The circumference and area of a circle are handwritten.

Circumference of the circle:

Formula for calculating the area of a circle:?

Or?

Note: R is the radius of the circle and D is the diameter of the circle.

Definition of a circle: The set of points whose distance from a fixed point on the same plane is equal to a fixed length is called a circle. This fixed point is called the center of the circle. The length of a circle is the circumference of a circle. Two circles that can overlap are called equal circles. A circle is a regular n polygon (n is an infinite positive integer), and its side length is infinitely close to 0 but can never be equal to 0.

? The nature of the circle:

? The (1) circle is an axisymmetric figure, and its symmetry axis is any straight line passing through the center of the circle. A circle is also a central symmetric figure, and its symmetric center is the center of the circle.

Vertical diameter theorem: the diameter perpendicular to the chord bisects the chord and bisects the two arcs opposite the chord.

Inverse theorem of vertical diameter theorem: bisecting the diameter of a chord (not the diameter) is perpendicular to the chord and bisecting two arcs opposite to the chord.

⑵ The properties and theorems of central angle and central angle.

(1) In the same circle or the same circle, if one of two central angles, two peripheral angles, two sets of arcs, two chords and the distance between two chords is equal, their corresponding other groups are equal respectively.

(2) In the same circle or equal circle, the circumferential angle of an equal arc is equal to half of the central angle it faces (the circumferential angle and the central angle are on the same side of the chord).

The circumferential angle of the diameter is a right angle. The chord subtended by a 90-degree circle angle is the diameter.

The formula for calculating the central angle is θ = (l/2π r) × 360 =180l/π r = l/r (radian).

That is, the degree of the central angle is equal to the degree of the arc it faces; The angle of a circle is equal to half the angle of the arc it faces.

(3) If the length of an arc is twice that of another arc, then the angle of circumference and center it subtends is also twice that of the other arc.

⑶ Properties and theorems about circumscribed circle and inscribed circle.

① A triangle has a unique circumscribed circle and inscribed circle. The center of the circumscribed circle is the intersection of the perpendicular lines of each side of the triangle, and the distances to the three vertices of the triangle are equal;

(2) The center of the inscribed circle is the intersection of the bisectors of the inner angles of the triangle, and the distances to the three sides of the triangle are equal.

③R=2S△÷L(R: radius of inscribed circle, s: area of triangle, l: perimeter of triangle).

(4) The intersection of the connecting lines of two tangent circles. (line: a straight line with two centers connected)

⑤ The midpoint M of the chord PQ on the circle O, if the intersection point M is two chords AB and CD, and the chords AC and BD intersect PQ on X and Y respectively, then M is the midpoint of XY.

? (4) If two circles intersect, the line segment (or straight line) connecting the centers of the two circles vertically bisects the common chord.

? (5) The degree of the chord tangent angle is equal to half the degree of the arc it encloses.

? (6) The degree of the angle inside a circle is equal to half of the sum of the degrees of the arcs subtended by the angle.

? (7) The degree of the outer angle of a circle is equal to half of the difference between the degrees of two arcs cut by this angle.

? (8) The perimeters are equal, and the area of a circle is larger than that of a square, rectangle or triangle.