Area of Segment
2026-02-28 13:36 Diff
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Last updated on September 15, 2025

The area of a segment is the region bounded by an arc and the chord joining the endpoints of the arc in a circle. Calculating the area of a segment involves understanding parts of the circle and applying the correct formulas. In this section, we will explore how to find the area of a segment.

What is the Area of a 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.

Area of the Segment Formula

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.

How to Find the Area of a Segment?

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.

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Unit of Area of Segment

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).

Special Cases or Variations for the Area of Segment

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.

Tips and Tricks for Area of Segment

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.

Common Mistakes and How to Avoid Them in Area of Segment

It's common for students to make mistakes while finding the area of a segment. Let’s explore some common errors and how to avoid them.

Problem 1

The central angle of a segment in a circle with a radius of 10 cm is given as 90 degrees. What will be the area of the segment?

Okay, lets begin

We will find the area as 21.46 cm².

Explanation

With a central angle of 90 degrees and radius of 10 cm, calculate the sector area and subtract the triangle area: 

Sector Area = (90/360) × π × 10² = 78.54 cm² 

Triangle Area = 0.5 × 10² × sin(90) = 50 cm² 

Segment Area = Sector Area - Triangle Area = 78.54 - 50 = 28.54 cm²

Well explained 👍

Problem 2

A segment has a chord length of 8 cm and a radius of 5 cm. What is the area of the segment?

Okay, lets begin

We will find the area as 3.94 cm².

Explanation

Given a chord length of 8 cm and radius of 5 cm, use trigonometry to find the central angle, then compute the segment area.

Well explained 👍

Problem 3

The arc length of a segment is 6 cm, and the radius is 4 cm. What is the area of the segment?

Okay, lets begin

We find the area of the segment as 4.19 cm².

Explanation

Find the central angle using θ = (l/r), then calculate the sector and triangle areas.

Subtract the triangle area from the sector area to find the segment area.

Well explained 👍

Problem 4

Find the area of the segment in a circle with a radius of 7 cm and a central angle of 45 degrees.

Okay, lets begin

We will find the area as 4.51 cm².

Explanation

With a radius of 7 cm and central angle of 45 degrees, calculate:

Sector Area = (45/360) × π × 7² = 19.24 cm²

Triangle Area = 0.5 × 7² × sin(45) = 14.73 cm²

Segment Area = Sector Area - Triangle Area = 19.24 - 14.73 = 4.51 cm²

Well explained 👍

Problem 5

Help Sarah find the area of a segment if the radius is 6 m and the chord length is 10 m.

Okay, lets begin

We will find the area as 6.28 m².

Explanation

Use the chord length and radius to find the central angle, then calculate the segment area by subtracting the triangle area from the sector area.

Well explained 👍

FAQs on Area of Segment

1.Can the area of a segment be negative?

No, the area of a segment cannot be negative. The area of any shape is always positive.

2.How to find the area of a segment if the central angle and radius are given?

Calculate the sector area using (θ/360) × π × r² and subtract the triangle area using 0.5 × r² × sin(θ).

3.How to find the area of a segment if only the chord length and radius are given?

Use trigonometry to find the central angle, then apply the segment area formula.

4.What is the difference between the area of a sector and the area of a segment?

The area of a sector includes the triangular part formed by the chord, while the area of a segment excludes it.

5.How is the chord length used in segment calculations?

The chord length helps determine the triangle area within the segment, which is subtracted from the sector area to find the segment area.

Seyed Ali Fathima S

About the Author

Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.

Fun Fact

: She has songs for each table which helps her to remember the tables