


Circles
Circles are another popular planegeometry test topic.
Unlike polygons, all circles are the same shape and vary only in
size. Circles have certain basic characteristics, and test questions
will focus on your understanding of these properties.
Basic Definitions of Circles
A circle is the collection of all points
equidistant from a given point, called the center. A circle is named
after its center point. The distance from the center to any point
on the circle is called the radius, (r),
which is the most important measurement in a circle. If you know
the radius of a circle, you can figure out all its other characteristics.
The diameter (d) of a circle is twice
as long as the radius (d = 2r),
and stretches between endpoints on the circle, making sure to pass
through the center. A chord also extends from endpoint
to endpoint on the circle, but it does not necessarily pass through
the center. In the figure below, point C is the center
of the circle, r is the radius, and AB is
a chord.
Tangent Lines
A line that intersects the circle at only one point is
called a tangent line. The radius whose endpoint is
the intersection point of the tangent line and the circle is always
perpendicular to the tangent line.
Every point in space outside the circle can extend exactly
two tangent lines to the circle. The distance from the origin of
the two tangents to the points of tangency are always equal. In
the figure below, XY = XZ.
The Math IC often includes tangent lines in the test.
For example:

If RS is tangent to circle Q,
then QR is perpendicular to RS,
and therefore QRS is a 306090 triangle. Given
that QR = 4, we know that RS =
4, and the area of triangle QRS is ^{1}⁄_{2}(4)(4) = 8.
Central Angles and Inscribed Angles
An angle whose vertex is the center of the circle is called
a central angle.
The degree of the circle (the slice of pie) cut by a central
angle is equal to the measure of the angle. If a central angle is
25º, then it cuts a 25º arc in the circle.
An inscribed angle is an angle
formed by two chords in a circle that originate from a single point.
An inscribed angle will always cut out an arc in the circle
that is twice the size of the degree of the inscribed angle. If
an inscribed angle has a degree of 40º, it will cut an arc of 80º
in the circle.
If an inscribed angle and a central angle cut out the
same arc in a circle, the central angle will be twice as large as
the inscribed angle.
Circumference of a Circle
The circumference of a circle is the length
of the 360º arc that forms the circle. In other words, if you were
to trace around the edge of the circle, it is the distance from
a point on the circle back to itself. The circumference is the perimeter
of the circle. The formula for circumference is:
where r is the radius. The formula can
also be written C = πd, where d is
the diameter. Using the formula, try to find the circumference of
the circle below:
Plugging the radius into the formula, C = 2πr =
2π (3) = 6π.
Arc Length
An arc is part of a circle’s circumference.
An arc contains two endpoints and all the points on the circle between
the endpoints. By picking any two points on a circle, two arcs are created:
a major arc, which is by definition the longer arc, and a minor
arc, which is the shorter one.
Since the degree of an arc is defined by the central or
inscribed angle that intercepts the arc’s endpoints, you need only
know the measure of either of those angles and the measure of the
radius of the circle to calculate the arc length. The arc length
formula is:
where n is the measure of the degree
of the arc, and r is the radius. The formula could
be rewritten as arc length = ^{n}⁄_{360} C, where C is
the circumference of the circle.
A Math IC question might ask:

In order to figure out the length of arc AB,
you need to know the radius of the circle and the measure of , which is the inscribed angle that
intercepts the endpoints of AB. The question tells
you the radius of the circle, but it throws you a little curveball
by not providing you with the measure of . Instead, the question puts in a triangle and tells you the
measures of the other two angles in the triangle. Using this information
you can figure out the measure of . Since the three angles of a triangle
must add up to 180º, you know that:
Since angle c is an inscribed angle,
arc AB must be 120º. Now you can plug these values
into the formula for arc length
Area of a Circle
The area of a circle depends on the radius of the circle.
The formula for area is:
where r is the radius. If you know the
radius, you can always find the area.
Area of a Sector
A sector of a circle is the area enclosed
by a central angle and the circle itself. It’s shaped like a slice
of pizza. The shaded region in the figure below is a sector:
The area of a sector is related to the area of a circle
the same way that the length of an arc is related to circumference.
To find the area of a sector, simply find what fraction of 360º
the sector comprises and multiply this fraction by the area of the
circle.
where n is the measure of the central
angle which forms the boundary of the sector, and r is the
radius.
Try to find the area of the sector in the figure below:
The sector is bounded by a 70º central angle in a circle
whose radius is 6. Using the formula, the area of the sector is:
Polygons and Circles
You could potentially see a question or two on the Math
IC that involve polygons and circles in the same figure. Here’s
an example:

To find the length of major arc BE, you
must know two things: the measure of the central angle that intersects
the circle at its endpoints and the radius of the circle. Because ABCD is a
rectangle, and rectangles only have right angles, figuring out the
measure of the central angle is simple. is 90º, so the measure of the central
angle is 360º – 90º = 270º.
Finding the radius of the circle is a little tougher.
From the diagram, you can see that it is equal to the height of
the rectangle. To find the height of the rectangle, you can use
the fact that the area of the rectangle is 18, and the length is
6. Since A = bh, and you know the values of both a and b,
With a radius of 3, we can use the arc length formula
to find the length of major arc BE.
