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.
Here's the first rule about them: A radius whose endpoint is the
intersection point of the tangent line and the circle is always
perpendicular to the tangent line. See for yourself:
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 with triangles.
Using
the figure below find the area of the triangle.
In this question since
QR and
RS are
perpendicular, and angle RQS is
60°, triangle QRSis a
30-60-90
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 (Math
Sat - Triangles). RS is
the side opposite 60°, so it's the
side:
RS =
4
. The area of
triangle QRS is 1/2(4)(4
) = 8
.
:2 ratio (Math
Sat - Triangles). 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 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 of80º 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.
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.
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:
where n is the
measure of the central angle that forms the boundary of the sector,
and ris the
radius.
Post Comments
Astha said – Tue, 15 Jul 2008 11:28:20 -0000 ( Flag Edit Link )
Yes, you are right! It is better to use degrees. Radians can be really confusing and the calculator sometimes gives me bizarre answers too!