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Angles inside a Circle Alternate Interior Angles, Corresponding Angles Literal Equations Probability Introduction Graphing Slope, Undefined Slope, Flat Slope Sets and Venn Diagrams Line Graphs, Bar GraphsBefore looking at how to establish the area of a segment in a circle.

It's handy to have a run through of what exactly a segment means regarding a circle.

A segment in a circle is similar to a __ sector__,
but is slightly different.

Similar to the case of a sector inside a circle, a segment can be either a Minor Segment, or a Major Segment.

If the segment is larger than half the circle, it is a major segment, and if it is smaller, then it
is a minor segment.

In both cases, the segments are formed between a straight Chord line across the circle at some part,
and an Arc on the edge of the circle.

in a Circle

A minor segment inside a circle is actually a smaller part of a whole sector in the circle.

With a specific Chord line creating both the minor segment, and also creating a separate triangle
too.

The points of such triangle we have labelled

The two sides of the triangle that are NOT the Chord line, are the length of the radius of the circle, labelled

Now to work out the area of the minor segment, we would want to establish:

As what is left over, will be the area of the minor segment.

Illustrated in the image below.

Now working with degree measure, the Area of the Sector is given by: \boldsymbol{\frac{\theta \pi r^2}{360}}

Also the Area of the Triangle, being isosceles, is given by: \boldsymbol{\frac{r^2}{2}}

Giving us the formula for area of the segment: \boldsymbol{\frac{\theta \pi r^2}{360}}

__Though we can simplify this a bit.__

\boldsymbol{\frac{\theta \pi r^2}{360}} can be rewritten as \boldsymbol{\frac{r^2}{2}} \boldsymbol{\frac{\theta \pi}{180}}

Turning the formula into: \boldsymbol{\frac{r^2}{2}} \boldsymbol{\frac{\theta \pi}{180}} − \boldsymbol{\frac{r^2}{2}}

Which can be written as \boldsymbol{\frac{r^2}{2}}( \boldsymbol{\frac{\theta \pi}{180}}

If

It is \boldsymbol{\frac{r^2}{2}}(

These formulas shown above for the area of a minor segment, also help when trying to find the area of a major segment, as the sum would be:

Because what is left over from this sum, will be the area of the major segment.

Illustrated in the image below.

Some clear examples are now shown below.

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Here the angle of the relevant sector is given in radian measure, instead of degrees, the angle is
sized **1.83** radians.

So make sure that any calculator being used to establish the value of **sin***θ* is
set to "radians" and NOT "degrees".

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This is a major segment, so we work out the area of the non shaded minor segment first, and then take that away from the area of the whole circle.

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