# Area of a Segment

A **segment **is a portion of a circle which is cut off by a straight line not passing through the center. The segment smaller than the semi-circle is called the **minor segment** and the segment larger than the semi-circle is called the **major segment**.

**(a) The area of the minor segment when angle **** and radius **** are given:**

Area of segment area of sector area of

Now the area of the major segment area of circle area of the minor segment

__Example__:

A chord of a circle of radius cm makes an angle of at the center of the circle. Find the area of the major and minor segment.

__Solution__:

Given that , radius, cm

area of the sector

square cm

area of

square cm

Area of the minor segment area of sector area of

square cm

Area of the circle

square cm

Area of the major segment area of the circle area of the minor segment

square cm.

**(b) The area of a segment when the height and length of the chord of the segment are given:**

Let be the radius of the circle

be the height of the segment

be the length of the chord

We note that is a right triangle; the hypotenuse is and the other two sides are and

by Pythagorean Theorem

Solving for , and , we obtain the following formulas:

--- (1)

--- (2)

--- (3)

**Note:**

gives the height of the major segment

gives the height of the minor segment

Many formulas are given for finding the approximate area of a segment. Two of the common methods are:

**Method-I: **

**Note: **If the height of the segment is less than the radius of the circle, then .

**Method-II: **

__Example__:

If the chord of the segment of a circle is cm and the height of the segment is cm, find the radius of the circle.

__Solution__:

Given that, cm, cm

cm

__Example__:

Find the area of the segment whose chord is cm and whose height is cm.

__Solution__:

Given that cm, cm

square cm.