What is a Major Arc?

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Grade Level:

Class 6

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

Definition

What is it?

A Major Arc is the longer part of the circumference of a circle that is cut off by two points on the circle. Imagine cutting a round roti into two pieces; the major arc is the bigger piece of the crust.

Simple Example

Quick Example

Think of a clock face from 12 to 6. If you trace the path from 12, going through 1, 2, 3, 4, 5, and ending at 6, this is a semi-circle. Now, if you trace from 12, going through 11, 10, 9, 8, 7, and ending at 6, this longer path is a Major Arc. It's the 'long way around' between two points.

Worked Example

Step-by-Step

QUESTION: A circle has two points, A and B, on its circumference. The angle formed at the center of the circle by these points (called the central angle) is 120 degrees. What is the measure of the Major Arc AB?

STEP 1: Understand that a full circle measures 360 degrees.
---STEP 2: The central angle for the shorter arc (Minor Arc) is given as 120 degrees.
---STEP 3: The Major Arc is the remaining part of the circle after the Minor Arc is taken out.
---STEP 4: To find the Major Arc, subtract the Minor Arc's central angle from the total degrees in a circle.
---STEP 5: Calculation: Major Arc = 360 degrees - 120 degrees.
---STEP 6: Major Arc = 240 degrees.

ANSWER: The measure of the Major Arc AB is 240 degrees.

Why It Matters

Understanding arcs helps in designing curved structures like bridges or roads, and even in computer graphics for creating smooth shapes. Engineers use this to build stable structures, and game developers use it for animations. It's a foundational concept for careers in design, architecture, and even satellite navigation.

Common Mistakes

MISTAKE: Confusing Major Arc with Minor Arc, often picking the smaller arc when asked for the major one. | CORRECTION: Remember 'Major' means bigger or longer. Always identify the longer path between two points on the circle's circumference.

MISTAKE: Calculating the Major Arc using the wrong central angle. Sometimes students use the reflex angle instead of 360 - central angle. | CORRECTION: The Major Arc's angle is always 360 degrees minus the central angle of the Minor Arc. The central angle for the Major Arc itself is a reflex angle.

MISTAKE: Not realizing that a Major Arc is part of the circumference, not the area inside. | CORRECTION: An arc is a curved line segment. It's the 'crust' of the roti, not the whole roti piece.

Practice Questions

Try It Yourself

QUESTION: If the central angle of a Minor Arc is 90 degrees, what is the measure of the corresponding Major Arc? | ANSWER: 270 degrees

QUESTION: A circular Ferris wheel has seats at two points, P and Q. The shorter distance along the wheel's edge between P and Q (Minor Arc) covers an angle of 150 degrees at the center. What is the angle covered by the longer distance (Major Arc)? | ANSWER: 210 degrees

QUESTION: A pizza is cut into three slices. Two slices together form a Minor Arc with a central angle of 100 degrees. If the third slice completes the pizza, what is the central angle of the Major Arc formed by the first two slices? | ANSWER: 260 degrees

MCQ

Quick Quiz

Which of the following statements is true about a Major Arc?

It is always shorter than a Minor Arc.

It is the longer part of a circle's circumference between two points.

It is the area inside the circle.

Its central angle is always less than 90 degrees.

The Correct Answer Is:

A Major Arc is defined as the longer part of the circumference cut by two points. Option A is incorrect because it's longer. Option C describes a sector, not an arc. Option D is incorrect as its central angle is always greater than 180 degrees.

Real World Connection

In the Real World

When you see the curved path of a metro line or a highway flyover, the engineers designing it consider arcs. For example, a curved ramp connecting two roads might be designed using a Major Arc concept to ensure a smooth, gradual turn for vehicles, like those seen on Mumbai's Eastern Freeway or Delhi's elevated corridors.

Key Vocabulary

Key Terms

ARC: A part of the circumference of a circle | CIRCUMFERENCE: The total distance around a circle | CENTRAL ANGLE: The angle formed at the center of a circle by two radii | MINOR ARC: The shorter part of a circle's circumference between two points

What's Next

What to Learn Next

Great job learning about Major Arcs! Next, you can explore 'Minor Arcs' and 'Sectors of a Circle'. Understanding these will help you calculate lengths of arcs and areas of pizza-like slices, which is super useful in geometry!