


Circles
After so many cruel, rigid line drawings, it’s nice to
contemplate the
circle, nature’s roundest
creation:
This is a circle with center C.
All points along the edge are equidistant from C.
This means that line segments .
The equals sign with a squiggle over it is the symbol for congruence,
the geometrical way of saying that the three line segments are all
equal. Any line going from the center of the circle to the edge
is called a radius. , ,
and are all radii. If a line goes from
the edge, through the center, and continues on to the other edge,
it is called a diameter, and its length is equal to
two radii. The diameter bisects a circle. is
a diameter.
The circumference of a circle is the distance
around the outer edge, so circumference is like saying “perimeter
of a circle.” The circumference and area formulas for a circle are:
Both of these formulas contain the mathematical constant pi,
which has a value of around 3.14 and looks like this: ,
and the variable r for radius (in the circumference
formula you can also use the variable d for diameter,
which is equal to 2r). This means that if you’re
given the circumference of a circle, you can find its area by first
solving for r using the circumference formula and
then plugging that value of r into the area formula.
You can use the area of a circle to find its circumference by doing the
same thing: solving for r and then placing that
value into the proper formula. Testmakers think this is very clever,
so you should expect to do it on the SAT.
Tangent Lines
Properties of tangent lines are one of the
concepts covered on the new SAT. Look at line GJ.
It touches the circle at only one point, F. At
the point where a tangent touches a circle, the tangent line is
perpendicular to the circle.
Whenever a tangent hits a circle, it creates a right angle
with a radius of the circle. Earn some brownie points and think
of what the SAT could do with this fact. Draw a dotted line from G to C,
and draw another from J to C.
What you get is two right triangles.
Arcs, Sectors, and Some Other Concepts That Sound Harder
Than They Are
It’s sliceanddice time!
Angle DEF has its vertex at the center
of circle E, so it shouldn’t blow your mind that
this is called a central angle. Since is
a right angle, the arc (portion of the circle) DE sliced
out by this angle is equal to 90˚. An arc defined
by a central angle is always equal to the measure of that angle.
Contrast this to an inscribed angle,
which is an angle formed by two chords (lines from
one point on the circle to another that do not pass through the
center) with a vertex resting on the edge of the circle. Angle DGF is
an inscribed angle formed by chords GD and GF.
An inscribed angle is half the size of the arc it creates. In our
diagram above, we already know that arc DF is 90˚.
Therefore, angle DGF must be half of 90˚, so .
This little fact makes inscribed angles quite sneaky, so if you
do encounter one, it will probably be on a medium or hard item.
Arc lengths and sectors are
pretty straightforward concepts:
Arc length = portion of the circumference.
Sector = portion of the area.
You don’t have to be Stephen Hawking to realize that before
you solve for arc length, you have to solve for the circumference.
Circle E has a radius of 5, so the circumference
is:
Now, arc DF takes up 90˚, and 90˚ is onefourth of
the total degrees in a 360˚ circle, so (drumroll) . . . arc length DF will
be onefourth of the total circumference.
Sector is the area equivalent of arc length. Look at the
shaded portion of the circle, which takes up onefourth of the total
circle since the central angle is 90˚. To find the area of the shaded
sector, find the area and multiply it by whatever fraction of the
total circle the sector comprises.
Arc lengths and sectors are not basic geometry concepts,
and their weirdness is what makes them tough for many students.
But their bark is worse than their bite: once you learn what they
mean, finding the correct value simply requires multiplying the
total area or circumference by a fraction of the total circle.
