1. Definition of a circle: A circle is a plane figure surrounded by a curve.

2. Center of the circle: Fold a circular piece of paper twice, and the point where the crease intersects the center of the circle is called the center of the circle.

Usually represented by the letter o. Its distance to any point on the circle is equal.

3. Radius: The line segment connecting the center of the circle and any point on the circle is called radius. Generally represented by the letter R.

Separate the two feet of the compass, and the distance between the two feet is the radius of the circle.

4. Diameter: The line segment whose two ends pass through the center of the circle is called diameter. Usually represented by the letter d.

The diameter is the longest line segment in a circle.

5. The center of the circle determines the position of the circle, and the radius determines the size of the circle.

6. In the same circle or equal circle, there are countless radii and countless diameters. All radii are equal and all diameters are equal.

7. In the same or equal circle, the length of the diameter is twice that of the radius, and the length of the radius is the diameter.

Expressed in letters: d=2r or r=

8, axisymmetric graphics:

If a graph is folded in half along a straight line, the graphs on both sides can completely overlap, and this graph is axisymmetric.

The straight line where the crease lies is called the symmetry axis. (Any straight line passing through the center of the circle or a straight line with diameter)

9. Rectangles, squares and circles are symmetrical figures, and they all have axes of symmetry. These figures are all axisymmetric figures.

10, only 1 has an axis of symmetry: angle, isosceles triangle, isosceles trapezoid, sector and semicircle.

A figure with only two axes of symmetry is a rectangle.

A figure with only three axes of symmetry is an equilateral triangle.

Figures with only four axes of symmetry are: squares;

Figures with countless axes of symmetry are: circles and rings.

Second, the circumference of the circle.

1, circumference of a circle: The length of the curve around a circle is called the circumference of a circle. It is represented by the letter c.

2, pi experiment:

Make a mark on the circular paper, aim at the scale of ruler 0, and roll it on the ruler once to find out the circumference of the circle.

It is found that the general rule is that the ratio of the circumference to the diameter of a circle is a fixed number (π).

3. Pi: The ratio of the circumference to the diameter of any circle is a fixed number, which we call Pi.

Represented by the letter π(pai).

(1), the circumference of a circle is always greater than 3 times its diameter, and this ratio is a fixed number.

Pi π is an infinite acyclic decimal. π≈3. 14 is generally taken in the calculation.

(2) When judging, the ratio of circumference to diameter of a circle is π times, not 3. 14 times.

(3) The first person in the world to calculate pi was Chinese mathematician Zu Chongzhi.

4. The circumference formula of a circle: C=πdd=C÷π.

Or C=2πrr=C÷2π.

Draw the largest circle in a square, and the diameter of the circle is equal to the side length of the square.

Draw the largest circle in the rectangle, and the diameter of the circle is equal to the width of the rectangle.

6. Distinguish the circumference of a semicircle from that of a semicircle:

(1) Half circumference: equal to the circumference of a circle ÷2 Calculation method: 2πr÷2 is π r.

(2) The circumference of a semicircle: equal to half the circumference plus the diameter. Calculation method: πr+2r

Third, the ratio and the application of the ratio

(A), the meaning of the ratio

1, the meaning of ratio: the division of two numbers is also called the ratio of two numbers.

2. In the ratio of two numbers, the number before the comparison sign is called the first item of the ratio, and the number after the comparison sign is called the last item of the ratio. The quotient obtained by dividing the former term by the latter term is called the ratio.

For example,15:10 =15 ÷10 = (the ratio is usually expressed as a fraction, but it can also be expressed as a decimal or an integer).

∶∶∶∶

The ratio of the former to the latter.

3. The ratio can represent the relationship between two identical quantities, that is, the multiple relationship. You can also use the ratio of two different quantities to represent a new quantity. For example: distance-speed = time.

4. Discrimination rate and ratio

Ratio: indicates the relationship between two numbers, which can be written as ratio or fraction.

Ratio: equivalent to quotient, it is a number, which can be an integer, a fraction or a decimal.

According to the relationship between fraction and division, the ratio of two numbers can also be written as a fraction.

6, the relationship between ratio and division, fraction:

Compared with the previous ratio symbol ":",the latter ratio

Division quotient of divisor "".

Fractional numerator dividing line "-"denominator fractional value

7. The difference between ratio, division and fraction: except is an operation, fraction is a number, and ratio represents the relationship between two numbers.

8. According to the relationship between ratio and division and fraction, it can be understood that the latter term of ratio cannot be 0.

In the sports competition, the scores of the two teams are 2: 0, etc. This is just a form of scoring, which does not represent the division of two numbers.

(B) The basic nature of the ratio

1, according to the relation of ratio, division and fraction:

The property that the quotient is invariant: the dividend and divisor are multiplied or divided by the same number at the same time (except 0), and the quotient is invariant.

The basic property of a fraction: when the numerator and denominator of the fraction are multiplied or divided by the same number at the same time (except 0), the value of the fraction remains unchanged.

The basic nature of the ratio: the first and last items of the ratio are multiplied or divided by the same number at the same time (except 0), and the ratio remains unchanged.

2. The simplest integer ratio: the first and last terms of the ratio are integers and prime numbers, so this ratio is the simplest integer ratio.

3. According to the basic properties of the ratio, the ratio can be reduced to the simplest integer ratio.

4. Simplified ratio:

① Divide the first and last terms of the ratio by their greatest common factor.

(1)② the ratio of two fractions: multiply the last item in the previous paragraph by the least common multiple of the denominator at the same time, and then simplify it by simplifying the integer ratio.

③ Proportion of two decimal places: move the decimal point position to the right, first change it into integer proportion and then simplify it.

(2) Using the method of calculating the ratio. Note: The final result should be written in the form of ratio.

For example:15:10 =15 ÷10 = = 3: 2.

5. Proportional allocation: allocate a quantity according to a certain proportion. This method is usually called proportional distribution.

If the ratio of two quantities is known, let these two quantities be.

6. The distance is fixed, and the speed ratio is inversely proportional to the time ratio. (For example, for the same distance, the speed ratio is 4: 5 and the time ratio is 5: 4).

The total amount of work is certain, and the work efficiency is inversely proportional to the working hours.

(For example, the total amount of work is the same, the working time ratio is 3: 2, and the working efficiency ratio is 2: 3)