Overview of the concept of circle

1. Definition of a circle: a curve graph on a plane.

2. 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. The center of the circle is generally represented by the letter O, and 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. The radius is generally represented by the letter R. If the two feet of a compass are separated, the distance between the two feet is the radius of the circle.

4. The center of the circle determines the position of the circle and the radius determines the size of the circle.

5. Diameter: The line segment whose two ends pass through the center of the circle is called diameter. The diameter is usually indicated by the letter d.

6. In the same circle, all radii are equal and all diameters are equal.

7. The same circle has countless radii and countless diameters.

8. The diameter of the same circle is twice the radius, and the radius is half the diameter.

Represented by letters: D = 2RR = D

Expressed in words: radius = diameter ÷2 diameter = radius ×2.

9. Circumference: The length of the curve around a circle is called circumference.

10. The circumference of a circle is always greater than 3 times the diameter, and this ratio is a fixed number. We call the ratio of the circumference to the diameter of a circle pi, which is expressed by letters. Pi is an infinite cyclic decimal. In the calculation, take 3. 14. The first person in the world to calculate pi was Chinese mathematician Zu Chongzhi.

1 1. The circumference formula of a circle: C= d or c = 2 r.

Perimeter = × Diameter Perimeter = × Radius ×2

12. Area of the circle: The area occupied by the circle is called the area of the circle.

13. Cut a circle into an approximate rectangle. The length of the cut rectangle is equal to half the circumference of the circle, which is indicated by the letter (R), and the width is equal to the radius of the circle, which is indicated by the letter (R). Because the area of a rectangle = length× width, and the area of a circle = r×r R. The area formula of a circle: s = r? .

14. Formula of circular area: s = r? Or S= (d 2)? Or S= (C 2)?

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

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

17. A circular ring, the radius of the outer circle is r, the radius of the inner circle is r, and its area is S= R? - r？ Or S= (R? -r？ )。

(where r = the width of the ring. )

19. The circumference of a semicircle is equal to half the circumference plus the diameter. The difference between the circumference of a semicircle and that of a semicircle is that a semicircle has a diameter and a semicircle has no diameter.

The perimeter formula of a semicircle: c = d 2+d or c = r+2r.

Half of the circumference = r

20. Area of semicircle = area of circle 2 The formula is: s = r? 2

2 1. How many times the radius of the same circle is enlarged or reduced, the diameter and circumference are also enlarged or reduced by the same times. And the area is expanded or reduced by the square of the multiple.

For example, the radius, diameter and circumference of the same circle are enlarged by four times, and the area is enlarged by 16 times.

22. The radius ratio of two circles is equal to the diameter ratio and the circumference ratio, and the area ratio is equal to the square of the above ratio.

For example, if the radius ratio of two circles is 2: 3, then the diameter ratio and perimeter ratio of these two circles are both 2: 3, while the area ratio is 4: 9.

The ratio of circumference to diameter is: 1, and the ratio is.

The ratio of perimeter to radius is 2: 1, and the ratio is 2.

23. The radius of the circle is increased by one centimeter, and the circumference is increased by two centimeters;

When the diameter of a circle increases by one centimeter, its circumference increases by one centimeter.

24. In the same circle, the central angle accounts for a fraction of the central angle, and its sector area accounts for a fraction of the circular area; The right arc occupies a small part of the circumference.

25. When the perimeters of rectangle, square and circle are equal, the area of circle is the largest and the area of rectangle is the smallest.

26. Sector arc length formula: Sector area formula: S= r? (n is the degree of the central angle of the sector, and r is the radius of the circle where the sector is located)

27. Axisymmetric figure: If a figure is folded in half along a straight line and the figures on both sides can completely overlap, it is an axisymmetric figure. The straight line where the crease lies is called the symmetry axis.

28. The figures with symmetry axis are: angle, isosceles triangle, isosceles trapezoid, sector and semicircle.

A figure with two axes of symmetry is a rectangle.

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

A figure with four axes of symmetry is a square.

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

29. A straight line with a diameter is the symmetry axis of a circle.

Unit 2 Percentage Application Problem

(A) the basic concept of percentage

1. Definition of percentage: A number indicating that one number is a percentage of another number is called a percentage. Percentages are also called percentages or percentages.

Percent refers to the ratio relationship between two numbers, not the specific quantity, so percentage cannot take units.

2. Meaning of percentage: It means that one number is the percentage of another number.

For example, 25% means that one number is 25% of another.

3. Percentages are usually not written in fractional form, but expressed by adding "%"after the original molecule. The molecular part can be a decimal or an integer, which can be greater than 100, less than 100 or equal to 100.

4. Rules for decimal and percentage exchange:

To convert decimals into percentages, just move the decimal point two places to the right, followed by hundreds of semicolons;

To convert percentages to decimals, simply remove the percent sign and move the decimal point two places to the left.

5. Reciprocity rule of percentage and score:

When a fraction is converted into a percentage, it is generally converted into a decimal (except for three decimal places) and then converted into a percentage;

Divide the percentage into components, and rewrite the percentage into components first, so that the quotation that can be lowered can be made into the simplest score.

(2) percentage application problem

Percentage application problem (1)

What percentage do you want to increase? What percentage?

Formula: Increased percentage = increased parts-unit 1.

Percentage reduction = reduced parts/unit 1

For example, after 1 and 45 cubic centimeters of water freeze, the volume of ice is 50 cubic centimeters. What percentage is the volume of ice more than the original water?

Thinking of solving the problem: according to the formula, how much percentage is increased = increased part ÷ unit 1, first determine that the unit 1 is water, which is already known as 45: the increased part is unknown, and you can subtract 45 from 50 to get 5; Finally, the increase of using 5÷ unit 1 45 water is equivalent to an increase of several percentage points.

Calculation steps: Step 1: unit 1: water: 45 cubic centimeters.

Step 2: The added part: 50-45 = 5 cubic centimeters.

Step 3: Increase by several percent: 5 ÷ 45 =11.1%.

After 2.45 cubic centimeters of water turned into ice, its volume increased by 5 cubic centimeters. What percentage has the volume of ice increased compared with the original amount of water?

Thinking of solving the problem: according to the formula, how much percentage is increased = the increased part ÷ unit 1, first determine that the unit 1 is water, which is known to be 45: the increased part is 5 cubic centimeters; Finally, the increase of using 5÷ unit 1 45 water is equivalent to an increase of several percentage points.

Calculation steps: Step 1: unit 1: water: 45 cubic centimeters.

Step 2: The added part: 5 cubic centimeters.

Step 3: Increase by several percent: 5 ÷ 45 =11.1%.

3. When water turns into ice, its volume increases by 5 cubic centimeters, and the volume of ice is 50 cubic centimeters. What percentage is the volume of ice more than the original water?

Thinking of solving the problem: According to the formula, how much percentage is increased = increased part ÷ unit 1, and the unit 1 is water first. I don't know, but according to the topic "The volume of water increases by 5 cubic centimeters after freezing", we know that there is less water and more ice, so we can use 50-5 to calculate that the water is 45 cubic centimeters. The increased part is 5 cubic centimeters; ; Finally, the increase of using 5÷ unit 1 45 water is equivalent to an increase of several percentage points.

Calculation steps: Step 1: unit 1: water: 50-5 = 45 cubic centimeters.

Step 2: The added part: 5 cubic centimeters.

Step 3: Increase by several percent: 5 ÷ 45 =11.1%.

4. The method of "reducing a few percent and increasing a few percent" is exactly the same.

5. The same as the percentage increase is "how much more" and "how much more".

"A few percent growth" and so on.

The same as a few percent reduction is "a few percent less", "a few percent less" and "a few percent less".

Percentage application problem (2)

A number that is a few percent more than a number and a number that is a few percent less than a number.

For example, 1, Yide Primary School had 80 students last year, and the number of students this year has increased by 25% compared with last year. How many students are there this year?

Thinking of solving problems: Unit 1 Last year, I learned to use multiplication, which increased the use (1+25%).

Formula: 80×( 1+25%)

There were 80 students in Yide Primary School last year. The number of students this year is 25% less than last year. How many students are there this year?

Thinking of solving problems: Unit 1 Last year, I learned to use multiplication, which reduced the use (1-25%).

Formula: 80×( 1-25%)

3. Yide Primary School has 100 students this year, an increase of 25% over last year. How many students were there last year?

Solution: Unit 1 I didn't know how to use division last year, so I added (1+25%).

Formula: 100( 1+25%)

Yide Primary School has 100 students this year, which is 25% less than last year. How many students were there last year?

Thinking of solving problems: Unit 1 I didn't know how to use division last year, but I used it more (1-25%).

Formula: 100( 1-25%)

Percentage application problem (3) Solving percentage application problem with series equation

1, Xiaoming reads a book. On the first day, he read 25% of the book, and the next day, he read 20 pages. How many pages are there in this book?

Solution: Unit 1 A book is unknown, so you can choose equation or division to solve it.

According to "the first day is 20 pages more than the second day", we can know that the first day is more, the second day is less, and the first day MINUS the second day is equivalent to 20 pages more.

Equivalence: Page 1 day-Day 2 =20

Method 1: Solution: Make this book have x pages.

From "I read 25% of the whole book on the first day", we can know that the first day is equal to the whole book multiplied by 25%, which can be expressed by X. From "I read 20% of the whole book on the second day", we can know that the second day is equal to the whole book multiplied by 20%, which can be expressed by X. According to the equivalence relationship "the first day-the second day =20 pages", we can list the equation.

Method 2: "Read 20 pages more on the first day than on the second day", knowing that 20 pages is the difference between the first day and the second day. As long as 20 pages are divided by 20 pages in half, the unit of 1 is needed.

The formula is: 20 ÷ (25%-20%)

2. Xiaoming reads a book, reading 25% on the first day, 20% on the second day and 20 pages in two days. How many pages are there in this book?

Equivalence: From "I read 20 pages in two days * * *", we can know that the first day+two days =20 pages.

Equation method: Solution: If the book has X pages, the first day is 25%X, and the second day is 20% X.

The equation is listed as: 25%X+20%X=20.

Arithmetic method: From "I read 20 pages in two days * * *", we can know that 20 pages are the sum of the first day and the second day, just divide the bisection rate of 20 pages by the unit of 1.

The formula is: 20(25%+20%)

Xiaoming has read a book. On the first day, he read 25% of the book, and the next day, he read 20% of the book, with 20 pages left. How many pages are there in this book?

Equivalence: a book-two pages a day =20 pages.

Equation method: suppose the book has x pages, then the first day is 25%X and the second day is 20% X.

The column equation is: x-25% x-20% x = 20.

Arithmetic: 20( 1-25% X-20%)

Xiao Ming has read a book. On the first day, he read 25% of the book. The next day he read 10 more pages than the first day, leaving 20 pages. How many pages are there in this book?

Equation method: suppose the book has x pages, then the first day is 25%X, and the second day is (25%X+ 10) pages.

The column equation is: x-25% x-(25% x+ 10) = 20.

Percentage application problem (4) calculation of interest

1. principal: the money in the bank is called principal.

2. Interest: The excess money paid by the bank when withdrawing money is called interest.

Interest = principal × interest rate× time

3. Before June 9, 2008, 10, the state stipulated that deposit interest should be taxed at the rate of 20%. There is no tax in debt interest. Interest tax shall be exempted after June 9, 2008. Therefore, unless otherwise specified, interest tax is not calculated.

4. Interest rate: The ratio of interest to principal is called interest rate.

5. Calculation formula of after-tax interest of bank deposits: after-tax interest = interest × (1-20%)

6. Calculation formula of debt interest: interest = principal × interest rate × time.

7. Principal and interest: The sum of principal and interest is called principal and interest.

8. Taxable amount: The tax paid is called taxable amount.

9. Tax rate: The ratio of taxable amount to various incomes is called tax rate.

10. Calculation of tax payable: tax payable = income x tax rate.

For example, Miss Li will deposit 2000 yuan in the bank for five years at an annual interest rate of 4. 14%. What is Miss Li's principal and interest at maturity?

Solution: "How much principal and interest * * *" should be 2000 yuan of principal plus interest.

Step 1: Calculate interest according to "interest = principal × interest rate× time"

Interest: 2000×4. 14%×5=4 14 yuan.

Step 2: principal+interest: 2000+4 14=24 14 yuan.

For example, Miss Li will deposit 2000 yuan in the bank for five years at an annual interest rate of 4. 14%. What is Miss Li's principal and interest at maturity? (If interest is taxed at 20%)

Solution: "How much principal and interest * * *" should be 2000 yuan of principal plus interest.

Step 1: Calculate interest according to "interest = principal × interest rate× time"

Interest: 2000×4. 14%×5=4 14 yuan.

Step 2: Calculate the after-tax interest: 414× (1-20%) = 331.2 yuan.

Principal+interest: 2000+33 1.2=233.2 yuan.

Chapter III Transformation of Graphics

1, three methods of graphic transformation:

The first translation: several translations about the direction of the Ming Dynasty (up, down, left and right).

Second turn: you need to specify which point to turn, clockwise or counterclockwise, and how many degrees to turn (90 degrees, 180 degrees, 270 degrees).

The third is symmetrical figure: it is necessary to explain which straight line is the symmetrical figure of which figure.

2. Calculation of the number of matches and handshakes

Step 1: First, figure out how many people (or teams) will take part in the competition. How many people shake hands.

Step 2: Count the number of matches and handshakes. If there are five people, it will increase from 1 to 4; if there are six people, it will increase from 1 to 5; if there are eight people, it will increase from 1 to 7; if there are 100 people, it will increase from 1 to 99.

2. Calculate the starting line.

Assume that the bend radius of the first lane is 36m, and the runway width of each lane is1.2m. ..

Then: curve radius of the second lane = curve radius of the first lane+runway width =36+ 1.2.

The curve radius of the third lane = the curve radius of the first lane+runway width+runway width =36+ 1.2+ 1.2.

The curve radius of the fourth lane = the curve radius of the first lane+runway width+runway width 1.2m+ runway width = 36+ 1.2+ 1.2.

The curve radius of the fifth lane = the curve radius of the first lane+runway width+runway width+runway width = 36+1.2+1.2+1.2+1.2+65438+1.

How many meters is the difference between the starting points of two different lanes? Step 1: Calculate how many laps to run first. Step 2: Calculate the circumference of the circle formed by two semi-circular runways. Step 3: Subtract the perimeter of the two roads, and you can get how many meters the difference between the starting points of the two roads is. Step 4: Use this difference × the number of laps run.

Understanding of the fourth unit ratio

(A) the basic concept of ratio

1. The division of two numbers is also called the ratio of two numbers. The quotient obtained by dividing the former term by the latter term is called the ratio.

2. Ratios are usually expressed by fractions, decimals and integers.

3. The last item of the ratio cannot be 0.

4. Compared with division, the former term of ratio is equivalent to dividend, the latter term is equivalent to divisor, and the ratio is equivalent to quotient;

5. According to the relationship between fraction and division, the former term of ratio is equivalent to numerator, the latter term is equivalent to denominator, and the ratio is equivalent to the value of fraction.

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

(2) Find the ratio

1. Find the ratio: divide the previous term of the ratio by the later term.

(3) Simplify the ratio

1. Simplified ratio: divide the former term of the ratio by the latter term of the ratio to find the ratio of the fraction, and then change the ratio of the fraction to the ratio.

(D) the application of the ratio

The first application of 1, ratio: Given the sum of two or more quantities and the ratio of these two or more quantities, what are these two or these two quantities?

For example, there are 60 students in the sixth grade, and the ratio of male to female is 5: 7. How many boys and girls are there?

Topic analysis: 60 people are the sum of the number of boys and girls.

Problem solving ideas: Step 1, get each copy: 60÷(5+7)=5 people.

Step 2, find boys and girls: boys: 5×5=25, girls: 5×7=35.

2. The second application of the ratio: knowing the number of one, the ratio of two or more numbers, what are the other numbers?

For example, there are 25 boys in grade six, with a ratio of 5: 7. How many girls are there? How many people are there in the class?

Topic analysis: 25 Boys is one of them.

Thinking of solving problems: Step 1, find each copy: 25÷5=5 people.

The second step is to find girls: girls: 5×7=35 people. Class: 25+35=60 people

3. The third application of ratio: knowing the difference between two quantities and the ratio of two or more numbers, what are these two or more numbers?

For example, in the sixth grade, there are 20 more boys than girls (or 20 fewer girls than boys), and the ratio of boys to girls is 7: 5. How many boys and girls are there? How many people are there in the class?

7. Required quantity = known quantity ×

7. The application of ratio in geometry;

(1) Given the circumference of a rectangle, the length-width ratio is a: b, and the length, width and area are found.

Length = perimeter ÷2× width = perimeter ÷2× area = length× width.

(2) Know the ratio of the side length to the length, width and height of a cuboid A: B: C ... Find out the length, width, height and volume.

Length = Length 4x Width = Length 4x

Height = side length ÷4× volume = length× width× height

(3) Given the ratio of three angles of a triangle as A: B: C, find the degrees of three inner angles.

These three angles are:

180× 180× 180×

(4) Given the perimeter of a triangle, the length ratio of the three sides is A: B: C, and find the length of the three sides.

These three aspects are:

Perimeter × perimeter × perimeter ×

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