What is the Area of a Segment?

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Grade Level:

Class 6

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

Definition

What is it?

The area of a segment is the space enclosed by an arc (a curved part of a circle) and the chord connecting the endpoints of that arc. Imagine a pizza slice, then cut off the triangle part – the remaining curved crust piece is a segment.

Simple Example

Quick Example

Think of a round biscuit. If you make a straight cut across it, you divide the biscuit into two pieces. Each piece is a segment. The area of a segment tells you how much biscuit is in one of those pieces.

Worked Example

Step-by-Step

Let's find the area of a segment of a circle with radius 7 cm and a central angle of 90 degrees.

Step 1: Understand the formula. Area of Segment = Area of Sector - Area of Triangle.
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Step 2: Calculate the Area of the Sector. Formula: (theta / 360) * pi * r^2. Here, theta = 90 degrees, r = 7 cm, pi = 22/7.
Area of Sector = (90 / 360) * (22/7) * 7 * 7 = (1/4) * (22/7) * 49 = (1/4) * 22 * 7 = 11 * 7 / 2 = 77/2 = 38.5 sq cm.
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Step 3: Calculate the Area of the Triangle. Since the central angle is 90 degrees, it's a right-angled triangle. Formula: (1/2) * base * height. Here, base = radius = 7 cm, height = radius = 7 cm.
Area of Triangle = (1/2) * 7 * 7 = (1/2) * 49 = 24.5 sq cm.
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Step 4: Subtract the triangle area from the sector area to get the segment area.
Area of Segment = 38.5 sq cm - 24.5 sq cm = 14 sq cm.
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Answer: The area of the segment is 14 square cm.

Why It Matters

Understanding segment areas is crucial in fields like engineering and design. Civil engineers use it when designing curved roads or bridges, and architects use it for unique building shapes. It even helps game developers create realistic curved objects in virtual worlds.

Common Mistakes

MISTAKE: Confusing a segment with a sector. | CORRECTION: A sector is like a pizza slice (includes the center), while a segment is the part left after removing the triangle from a sector (it doesn't include the center).

MISTAKE: Using the wrong formula for the triangle area within the sector. | CORRECTION: If the central angle is not 90 degrees, the triangle area is not simply (1/2) * r * r. You might need trigonometry (1/2 * r^2 * sin(theta)) which you will learn later, or divide it into two right triangles.

MISTAKE: Forgetting to use consistent units (e.g., mixing cm and m). | CORRECTION: Always ensure all measurements (radius, area) are in the same units before calculating. Convert if necessary.

Practice Questions

Try It Yourself

QUESTION: A circle has a radius of 10 cm. If a sector has a central angle of 90 degrees, what is the area of the segment formed by this sector? (Use pi = 3.14) | ANSWER: 28.5 sq cm

QUESTION: A circular park has a radius of 14 meters. A straight path cuts across a section of the park, creating a segment with a central angle of 60 degrees. What is the area of this segment? (Use pi = 22/7 and sqrt(3) = 1.732) | ANSWER: Approximately 17.89 sq meters

QUESTION: A circular disc of radius 21 cm has a segment cut out. If the area of the sector forming this segment is 231 sq cm and the area of the triangle within this sector is 100 sq cm, what is the area of the segment? | ANSWER: 131 sq cm

MCQ

Quick Quiz

Which of these best describes a segment of a circle?

The entire circle

A part of the circle enclosed by an arc and a chord

A part of the circle enclosed by two radii and an arc

The circumference of the circle

The Correct Answer Is:

A segment is the region bounded by a chord and the arc it cuts off. Option C describes a sector, not a segment.

Real World Connection

In the Real World

Imagine a drone delivering a package in a circular flight path. If it needs to avoid a building by flying a straight path across a part of its usual circular route, the area 'cut off' by this straight path (the segment) is important for calculating fuel, time, and mapping its flight plan. This is used in logistics for companies like Zomato or Swiggy.

Key Vocabulary

Key Terms

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

What's Next

What to Learn Next

Great job understanding segments! Next, you can explore 'Area of Composite Shapes'. This will teach you how to find areas of figures made up of different basic shapes, including segments, which is super useful in real-life problems!