A circle is defined as the set of all points in a plane that are equidistant from a fixed point, called the center. The distance from the center to any point on the circle is the radius. A diameter is a line segment that passes through the center containing two points on the circle. The circumference is the perimeter or distance around the circle, while the area is the region within the circle boundary. Circles have analytic definitions and properties that can be described through equations.

2. A circle is the collection of points equidistant from a fixed point. The fixed point is
called the center. The distance from the center to any point on the circle is the
radius of the circle, and a segment containing the center whose end points are both
on the circle is a diameter of the circle. The radius, r, equals one-half the diameter,
d.

3. The standard equation for a circle is (x - h)2 + (y - k)2 = r2. The center is at (h,
k). The radius is r.
In a way, a circle is a special case of an ellipse. Consider an ellipse whose foci
are both located at its center. Then the center of the ellipse is the center of
the circle, a = b = r, and
e = = 0

4. In Euclidean geometry, 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 is the perimeter of the circle, and the interior of
the circle is called a disk. An arc is any continuous portion of a circle.
A circle is a special ellipse in which the two foci coincide (i.e., are the same
point). Circles are conic sections attained when a right circular cone is
intersected with a plane perpendicular to the axis of the cone.

5. ContentsContents
► Analytic resultsAnalytic results
 Equation of a circleEquation of a circle
 SlopeSlope
 Pi (π)Pi (π)
 CircumferenceCircumference
 Area enclosedArea enclosed

6. Analytic resultsAnalytic results
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
(x – a)2
+ (y – b)2
= r2
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
X2
+ y2
= r2
and its tangent will be
xx1 + yy1 = r2
where x1, y1 are the coordinates of the common point.
When expressed in parametric equations, (x, y) can be written using the
trigonometric functions sine and cosine as
x = a + r cost,
y = b + r sint
where t is a parametric variable, understood as the angle the ray to (x, y) makes with
the x-axis.

7. In homogeneous coordinates each conic section with equation of a circle is
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 polar coordinates the equation of a circle is
r2
– 2rr0 cos(θ - φ ) + r02 = a2

8. 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:
y´ = - x/y
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
y´ = (a – x) / (y – b)
provided that y ≠ b, of course.

9. Pi (π)
Pi or π is the ratio of a circle's Circumference to its Diameter.
π = C/D ≈ 3.141592654
The numeric value of π never changes.
π is always approximately 3.14159.
In modern English, it is pronounced /paɪ/ (as in apple pie).

10. Circumference
The distance around a circle is called its circumference. The distance across a
circle through its center is called its diameter. We use the Greek letter ∏
(pronounced Pi) to represent the ratio of the circumference of a circle to the
diameter. In the last lesson, we learned that the formula for circumference of a
circle is: . For simplicity, we use = 3.14. We know from the last lesson that the
diameter of a circle is twice as long as the radius. This relationship is expressed
in the following formula: .
d = 2 · r

11. Area
The area of a circle is the number of square units inside that circle. If each
square in the circle to the left has an area of 1 cm2, you could count the total
number of squares to get the area of this circle. Thus, if there were a total of
28.26 squares, the area of this circle would be 28.26 cm2 However, it is easier
to use one of the following formulas:
Where A is the area, and r is the radius. Let's look at some examples
involving the area of a circle. In each of the three examples below, we will
use ∏ = 3.14 in our calculations.