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Segment of a Circle

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A segment of a circle is the area defined by a line from one side of the circumference to the other.

Segments are divided into major and minor segments:

Major segments are the larger proportion of the circle.

Minor segments are the smaller proportion of the circle.

When working with the area of a segment of a circle, you should always remember the formula for the area of a circle: . This is the formula you use regardless of whether the angle is in radians or degrees.

When working out the area or circumference of a segment of a circle, the angle at the centre of the circle which defines the segment can be in either radians or degrees.

**Degrees**are**Radians**are another type of unit for angles. They are defined by the ratio of the radius of the circle to the arc length of the circle and denoted by . In radians, a full rotation is equal to .

To find the area of a segment of a circle (the blue part), you need to know the angle at the centre where the radii brackets the chord (x) and the radius:

To find the area of a minor segment of a circle when the angle at the centre (x) is in radians, the formula is:

To find the area of a major segment of a circle when the angle at the centre is in radians, the formula is:

Instead of trying to remember both formulas, it might be easier to remember the area of the major segment formula as a word equation:

Circle A has a minor segment which is highlighted in pink.

- Find the area of the minor segment.
- Find the area of the major segment.

- Start by defining the characteristics of the segment:
- Substitute into the formula:

Minor segment = 7.64 square units (3 sf)

b. Finding the area of the major segment

- Remember to find the major segment; you subtract the minor segment from the area of the circle.

To check, if you add both the minor and major segments together, you should get approximately the same as the area of the whole circle . Here, and minor segment + major segment = .

You still need to know the radius and the centre of the circle, but there is now a different formula.

The formula to find the minor segment of a circle, when the angle at the centre (x) is in degrees:

To find the major segment of a circle when the angle at the centre (x) is in degrees, the formula is:

Use the same principle as when the angle is in radians – you need to minus the minor segment from the whole area of the circle.

Circle B has a minor segment, and the angle at the centre defines the length of the segment. The angle is and the radius is 10 cm.

- What is the area of the minor segment of Circle B?
- What is the area of the major segment of Circle B?

a. Finding the minor segment of Circle B.

Identify all the key information required to calculate the area. Radius = 10 cm; angle at the center =

Substitute into the formula

Minor segment = 75.7 square units (3 sf)

b. Finding the major segment of Circle B.

- Substitute the key information into the major segment formula

Major segment = 239 square units (3 sf)

The method to calculate the arc length of a segment is the same for calculating the arc length of a sector.

- To find the arc length when the angle at the centre (x) that defines the segment is in radians:

A segment in Circle C has a radius of 7 cm with an angle of . What is the arc length of this segment?

- To find the arc length when the angle at the centre (x) that defines the segment is in degrees:

A segment in Circle D has a radius of 5 cm with an angle of . What is the arc length of this segment?

- A segment of a circle is the area bounded by the circumference and the chord. Segments can either be the major (the bigger proportion) or minor (the smaller proportion).
- To find the area of a minor segment of a circle, you either use where the angle (x) is in radians or where the angle (x) is in degrees.
- To find the area of a major segment, you subtract the area of the minor segment away from the area of the circle.
- Calculating the arc length of a segment is the same as calculating the arc length of a sector. To calculate the arc length of a segment where the angle (x) is in radians, you can do . If the angle (x) is in degrees, then you use .

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