, keep these tips in mind:

• Always verify the unit of measurement and convert if necessary.
   
• Understand the difference between sector area and segment area.
   
• Use trigonometric identities effectively when angles or side lengths are given.
   
• Don't forget to subtract the triangle area from the sector area to get the segment area.

• Segment: A region in a circle bounded by an arc and the chord connecting the arc's endpoints.

• Sector: A portion of a circle enclosed by two radii and the arc between them.

• Central Angle: The angle subtended at the center of the circle by two radii.

• Chord: A line segment whose endpoints lie on the circle.

• Trigonometry: A branch of mathematics dealing with the relations of the sides and angles of triangles.