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Like the circumference of a circle,
its area
is dependent only on the radius. The area of a
circle is equal to the product of pi and the radius squared (Πr2).
The formula for the area of a circle helps us calculate the area of circle
sectors and segments as
well. A circle sector's area in relation to the area of the whole circle is
much like that between an arc and the circumference.
A sector bound by a central angle of n
degrees is equal to (n/360) times the area of
the circle.
Figure %: A sector's area is related to the circle's area
The area of a circle segment is slightly more difficult to calculate. If given
the measure of the central angle or the measure of the arc of the segment, along
with the length of the chord that determines the segment, then it is possible to
calculate the difference in area between the sector that contains the segment
and the triangle formed by the central angle and
the chord. The area of a segment equals the area of
the sector containing it minus the area of the triangle within the sector. The
illustration makes this more clear.
Figure %: A sector's area minus a triangle's area equals a segment's area
Conclusion
With these new tools to calculate perimeter and area, we have another way to
make comparisons between figures. Very soon we'll be able to look at a figure
and use our knowledge of geometric relationships to understand a great deal
about that figure from very little given information. Perimeter and area
provide a great help in this endeavor; with an understanding of perimeter and
area, the good mathematician can look at certain existing conditions and deduce
that two figures with the same area must be congruent. This is one of the most
powerful ways to use geometry.
