a
circle is the
set of all
points in a plane at a fixed
distance, called the
radius, from a given point, the
centre.
Circles are simple closed
curves which divide the plane into an interior and exterior. The
circumference of a circle means the length of the circle, and the interior of the circle is called a
disk. An
arc is any
continuous portion of a circle.
Circle of radius r=1, centre (a, b)=(1.2, -0.5).
Equation of a circle
In an
x-
y coordinate system, the circle with centre (
a,
b) and radius
r is the set of all points (
x,
y) such that
The equation of the circle follows from the
Pythagorean theorem applied to any point on the circle.
If the circle is centred at the origin (0, 0), then this
formula can be simplified to
where x1, y1 are the coordinates of the common point.
-
where
t is a
parametric variable, understood as the angle the ray to (
x,
y) makes with the
x-axis.
- ax2 + ay2 + 2b1xz + 2b2yz + cz2 = 0.
It can be proven that a
conic section is a circle if and only if the point I(1,i,0) and J(1,-i,0) lie on the conic section. These points are called the
circular points at infinity.
In the
complex plane, a circle with a centre at
c and radius
r has the equation
| z ? c | 2 = r2. Since
, the slightly generalized equation
for real
p,
q and complex
g is sometimes called a
generalized circle. It is important to note that not all generalized circles are actually circles.
Slope
The
slope of a circle at a point (
x,
y) can be expressed with the following formula, assuming the centre is at the origin and (
x,
y) is on the circle:
More generally, the
slope at a point (
x,
y) on the circle
(x ? a)2 + (y ? b)2 = r2, i.e., the circle centred at (
a,
b) with radius
r units, is given by
provided that , of course.
Pi (
)
The numeric value of ? never changes.
In modern English, it is pronounced /pa?/ (as in apple pie).
Circumference
- Length of a circle's circumference is
- Alternate formula for circumference:
Given that the ratio circumference c to the Area A is
The
r and the
can be canceled, leaving
Therefore solving for c:
So the circumference is equal to 2 times the area, divided by the radius. This can be used to calculate the circumference when a value for
cannot be computed.
Diameter
- The diameter of a circle is a straight line through the center of the circle touching the circle at both sides.
The diameter of a circle is double its radius.
Area enclosed
Area of the circle =
× area of the shaded square
- The area enclosed by a circle is the radius squared, multiplied by ?.
Using a square with side lengths equal to the diameter of the circle, then dividing the square into four squares with side lengths equal to the radius of the circle, take the area of the smaller square and multiply by
.
Properties
Chord properties
- Chords equidistant from the centre of a circle are equal (length).
- Equal (length) chords are equidistant from the centre.
- The perpendicular bisector of a chord passes through the centre of a circle; equivalent statements stemming from the uniqueness of the perpendicular bisector:
- A perpendicular line from the centre of a circle bisects the chord.
- The line segment (Circular segment) through the centre bisecting a chord is perpendicular to the chord.
- If a central angle and an inscribed angle of a circle are subtended by the same chord and on the same side of the chord, then the central angle is twice the inscribed angle.
- If two angles are inscribed on the same chord and on the same side of the chord, then they are equal.
- If two angles are inscribed on the same chord and on opposite sides of the chord, then they are supplemental.
- For a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle.
- An inscribed angle subtended by a diameter is a right angle.
- The diameter is longest chord of the circle.
Tangent properties
- The line drawn perpendicular to the end point of a radius is a tangent to the circle.
- A line drawn perpendicular to a tangent at the point of contact with a circle passes through the centre of the circle.
- Tangents drawn from a point outside the circle are equal in length.
- Two tangents can always be drawn from a point outside of the circle.
Theorems
- The chord theorem states that if two chords, CD and EF, intersect at G, then . (Chord theorem)
- If a tangent from an external point D meets the circle at C and a secant from the external point D meets the circle at G and E respectively, then . (tangent-secant theorem)
- If two secants, DG and DE, also cut the circle at H and F respectively, then . (Corollary of the tangent-secant theorem)
- The angle between a tangent and chord is equal to the subtended angle on the opposite side of the chord. (Tangent chord property)
- If the angle subtended by the chord at the centre is 90 degrees then l = ?(2) × r, where l is the length of the chord and r is the radius of the circle.
- If two secants are inscribed in the circle as shown at right, then the measurement of angle A is equal to one half the difference of the measurements of the enclosed arcs (DE and BC). This is the secant-secant theorem.
Inscribed angles
An
inscribed angle ? is exactly half of the corresponding
central angle ? (see Figure). Hence, all inscribed angles that subtend the same arc have the same value (cf. the blue and green angles
? in the Figure). Angles inscribed on the arc are supplementary. In particular, every inscribed angle that subtends a
diameter is a
right angle.
An alternative definition of a circle
Apollonius' definition of a circle
Apollonius of Perga showed that a circle may also be defined as the set of points having a constant
ratio of distances to two foci, A and B.
The proof is as follows. A line segment PC bisects the interior angle APB, since the segments are similar:
Analogously, a line segment PD bisects the corresponding exterior angle. Since the interior and exterior angles sum to
, the angle CPD is exactly
, i.e., a
right angle. The set of points P that form a right angle with a given line segment CD form a circle, of which CD is the diameter.
As a point of clarification, note that C and D are determined by A, B, and the desired ratio; i.e. A and B are not arbitrary points lying on an extension of the diameter of an existing circle.
Calculating the parameters of a circle
Given three non-collinear points lying on the circle
Radius
The radius of the circle is given by
Center
The center of the circle is given by
where
Plane unit normal
A unit normal of the plane containing the circle is given by
Parametric Equation
Given the radius, r , center, Pc, a point on the circle, P0 and a unit normal of the plane containing the circle, , the parametric equation of the circle starting from the point P0 and proceeding counterclockwise is given by the following equation:
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