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Study Notes 019:
By William E. Steinman:
Geometry 12 Sectors and Segments:
A sector of a circle is a part of a circle bounded by two radii and their intercepted arc. In the drawing below, oab is a sector.
We can derive the equation for the length of the arc as follows.
Since the circumference of a circle is 3600 by definition, the length of the arc above is ab, which is equal to x/360. l = x/360.
We know that the circumference of a circle is equal to 2 πr.
Therefore, the length of the arc l = x/360(2 πr) = x πr/180.
Yippee!
The equation for the area of a sector is similarly derived.
The area is equal to the x/360 times the area of the circle.
So the area of he sector is x/360(πr2) = xπr2/360.
Alimentary my dear Gaston.
A segment of a circle is a part of a circle bounded by a chord and its arc.
A minor segment of a circle is the smaller of the two segments so formed.
In the drawing below, the segment abc is a minor segment of the circle.
Since we can find the area of a sector and the area of a triangle we should be able to define an equation for the area of a minor segment.
The area of a minor segment of a circle is equal to the area of its sector less the area of the triangle formed by the radii and the chord.
The area of the triangle we know is ½ rc
Therefore, the area of the minor segment am is (xπr2/360) - ½ rc.
Ain’t this fun?
From all of the forgoing crap we can state a general rule for areas of complex or compound figures. We simply find the areas of the individual components and add or subtract as necessary.
One example should suffice.
To find the area of this drawing we must find the area of the rectangle and add to it ½ the area of the circle.
We know the area of a rectangle is equal to the length times the height.
As = hl.
The area of a circle is πr2 or Ac = πr2.
Therefore the area of the compound figure Af = hl + ½ πr2.
Let’s talk about loci.
In math we define locus as the set or configuration of all points whose coordinates satisfy a single equation or one or more algebraic conditions.
More generally, a locus of points is a set of points, and only those points, that satisfy the given conditions.
A simple example is the points on a circle of radius n.
The locus of points that are n distance from the point x lie on a circle of radius n. Only the points on the circle satisfy the condition. No other points do.
Some other examples may help.
Make drawings if you need to.
The locus of points equidistant from two given points is the perpendicular bisector of the line segment joining the two points.
The locus of points equidistant from two parallel lines is a line parallel to the two lines and midway between them.
The locus of points equidistant from the sides of a given angle is the dissector of the angle.
And a whole lot of etceteras.
Some stodgy professors may call the above principles, but they are really nothing more than examples of loci.
Enough already!