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


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.