Formula of minor segment

General
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formula of minor segment

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Formula of Minor Segment

Answer:

To understand the formula for the area of a minor segment in a circle, we first need to establish some basic concepts related to circle geometry.

Introduction to Circle Segments

A segment in a circle is the region bounded by a chord and the arc that subtends the chord. When discussing minor and major segments, the minor segment is the smaller portion of the circle. Understanding this relationship involves knowing several components:

  1. Circle: A set of points on a plane equidistant from a fixed point (center).
  2. Radius: A line segment from the center of the circle to any point on the circle.
  3. Chord: A line segment with both endpoints on the circle.
  4. Arc: A continuous part of the circumference of a circle.
  5. Central Angle: An angle whose vertex is at the center of the circle and whose sides are radii intersecting the circle.

Formula for the Area of a Minor Segment

To find the area of a minor segment, follow these steps:

  1. Compute the area of the sector formed by the central angle.
  2. Find the area of the triangular region within the sector.
  3. Subtract the triangular area from the sector area to get the segment area.

Step 1: Sector Area Calculation

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

\text{Sector Area} = \frac{1}{2} r^2 \theta

Step 2: Triangle Area Calculation

The triangle in question here is the triangle formed by the center of the circle and the ends of the chord. If the angle subtended by the chord at the center is \theta, the triangle’s area can be calculated using the sine function as follows:

\text{Triangle Area} = \frac{1}{2} r^2 \sin(\theta)

Step 3: Minor Segment Area Calculation

Subtract the area of the triangle from the area of the sector to find the area of the minor segment:

\text{Minor Segment Area} = \text{Sector Area} - \text{Triangle Area} = \left(\frac{1}{2} r^2 \theta\right) - \left(\frac{1}{2} r^2 \sin(\theta)\right) = \frac{1}{2} r^2 (\theta - \sin(\theta))

Key Points to Remember

  • The angle \theta used in these calculations must be in radians.
  • In cases where the angle is given in degrees, convert it to radians by multiplying with \frac{\pi}{180}.
  • The derived formula applies specifically to the minor segment of the circle, given that the angle is less than \pi radians (or 180 degrees).

This comprehensive approach helps in deriving the formula for the area of a minor segment using the principles of geometry.

Final Answer:

The formula for the area of a minor segment in a circle, defined by the angle \theta (in radians), is given by:

\text{Minor Segment Area} = \frac{1}{2} r^2 (\theta - \sin(\theta))

Here, r is the radius of the circle, and \theta is the angle subtended by the segment at the circle’s center.