To calculate the area of a major segment of a circle, we first need to understand the structure of a segment. A segment is the region of a circle bounded by a chord and the arc that subtends the chord.

The major segment refers to the larger part of the circle, while the minor segment is the smaller region.

To find the area of the major segment, follow these steps:

1. Formula for the Area of a Segment:

The area of the segment (minor segment) is given by:

$A_{\text{segment}} = A_{\text{sector}} - A_{\text{triangle}}$

where:

• $A_{\text{sector}}$ is the area of the sector.
• $A_{\text{triangle}}$ is the area of the triangle formed by the chord and the two radii.

2. Find the Area of the Sector:

The area of the sector with central angle $\theta$ (in radians) and radius $r$ is given by:

$A_{\text{sector}} = \frac{1}{2} r^2 \theta$

3. Find the Area of the Triangle:

The area of the triangle formed by the chord and the radii is:

$A_{\text{triangle}} = \frac{1}{2} r^2 \sin(\theta)$

4. Subtract to Find the Minor Segment Area:

Now, subtract the area of the triangle from the sector to find the area of the minor segment:

$A_{\text{minor}} = \frac{1}{2} r^2 \left( \theta - \sin(\theta) \right)$

5. Find the Area of the Major Segment:

The area of the major segment is the total area of the circle minus the area of the minor segment:

$A_{\text{major}} = \pi r^2 - A_{\text{minor}}$

Key Points:

• $\theta$ is the angle subtended by the chord at the center of the circle.
• Ensure that $\theta$ is in radians for these formulas to work correctly.

Let me know if you need further clarification, or I can solve an example using specific values for $r$ and $\theta$.

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Related Questions:

1. How do you convert degrees to radians?
2. What is the relationship between chord length and the angle $\theta$?
3. How do you calculate the length of the arc in a sector?
4. How do you calculate the area of a sector if $\theta$ is in degrees?
5. How does the area of a major segment change as the angle $\theta$ approaches $\pi$?

Tip:

To avoid errors, always ensure that the angle $\theta$ is in radians when using the formulas for sectors and segments.