What is the Area of a Segment of a Circle?

S3-SA2-0301

Grade Level:

Class 6

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The area of a segment of a circle is the space enclosed by an arc and the chord connecting the endpoints of that arc. Imagine a pizza slice, then remove the triangle part that goes to the center; the remaining crusty bit is a segment.

Simple Example

Quick Example

Think of a round 'chapati'. If you cut a straight line across it (a chord), you'll get two pieces. Each of these pieces is a segment. The 'area of a segment' would be how much chapati is in one of those pieces.

Worked Example

Step-by-Step

Let's find the area of a segment of a circle with a radius of 10 cm, where the central angle is 60 degrees. (Use pi = 3.14)

---1. First, find the area of the sector. The formula for the area of a sector is (theta / 360) * pi * r^2.
Here, theta = 60 degrees, r = 10 cm. So, Area of sector = (60 / 360) * 3.14 * 10 * 10 = (1/6) * 3.14 * 100 = 314 / 6 = 52.33 sq cm.

---2. Next, find the area of the triangle formed by the two radii and the chord. Since the central angle is 60 degrees and the two sides are radii (equal), it's an equilateral triangle. The formula for an equilateral triangle is (sqrt(3)/4) * side^2. Here, side = radius = 10 cm.
So, Area of triangle = (1.732 / 4) * 10 * 10 = (1.732 / 4) * 100 = 173.2 / 4 = 43.3 sq cm.

---3. Finally, subtract the area of the triangle from the area of the sector to get the area of the segment.
Area of segment = Area of sector - Area of triangle = 52.33 - 43.3 = 9.03 sq cm.

---Answer: The area of the segment is 9.03 sq cm.

Why It Matters

Understanding segment areas is key for engineers designing curved structures like tunnels or bridges. It helps in AI for image recognition (identifying parts of circular objects) and in physics for calculating forces on curved surfaces. Even game developers use it to create realistic circular movements and shapes.

Common Mistakes

MISTAKE: Confusing a segment with a sector. | CORRECTION: A sector is like a pizza slice (includes the center point). A segment is the part left after you cut off the triangle from a sector (it does not include the center point).

MISTAKE: Forgetting to subtract the area of the triangle from the area of the sector. | CORRECTION: The segment is the *difference* between the sector and the triangle. Always remember to subtract the triangle's area.

MISTAKE: Using the wrong formula for the triangle's area, especially when the angle isn't 90 or 60 degrees. | CORRECTION: For a general triangle with two sides 'r' and included angle 'theta', the area is (1/2) * r^2 * sin(theta). For 60 degrees, it's an equilateral triangle. For 90 degrees, it's (1/2) * base * height.

Practice Questions

Try It Yourself

QUESTION: A circle has a radius of 7 cm. A sector of this circle has a central angle of 90 degrees. Find the area of the segment formed by this sector. (Use pi = 22/7) | ANSWER: Area of sector = (90/360) * (22/7) * 7 * 7 = (1/4) * 22 * 7 = 38.5 sq cm. Area of triangle = (1/2) * 7 * 7 = 24.5 sq cm. Area of segment = 38.5 - 24.5 = 14 sq cm.

QUESTION: The radius of a circle is 14 cm and the central angle of a sector is 120 degrees. Calculate the area of the minor segment. (Use pi = 22/7, sqrt(3) = 1.732) | ANSWER: Area of sector = (120/360) * (22/7) * 14 * 14 = (1/3) * 22 * 2 * 14 = 205.33 sq cm. Area of triangle = (1/2) * 14 * 14 * sin(120) = (1/2) * 196 * (sqrt(3)/2) = 49 * 1.732 = 84.868 sq cm. Area of segment = 205.33 - 84.868 = 120.462 sq cm.

QUESTION: A circular park has a radius of 20 meters. A straight path (chord) is built across it, such that it forms a sector with a central angle of 45 degrees. What is the area of the smaller grassy segment created by this path? (Use pi = 3.14, sin(45) = 0.707) | ANSWER: Area of sector = (45/360) * 3.14 * 20 * 20 = (1/8) * 3.14 * 400 = 157 sq meters. Area of triangle = (1/2) * 20 * 20 * sin(45) = (1/2) * 400 * 0.707 = 200 * 0.707 = 141.4 sq meters. Area of segment = 157 - 141.4 = 15.6 sq meters.

MCQ

Quick Quiz

What is the correct formula to find the area of a minor segment?

Area of sector + Area of triangle

Area of sector - Area of triangle

Area of circle - Area of sector

Area of circle - Area of triangle

The Correct Answer Is:

The area of a minor segment is found by subtracting the area of the triangle (formed by the chord and two radii) from the area of the corresponding sector. Options A, C, and D do not represent this relationship.

Real World Connection

In the Real World

When ISRO designs satellite dishes or antennae, they often deal with segments of circles to optimize signal reception. In construction, architects might calculate the area of glass segments for unique window designs in modern buildings or for curved parts of stadiums. Even in making 'jalebis', the amount of batter for a specific curve could relate to segment area!

Key Vocabulary

Key Terms

SEGMENT: The region bounded by a chord and an arc | SECTOR: The region bounded by two radii and an arc | CHORD: A straight line connecting two points on a circle | ARC: A part of the circumference of a circle | RADIUS: The distance from the center to any point on the circle

What's Next

What to Learn Next

Great job understanding segments! Next, you can explore the 'Perimeter of a Segment' to learn about the length of its boundary. After that, 'Area of Composite Shapes' will help you combine these concepts to find areas of more complex figures.