What is Area of a Segment?

S3-SA2-0150

Grade Level:

Class 7

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

Definition

What is it?

A segment of a circle is the region enclosed by an arc and the chord connecting the endpoints of that arc. The area of a segment is the space it occupies within the circle, like a slice of pizza where the crust is the arc and the straight cut is the chord.

Simple Example

Quick Example

Imagine you have a round roti (chapati) and you make one straight cut across it, but not through the center. The smaller piece you cut off is a segment. The area of this piece is what we are talking about.

Worked Example

Step-by-Step

Let's find the area of a segment of a circle with radius 10 cm and a central angle of 60 degrees.

Step 1: Calculate the area of the sector formed by the central angle. The formula is (theta / 360) * pi * r^2.
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Step 2: Here, theta = 60 degrees, r = 10 cm. Area of sector = (60 / 360) * (22/7) * (10)^2 = (1/6) * (22/7) * 100 = 100 * 22 / 42 = 2200 / 42 = 52.38 sq cm (approx).
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Step 3: Calculate the area of the triangle formed by the two radii and the chord. Since the central angle is 60 degrees and the two sides (radii) are equal, it's an equilateral triangle.
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Step 4: Area of an equilateral triangle = (sqrt(3) / 4) * side^2. Here, side = radius = 10 cm.
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Step 5: Area of triangle = (1.732 / 4) * (10)^2 = (1.732 / 4) * 100 = 173.2 / 4 = 43.3 sq cm (approx).
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Step 6: Subtract the area of the triangle from the area of the sector to find the area of the segment.
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Step 7: Area of segment = Area of sector - Area of triangle = 52.38 - 43.3 = 9.08 sq cm.
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Answer: The area of the segment is approximately 9.08 sq cm.

Why It Matters

Understanding segments is key for engineers designing curved structures like bridges or tunnels. It helps data scientists analyze circular data patterns, and physicists calculate forces on curved objects. It's used in architecture, game development, and even creating cool graphics.

Common Mistakes

MISTAKE: Confusing a segment with a sector. | CORRECTION: A sector is like a pizza slice (arc + two radii). A segment is like a pizza slice with the straight cut (arc + chord). Always visualize the difference.

MISTAKE: Using the wrong formula for the triangle's area. | CORRECTION: The triangle formed by the radii and chord might not always be equilateral or right-angled. Use the general formula (1/2) * r^2 * sin(theta) or (1/2) * base * height, depending on the angle and triangle type.

MISTAKE: Forgetting to convert angles to radians if using the (1/2)r^2(theta - sin theta) formula. | CORRECTION: When using the formula Area = (1/2)r^2(theta - sin theta), theta MUST be in radians. If using degrees, stick to (theta/360)*pi*r^2 for sector and (1/2)*r^2*sin(theta) for triangle.

Practice Questions

Try It Yourself

QUESTION: A circle has a radius of 7 cm. A segment is formed by a central angle of 90 degrees. Find the area of this segment. (Use pi = 22/7). | ANSWER: Area of sector = (90/360) * (22/7) * 7^2 = (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: Calculate the area of the minor segment of a circle with a radius of 14 cm, where the chord subtends an angle of 120 degrees at the center. (Use pi = 22/7, sqrt(3) = 1.732). | ANSWER: Area of sector = (120/360) * (22/7) * 14^2 = (1/3) * (22/7) * 196 = (1/3) * 22 * 28 = 616/3 = 205.33 sq cm. Area of triangle = (1/2) * 14^2 * sin(120) = (1/2) * 196 * (sqrt(3)/2) = 98 * 0.866 = 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 pathway of length 20 meters is built inside the park, dividing it into two segments. Find the area of the smaller segment. (Hint: The triangle formed is equilateral). | ANSWER: Since the chord length (20m) is equal to the radius (20m), the triangle formed by the two radii and the chord is equilateral, meaning the central angle is 60 degrees. Area of sector = (60/360) * pi * 20^2 = (1/6) * 3.14 * 400 = 209.33 sq m. Area of triangle = (sqrt(3)/4) * 20^2 = (1.732/4) * 400 = 173.2 sq m. Area of smaller segment = 209.33 - 173.2 = 36.13 sq m.

MCQ

Quick Quiz

Which of these correctly describes a segment of a circle?

The region between two radii and an arc.

The region between a chord and an arc.

The entire boundary of the circle.

The distance around the circle.

The Correct Answer Is:

A segment is defined as the region bounded by an arc and the chord connecting its endpoints. Option A describes a sector, not a segment.

Real World Connection

In the Real World

Imagine you're an architect designing a circular window with a straight decorative bar across it. To calculate the amount of glass needed for the smaller section, you'd use the area of a segment. Similarly, city planners might use this to calculate the area of a specific part of a circular garden or roundabout cut by a straight path.

Key Vocabulary

Key Terms

SEGMENT: The region bounded by an arc and a chord | 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 | SECTOR: The region bounded by two radii and an arc

What's Next

What to Learn Next

Great job understanding segments! Next, you can explore 'Area of a Sector' and 'Area of a Quadrilateral'. These concepts build on your knowledge of circles and shapes, which are super useful in higher math and science.