Definition of Sectors
Exactly.
A pizza slice, geometrically, consists of two radii of a circle (the whole pizza) plus the area enclosed by those two radii and the circumference.
But isn't a sector just the same thing as a triangle?
No, a sector and a triangle are actually different shapes.
Exactly. A sector may look a bit like a triangle, but it's really not one, because a triangle has three straight sides, whereas a sector has only two. The third side of a sector is curved, because it is part of the circumference of a circle.
A second type of shape you can carve out of a circle is called a **quadrant**. It's defined by a central right angle.
How do you think a quadrant differs from a sector?
No, it doesn't depend. The answer, yes or no, is always the same.
No, a sector can be formed by any central angle.
Not exactly. "This shape is a sector" conveys different information than "This shape is a quadrant."
Exactly.
The question you were just asked is a bit misleading. A quadrant actually is a sector. It's a special kind of sector, defined by a central angle that measures 90°. A sector can be defined by any central angle.
How many different quadrants do you think could be carved out of a given circle?
No, you can certainly get at least four. Think of cutting a pizza four ways by making two cuts along perpendicular diameters. Those are quadrants.
Surprisingly, no!
You probably know what a __semicircle__ is. It's half of a circle (_semi-_ means "half") defined by a diameter. A semicircle is twice as large as a quadrant. Since a diameter consists of two radii, a semicircle is another special kind of sector.
Finally, a __segment__ is the region between any chord of a circle and the circumference. The segment and the semicircle are shown in the image below.
Now for a hard question. Quadrants and semicircles are special cases of the sector. Are segments a special kind of sector, too?
No. A sector is defined by two radii, which radiate from a circle's center. A segment is defined by a chord, which may not run through the center. If it does not, the segment is not a sector.
Close but not quite. Beware of overly general statements!
To sum up:
[[summary]]
You should be aware of a number of interesting shapes that can be carved out of circles, and which might appear on the GRE. The first of these is called a __sector__, and you encounter instances of it every time you cut up a pizza or a pie.
Which of the following is a good geometric description of the shape of a pizza slice?
Not quite. That's not specific enough to be a good description.
No, the edges of a pizza slice are not chords of a circle. They do not connect two points on the circumference.
Exactly! In fact, from any given circle, an infinite number of different quadrants could be extracted.
To see why, think how many radii you can draw for a given circle. You can probably see that it's an infinite number, since from the center, you can go in any direction. Well, for any radius you draw, you can create a different quadrant. Just draw another radius perpendicular to the first. The area defined by those two radii is a quadrant.
Exactly right!
Most segments are not sectors. A sector is defined by two radii. Radii emanate from a circle's center. A segment is defined by a chord. A chord may not run through the center. A segment defined by a chord that does not run through the circle's center is not a sector.
However, a chord may run through the center. If it does, it is a diameter. A diameter consists of two radii and therefore defines a sector. More specifically, it defines a semicircle. A semicircle is both a sector and a segment. Conversely, anything that is both a sector and a segment is necessarily a semicircle.
Yes, but the terms are used in different contexts.
No, because it has a curved side.
It depends.
Sectors must be formed by a central angle less than 90°.
A quadrant is a special kind of sector.
There is no difference.
Less than 4
4
More than 4
Yes
No
Sometimes
Part of a circle
Two radii of a circle, plus the area enclosed by them and the circumference
Two chords of a circle that meet at one vertex, plus the area enclosed by them and the circumference
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