


Circles
A circle is the collection of 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),
the most important measurement in a circle. If you know a circle’s
radius, 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, passing 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
Tangents are lines that intersect a circle at only one
point. Tangents are a new addition to the SAT. You can bet that
the new SAT will make sure to cram at least one tangent question
into every test.
Just like everything else in geometry, tangent lines are
defined by certain fixed rules. Know these rules and you’ll be able
to handle anything the SAT throws at you. Here’s the first: A radius
whose endpoint is the intersection point of the tangent line and
the circle is always perpendicular to the tangent line. See?
And the second rule: 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.
Tangents and Triangles
Tangent lines are most likely to appear in conjunction
with triangles.

You can answer this question only if you know
the rules of circles and tangent lines. The question doesn’t tell
you that QR is the radius of the circle;
you just have to know it: Because the circle is named circle Q,
point Q must be the center of the
circle, and any line drawn from the center to the edge of the circle
is the radius. The question also doesn’t tell you that QR is
perpendicular to RS. You have to know
that they’re perpendicular because QR is
a radius and RS is a tangent that
meet at the same point.
If you know how to deduce those key facts about this circle,
then the actual math in the question is simple. Since QR and RS are
perpendicular, and angle RQS is 60°, triangle QRS is
a 306090 triangle. The image tells you that side QR,
the side opposite the 30° angle equals 4.
Side QR is the height of the triangle.
To calculate the area, you just have to figure out which of the
other two sides is the base. Since the height and base of the triangle
must be perpendicular to each other, side RS must
be the base. To find RS, use the 1::2 ratio. RS is
the side opposite 60°, so it’s the side: RS =
4. 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 originating
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 40, 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 is the perimeter of the circle. The
formula for circumference of a circle is
where r is the radius. The
formula can also be written C = πd,
where d is the diameter. 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 a 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, the shorter one.
Since the degree of an arc is defined by the central or
inscribed angle that intercepts the arc’s endpoints, you can calculate
the arc length as long as you know the circle’s radius and the measure
of either the central or inscribed angle.
The arc length formula is
where n is the measure of
the degree of the arc, and r is the
radius.
Here’s the sort of question the SAT 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 , 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. Like we said,
only a little curveball: You can easily figure out the measure of because, as you (better) know, the
three angles of a triangle add up to 180º.
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
If you know the radius of a circle, you can figure out
its area. The formula for area is:
where r is the radius. So
when you need to find the area of a circle, your real goal is to
figure out the radius.
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:
There are no analogies on the SAT anymore, but here’s
one anyway: The area of a sector is related to the area of a circle
just as the length of an arc is related to the circumference. To
find the area of a sector, find what fraction of the sector makes up and multiply
this fraction by the area of the circle.
where n is the measure of
the central angle that 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 70° central angle in a circle
whose radius is 6. Using the formula, the area of the
sector is
Polygons and Circles
We’ve talked already about triangles in circle problems.
But all kinds of polygons have also been known to make cameos on
SAT circle questions. Here’s an example:

To find the length of minor arc BE,
you have to know two things: the radius of the circle and the measure
of the central angle that intersects the circle at points B and E. Because ABCD is
a rectangle, and rectangles only have right angles, figuring out
the measure of the central angle is simple. is 90°, so the measure
of the central angle is 90°.
Finding the radius of the circle is a little tougher.
From the diagram, you can see that the radius 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,
Now that you’ve got the radius and measure of the angle,
plug them into the arc length formula to find the length of minor
arc BE.
