What is the Area of a Segment? | Simple Explanation for Class 10

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What is the Area of a Segment (Trigonometric Context)?

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The area of a segment is the area of the region enclosed by a chord and the arc it cuts off from a circle. In a trigonometric context, we use angles and trigonometric ratios (like sine) to calculate this area.

Simple Example

Quick Example

Imagine a round pizza cut into slices. If you take one slice (a sector) and then cut a straight line across the crust to remove the triangular part, the remaining curved piece of pizza is a segment. We want to find how much pizza is in that curved piece.

Worked Example

Step-by-Step

Let's find the area of a segment of a circle with a radius of 10 cm and a central angle of 60 degrees. (Use pi = 3.14 and sin 60 degrees = 0.866)

Step 1: Calculate the area of the sector. The formula is (theta / 360) * pi * r^2.
Area of sector = (60 / 360) * 3.14 * (10)^2
Area of sector = (1/6) * 3.14 * 100
Area of sector = 314 / 6 = 52.33 cm^2

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Step 2: Calculate the area of the triangle formed by the two radii and the chord. The formula is (1/2) * r^2 * sin(theta).
Area of triangle = (1/2) * (10)^2 * sin(60)
Area of triangle = (1/2) * 100 * 0.866
Area of triangle = 50 * 0.866 = 43.3 cm^2

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Step 3: 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
Area of segment = 52.33 - 43.3
Area of segment = 9.03 cm^2

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Answer: The area of the segment is 9.03 cm^2.

Why It Matters

Understanding segment areas is crucial in fields like engineering for designing curved parts of machines or buildings. In AI/ML, it helps in image processing for recognizing shapes. Even in space technology, calculating areas of curved surfaces on satellites uses similar principles.

Common Mistakes

MISTAKE: Using the angle in degrees directly in the sine function without converting it to radians if the calculator is set to radians. | CORRECTION: Always check your calculator's mode (degrees or radians) or convert the angle (degrees * pi / 180) before using the sine function.

MISTAKE: Forgetting to subtract the triangle's area from the sector's area, or sometimes subtracting the sector from the triangle. | CORRECTION: Remember the segment is the 'leftover' part of the sector after removing the triangle. So, it's always Area of Sector - Area of Triangle.

MISTAKE: Using the wrong formula for the area of the triangle, especially when the angle is not 90 degrees. | CORRECTION: For a triangle formed by two radii and a chord, the area is (1/2) * r^2 * sin(theta), where theta is the central angle.

Practice Questions

Try It Yourself

QUESTION: A circular park has a radius of 7 meters. A straight path cuts across the park, forming a segment. If the central angle subtended by the chord is 90 degrees, find the area of the segment. (Use pi = 22/7) | ANSWER: Area of sector = (90/360) * (22/7) * 7^2 = (1/4) * 22 * 7 = 38.5 sq meters. Area of triangle = (1/2) * 7^2 * sin(90) = (1/2) * 49 * 1 = 24.5 sq meters. Area of segment = 38.5 - 24.5 = 14 sq meters.

QUESTION: Calculate the area of a segment of a circle with a radius of 12 cm, where the central angle is 120 degrees. (Use pi = 3.14 and sin 120 degrees = 0.866) | ANSWER: Area of sector = (120/360) * 3.14 * 12^2 = (1/3) * 3.14 * 144 = 150.72 sq cm. Area of triangle = (1/2) * 12^2 * sin(120) = (1/2) * 144 * 0.866 = 72 * 0.866 = 62.352 sq cm. Area of segment = 150.72 - 62.352 = 88.368 sq cm.

QUESTION: A circular plate of radius 14 cm has a portion removed by a straight cut. If the length of the chord formed by the cut is 14 cm, find the area of the smaller segment. (Hint: Find the central angle first using the triangle formed by radii and chord. Use pi = 22/7 and sqrt(3) = 1.732) | ANSWER: Since the chord length is equal to the radius (14 cm), the triangle formed by the two radii and the chord is an equilateral triangle. Therefore, the central angle (theta) is 60 degrees. Area of sector = (60/360) * (22/7) * 14^2 = (1/6) * 22 * 2 * 14 = 102.67 sq cm. Area of triangle = (sqrt(3)/4) * side^2 = (1.732/4) * 14^2 = 0.433 * 196 = 84.868 sq cm. Area of segment = 102.67 - 84.868 = 17.802 sq cm.

MCQ

Quick Quiz

What is the formula for the area of a segment of a circle with radius 'r' and central angle 'theta' (in degrees)?

(theta / 360) * pi * r^2 + (1/2) * r^2 * sin(theta)

(theta / 360) * pi * r^2 - (1/2) * r^2 * sin(theta)

pi * r^2 - (1/2) * r^2 * sin(theta)

(1/2) * r^2 * sin(theta)

The Correct Answer Is:

The area of a segment is found by subtracting the area of the triangle formed by the radii and the chord from the area of the corresponding sector. Option B correctly represents this: Area of Sector - Area of Triangle.

Real World Connection

In the Real World

Imagine an ISRO satellite camera taking pictures of Earth. To calculate the exact land area covered by a curved cloud formation or a lake with a curved boundary, engineers use segment area calculations. This helps in weather forecasting and mapping resources accurately.

Key Vocabulary

Key Terms

SEGMENT: The region of a circle enclosed by a chord and the arc it cuts off. | CHORD: A straight line segment whose endpoints both lie on the circle. | SECTOR: The region of a circle enclosed by two radii and an arc. | CENTRAL ANGLE: The angle formed at the center of the circle by two radii.

What's Next

What to Learn Next

Great job understanding segments! Next, you can explore the concept of 'Area of a Major Segment and Minor Segment'. This will help you understand how the segment area changes based on the angle and whether it's the smaller or larger part of the circle.