Formulas of perimeter, area and volume of graphics and related knowledge.

Rectangular perimeter = (length+width) ×2

Rectangular area = length × width

Square perimeter = side length × 4

Square area = side length × side length

Triangle area = base × height ÷2

Parallelogram area = base × height

Trapezoidal area = (upper bottom+lower bottom) × height ÷2

The circumference of a circle is equal to ∈× diameter or ∈× radius× 2, that is, C =∏d or c = 2 ∏ r.

The area of a circle is equal to 3. 14× the square of the radius.

The area of the ring is equal to 3. 14× (the square of the long radius-

Square with small radius)

The circumference of a semicircle = half of the circumference+diameter.

That is ∏ r+2 r

The surface area of a cuboid = (length× width+length× height+width× height) × 2.

Cuboid volume = length × width × height

or

Bottom area × height

Surface area of cube = side length × side length × 6

Volume of cube = side length × side length × side length

Surface area of cylinder =2 bottom areas+side area.

Transverse area = bottom circumference × height.

Volume of cylinder = bottom area × height

Volume of cone = bottom area × height ÷ 3

Both cuboids and cubes have 6 faces, 8 vertices and 12 edges.

The three sides that intersect at the same vertex are called the length, width and height of a cuboid.

A cube can be regarded as a special cuboid.

At least eight identical small cubes are needed to form a big cube.

The upper and lower bottom surfaces of the cylinder are round and have equal areas.

The sides of a cylinder are rectangular, the length of which is the circumference of the bottom of the cylinder, and the height of which is the height of the cylinder.

The bottom of the cone is also round, and the side is fan-shaped.

The volume of a cylinder is three times that of a cone of equal height.

The radius of the big circle is the diameter of the small circle, and the area of the big circle is four times that of the small circle.

Cut out the largest circle in the square, and the side length of the square is the diameter of the circle.

Cut out the largest circle in a rectangle, and the width of the rectangle is the diameter of the circle.

After drawing the rectangle into a parallelogram, the area is smaller than before.

The perimeter of the rectangle should be divided by 2 first, and then distributed in proportion; The sum of the edges of a cuboid can only be distributed if it is divided by 4.

The radius, perimeter and area of a circle are enlarged three times and nine times respectively.

The side length of the cube is enlarged by 3 times, the surface area is enlarged by 9 times and the volume is enlarged by 27 times.

The radius of the bottom surface of a cylinder or cone is enlarged by 2 times, and the volume is enlarged by 4 times.

Common statistical charts include bar chart, line chart and fan chart.

The characteristic of bar graph is that it is easy to see the quantity of each kind; The characteristic of broken-line statistical chart is that it can not only see the quantity of various quantities, but also clearly show the change of quantity; The characteristic of fan-shaped statistical chart is that it can clearly show the relationship between the number of each part and the total.

A rudimentary knowledge of geometry

A straight line has no end points, both ends can extend indefinitely, and the length cannot be measured.

Ray has an endpoint, one end can extend indefinitely, and the length cannot be measured.

A line segment has two endpoints and cannot be extended. Its length can be measured.

You can draw countless straight lines after one o'clock, and a straight line after two o'clock.

In the same plane, the mutual positions of two straight lines intersect and are parallel.

On the same plane, two lines that do not intersect are called parallel lines.

A figure consisting of a vertex and two rays emanating from the vertex is called an angle.

An angle greater than 0 degrees and less than 90 degrees is called an acute angle; An angle greater than 90 degrees and less than 180 degrees is called an obtuse angle.

The sum of the internal angles of the triangle is 180 degrees; The sum of the internal angles of a quadrilateral is 360 degrees.

The right angle is 90 degrees, the straight angle is 180 degrees, and the fillet is 360 degrees.

Triangle can be divided into right triangle, acute triangle and obtuse triangle according to angle.

Triangle can be divided into equilateral triangle, isosceles triangle and equilateral triangle according to side length. All three sides of an equilateral triangle are equal and all three angles are 60 degrees.

Rectangular and square are special parallelograms.

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

Triangles are stable and parallelograms are easily deformed.

In the case of equal base and equal height, the area of triangle is half that of parallelogram.

A circle is a curved figure on a plane, and the length of the curve around the circle is called the circumference of the circle; The size of the plane where the circle lies is called the area of the circle.

The line segment from the center of the circle to any point on the circle is called the radius of the circle.

The line segment whose two ends pass through the center of the circle is called the diameter of the circle.

The angle of the vertex at the center of the circle is called the central angle; The longest line segment in a circle is the diameter.

A circle has countless radii and diameters.

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

In the same circle, the diameter is twice the radius.

The ratio of the circumference to the diameter of a circle is called pi, which is expressed by the letter ∏, and was first calculated by Zu Chongzhi. ∏≈ 3. 14

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

The size of the sector is determined by the radius and the central angle.

The distance between the two corners of a compass refers to the radius of a circle.

If a graph is folded in half along a straight line, the graphs on both sides can completely overlap. This figure is called an axisymmetric figure, and the straight line where the crease is located is called an axis of symmetry.

A circle has countless symmetry axes, a rectangle has two symmetry axes, a square has four symmetry axes, an isosceles triangle has one symmetry axis, an equilateral triangle has three symmetry axes, an isosceles trapezoid has one symmetry axis, and a semicircle or a sector has one symmetry axis.

Quantitative measurement

The commonly used units of length are kilometers, meters, decimeters, centimeters and millimeters.

Commonly used area units are square kilometers, hectares, square meters, square decimeters and square centimeters.

Commonly used unit of volume are cubic meters, cubic decimeters and cubic centimeters.

The commonly used unit of volume are liters and milliliters. 1 l = 1000 ml.

Cubic decimeter is liter, cubic centimeter is milliliter.

The commonly used weight units are tons, kilograms and grams.

The commonly used RMB units are yuan, jiao and fen.

Common time units are century, year, month, day, hour, minute and second.

1 century = 100, 1 year =1February, big month 3 1 day, abortion 30 days.

A year has 12 months, which is divided into four quarters, and each quarter has three months.

There are three normal years and a leap year every four years. There are 28 days in February in a normal year and 29 days in February in a leap year.

Basic knowledge of algebra

Equations with unknowns are called equations.

The process of solving an equation is called solving an equation.

The division of two numbers is also called the ratio of two numbers; The formula that two ratios are equal is called proportion.

The last item of the ratio cannot be 0.

The quotient of the former term divided by the latter term is called the ratio. The ratio can be an integer, decimal or fraction.

The basic property of the ratio is that both the former term and the latter term are multiplied or divided by the same number (except 0), and the ratio remains unchanged.

In proportion, the product of two internal terms is equal to the product of two external terms, which is called the basic property of proportion.

The ratio of the distance on the map to the actual distance is called the scale.

There are two scales: numerical scale and line scale.

Two related quantities, one of which changes and the other changes with it. If the ratio of the corresponding two numbers in these two quantities is certain, these two quantities are called multiplied proportional quantities, and the relationship between them is called proportional relationship. That is: x ÷ y = k (certain)

Two related quantities, one of which changes and the other changes with it. If the product of the corresponding two numbers in these two quantities is certain, these two quantities are called multiplied and inversely proportional quantities, and the relationship between them is called inversely proportional relationship. Namely: x × y = k (certain)

The radius and area of a circle are out of proportion to its circumference.

The area of a triangle is constant, and the base is inversely proportional to the height.

The scale is fixed, and the distance on the map is proportional to the actual distance.

A commodity is reduced by 10% first and then increased by 10%, and the price is lower than the original.

A is 25% more than B, and B is 20% less than A. ..

Numbers and digital operations

When we count objects, 1, 2, 3 ... the numbers used to represent the number of objects are called natural numbers. 0 is also a natural number, the smallest natural number, and there is no largest natural number. Natural numbers are all integers.

Divide the unit "L" into several parts on average, and the number representing such a part or parts is called a fraction. The number representing one of them is the decimal unit of this fraction.

When two integers are divided, their quotient can be expressed as a fraction. Namely: a÷b = (b≠0)

A fraction whose numerator and denominator are prime numbers is called simplest fraction.

The reciprocal of true fraction must be greater than 1, and the reciprocal of false fraction is not necessarily less than 1.

The numerator and denominator of a fraction are multiplied or divided by the same number (except 0) at the same time, and the size of the fraction remains the same, which is called the basic property of the fraction.

Add "0" or remove "0" at the end of the decimal, and the size of the decimal remains the same, which is the so-called basic nature of the decimal.

A decimal, starting from somewhere in the decimal part, and one or several numbers are repeated in turn. Such decimals are called cyclic decimals.

The cyclic segment starting from the first digit of the decimal part is called pure cyclic decimal; Loop segments that do not start with the first decimal part are called mixed loop decimals.

A number indicating that one number is a percentage of another number is called a percentage, also called a percentage or a percentage. Percentages have no units.

When the integer A is divided by the integer b( b≠0), the quotient is exactly an integer with no remainder, so we say that A can be divisible by B or B can be divisible by A. ..

If A is divisible by B, we say that A is a multiple of B and B and a divisor of A. ..

The divisor of a number is finite, in which the smallest divisor is 1 and the largest divisor is itself.

The number of multiples of a number is infinite, in which the minimum multiple is itself and there is no maximum multiple.

If a number has only 1 and two divisors of itself, it is called a prime number.

If a number has other divisors besides 1 and itself, then it is called a composite number.

Writing a composite number into the form of multiplying several prime numbers is called decomposing prime factors.

The common multiple of several numbers is called the common multiple of these numbers, and the smallest is called the least common multiple of these numbers.

The common divisor of several numbers is called the common divisor of these numbers, and the largest number is called the greatest common divisor of these numbers.

The common divisor is only 1, which is called a prime number.

Numbers that are divisible by 2 are called even numbers, and numbers that are not divisible by 2 are called odd numbers. Natural numbers are either even or odd.

The smallest even number is 0, the smallest odd number is 1, the smallest prime number is 2, and the smallest composite number is 4.

Except 0 and 2, all even numbers are composite numbers.

The smallest two digits divisible by 2, 3 and 5 at the same time is 30, and the smallest three digits are 120.

If an expression only contains operations of the same level, it should be calculated from left to right. If there are two levels of operation, multiply and divide first, then add and subtract. If there are brackets, count the inside of the brackets first, and then the outside of the brackets.

Two numbers whose product is 1 are called reciprocal.

The number A divided by the number B (except 0) is equal to the reciprocal of the number A multiplied by the number B.

Interest = principal × interest rate× time

After-tax interest = principal × interest rate× time× 80%

concept

How to read and write numbers

1. integer reading method: from high to low, read step by step. When reading the 110 million level, first read according to the reading method of the 100 million level, and then add a word "100 million" or "10 thousand" at the end. The zeros at the end of each stage are not read, and only a few zeros of other digits are read.

2. Writing of integers: from high to low, writing step by step. If there is no unit on any number, write 0 on that number.

3. Decimal reading method: When reading decimals, the integer part is read by integer reading method, the decimal point is read as "dot", and the decimal part reads the numbers on each digit from left to right in sequence.

4. Decimal writing: When writing decimals, the integer part is written as an integer, the decimal point is written in the lower right corner of each digit, and the decimal part is written on each digit in turn.

5. How to read fractions: When reading fractions, read the denominator first, then the "fraction", and then the numerator. Both numerator and denominator read integers.

6. How to write the fraction: write the fraction first, then the denominator, and finally the numerator and the integer.

7. Reading method of percentage: When reading percentage, read the percentage first, and then read the number before the percentage symbol. When reading, read it as an integer.

8. Writing of percentage: percentage is usually expressed by adding a percent sign "%"after the original molecule instead of a fraction.

(2) The number of rewrites

In order to facilitate reading and writing, a large multi-digit number is often rewritten as a number in units of "10,000" or "100 million". Sometimes, if necessary, you can omit the number after a certain number and write it as an approximation.

1. exact number: in real life, for the convenience of counting, larger numbers can be rewritten into numbers in units of ten thousand or hundreds of millions. The rewritten number is the exact number of the original number. For example, 1254300000 is rewritten into ten thousand, and the number is125430000; Rewritten into a number of 65.438+025.43 billion in units of hundreds of millions.

2. Approximation: According to the actual needs, we can also use a similar number to represent a larger number and omit the mantissa after a certain number. For example: 13024900 15 The mantissa after omitting 100 million is1300 million.

3. Rounding method: If the highest digit of the mantissa to be omitted is 4 or less, the mantissa is removed; If the digit with the highest mantissa is 5 or more, the mantissa is truncated and 1 is added to its previous digit. For example, the mantissa after omitting 3.459 billion is about 350,000. After omitting 472509742 billion, the mantissa is about 4.7 billion.

4. Size comparison

1. Compare the sizes of integers: compare the sizes of integers, and the number with more digits will be larger. If the numbers are the same, view the highest number. If the number in the highest place is larger, the number is larger. The number in the highest bit is the same. Just look at the next bit, and the bigger the number, the bigger it is.

2. Compare the sizes of decimals: first look at their integer parts, and the larger the integer part, the larger the number; If the integer parts are the same, the tenth largest number is larger; One tenth of the numbers are the same, and the number with the largest number in the percentile is the largest. ...

3. Compare the scores: the scores with the same denominator and the scores with large numerator are larger; For numbers with the same numerator, the score with smaller denominator is larger. If the denominator and numerator of a fraction are different, divide the fraction first, and then compare the sizes of the two numbers.

(3) the number of mutual

1. Decimal component number: There are several decimals, so writing a few zeros after 1 as denominator and removing the decimal point after the original decimal point as numerator can reduce the number of quotation points.

2. Fractions become decimals: numerator divided by denominator. Those that are divisible are converted into finite decimals, and some that are not divisible are converted into finite decimals. Generally three decimal places are reserved.

3. A simplest fraction, if the denominator does not contain other prime factors except 2 and 5, this fraction can be reduced to a finite decimal; If the denominator contains prime factors other than 2 and 5, this fraction cannot be reduced to a finite decimal.

4. Decimal percentage: Just move the decimal point to the right by two places, followed by hundreds of semicolons.

5. Decimal percentage: Decimal percentage, just remove the percent sign and move the decimal point two places to the left.

6. Convert fractions into percentages: usually, first convert fractions into decimals (three decimal places are usually reserved when they are not used up), and then convert decimals into percentages.

7. Decimalization of percentage: First, rewrite percentage into component quantity and put forward a quotation that can be simplified to the simplest score.

(4) Divisibility of numbers

1. Usually a composite number is decomposed into prime factors by short division. Divide this complex number by a prime number until the quotient is a prime number, and then write the divisor and quotient in the form of multiplication.

2. The way to find the greatest common divisor of several numbers is to divide the common divisors of these numbers continuously until the quotient obtained is only the common divisor of 1, and then multiply all the common divisors to get the product, which is the greatest common divisor of these numbers.

3. The method of finding the least common multiple of several numbers is: divide by the common divisor of these numbers (or part of them) until it is coprime (or pairwise coprime), and then multiply by all the divisors and quotients to get the product, which is the least common multiple of these numbers.

4. Two numbers that become coprime relations: 1 and any natural number coprime; Two adjacent natural numbers are coprime; When the composite number is not a multiple of the prime number, the composite number and the prime number are coprime; When the common divisor of two composite numbers is only 1, these two composite numbers are coprime.

(5) Approximate points and general points

Reduction method: divide the denominator by the common divisor of the denominator (except 1); Usually, we have to separate it until we get the simplest score.

General division method: first find the least common multiple of the denominator of the original fraction, and then turn each fraction into a fraction with this least common multiple as the denominator.

Chapter I Number and Number Operation

(1) integer

The meaning of integer

Natural numbers and 0 are integers.

natural number

When we count objects, 1, 2, 3 ... the numbers used to represent the number of objects are called natural numbers.

There is no object, which is represented by 0. 0 is also a natural number.

finger

One, ten, one hundred, one thousand, ten thousand, one hundred thousand, one million, ten million, one hundred million ... are all counting units.

The propulsion rate between every two adjacent counting units is 10. This counting method is called decimal counting method.

finger

Counting units are arranged in a certain order, and their positions are called numbers.

Divisibility of numbers

When the integer A is divided by the integer b(b ≠ 0), the quotient is an integer with no remainder, so we say that A is divisible by B, or that B is divisible by A. ..

If the number A is divisible by the number B (b ≠ 0), then A is called a multiple of B, and B is called a divisor of A (or a factor of A). Multiplication and divisor are interdependent.

Because 35 is divisible by 7, 35 is a multiple of 7, and 7 is a divisor of 35.

The divisor of a number is finite, in which the smallest divisor is 1 and the largest divisor is itself. For example, the divisor of 10 is 1, 2,5, 10, where the smallest divisor is 1 0 and the largest divisor is 10.

The number of multiples of a number is infinite, and the smallest multiple is itself. The multiple of 3 is: 3, 6, 9, 12 ... The minimum multiple is 3, but there is no maximum multiple.

Numbers in units of 0, 2, 4, 6 and 8 can be divisible by 2, for example, 202, 480 and 304 can be divisible by 2. .

Numbers in units of 0 or 5 can be divisible by 5, for example, 5,30,405 can be divisible by 5. .

The sum of the numbers in each bit of a number can be divisible by 3, so this number can be divisible by 3. For example, 12,108,204 can all be divisible by 3.

The sum of each digit of a number can be divisible by 9, and so can this number.

A number divisible by 3 may not be divisible by 9, but a number divisible by 9 must be divisible by 3.

The last two digits of a number can be divisible by 4 (or 25), and this number can also be divisible by 4 (or 25). For example,16,404 and 1256 can all be divisible by 4, and 50,325,500 and 1675 can all be divisible by 25.

The last three digits of a number can be divisible by 8 (or 125), and this number can also be divisible by 8 (or 125). For example,1168,4600,5000, 12344 can all be divisible by 8, and 1 125,13375,5000 can all be/kloc-.

A number divisible by 2 is called an even number.

Numbers that are not divisible by 2 are called odd numbers.

0 is also an even number. Natural numbers can be divided into odd and even numbers according to their divisibility by 2.

A number with only two divisors of 1 is called a prime number (or prime number), and the prime numbers within 100 are: 2, 3, 5, 7,1,13, 17.

If a number has other divisors besides 1 and itself, then it is called a composite number. For example, 4, 6, 8, 9 and 12 are all complex numbers.

1 is not a prime number or a composite number, and natural numbers are either prime numbers or composite numbers except 1. If natural numbers are classified according to the number of their divisors, they can be divided into prime numbers, composite numbers and 1.

Every composite number can be written as the product of several prime numbers. Every prime number is a factor of this composite number, which is called the prime factor of this composite number. For example, 15=3×5, and 3 and 5 are called prime factors of 15.

Multiplying a composite number by a prime factor is called prime factor decomposition.

For example, decompose 28 into prime factors.

The common divisor of several numbers is called the common divisor of these numbers. The largest one is called the greatest common divisor of these numbers. For example, the divisor of 12 is 1, 2, 3, 4, 6,12; The divisors of 18 are 1, 2,3,6,9 and 18. Where 1, 2,3,6 are the common divisors of 12 and 1 8, and 6 is their greatest common divisor.

The common divisor is only 1, which is called prime number. There are the following situations:

1 is coprime with any natural number.

Two adjacent natural numbers are coprime.

Two different prime numbers are coprime.

When the composite number is not a multiple of the prime number, the composite number and the prime number are coprime.

When the common divisor of two composite numbers is only 1, these two composite numbers are coprime. If any two numbers are coprime, they are said to be coprime.

If the smaller number is the divisor of the larger number, then the smaller number is the greatest common divisor of these two numbers.

If two numbers are prime numbers, their greatest common divisor is 1.

The common multiple of several numbers is called the common multiple of these numbers, and the smallest is called the least common multiple of these numbers. For example, the multiple of 2 is 2,4,6,8, 10, 12, 14, 16, 18. ...

The multiple of 3 is 3,6,9, 12, 15, 18 ... where 6, 12, 18 ... are the common multiples of 2 and 3, and 6 is their least common multiple. .

If the larger number is a multiple of the smaller number, the larger number is the least common multiple of the two numbers.

If two numbers are prime numbers, then the product of these two numbers is their least common multiple.

The common divisor of several numbers is finite, while the common multiple of several numbers is infinite.

(B) the meaning of decimal

Divide the integer 1 into 10, 100, 1000 ... a tenth, a percentage, a thousandth ... can be expressed in decimals.

One decimal place indicates a few tenths, two decimal places indicate a few percent, and three decimal places indicate a few thousandths. ...

Decimal system consists of integer part, decimal part and decimal part. The point in the number is called the decimal point, the number to the left of the decimal point is called the integer part, and the number to the right of the decimal point is called the decimal part.

In decimals, the series between every two adjacent counting units is 10. The propulsion rate between the highest decimal unit "one tenth" of the decimal part and the lowest unit "one" of the integer part is also 10.

Classification of decimals

Pure decimals: Decimals with zero integer parts are called pure decimals. For example, 0.25 and 0.368 are pure decimals.

With decimals: decimals whose integer part is not zero are called with decimals. For example, 3.25 and 5.26 are all decimals.

Finite decimals: The digits in the decimal part are finite decimals, which are called finite decimals. For example, 4 1.7, 25.3 and 0.23 are all finite decimals.

Infinite decimal: The digits in the decimal part are infinite decimal, which is called infinite decimal. For example: 4.33...3. 145438+05926 ...

Infinite acyclic decimal: the decimal part of a number with irregular arrangement and unlimited digits. Such decimals are called infinite cyclic decimals. For example: ∈

Cyclic decimal: the decimal part of a number, in which one or several numbers appear repeatedly in turn, is called cyclic decimal. For example: 3.555 … 0.0333 …12.15438+009 …

The decimal part of cyclic decimal is called the cyclic part of cyclic decimal. For example, the period of 3.99 ... is "9", and the period of 0.5454 ... is "54".

Pure cyclic decimal: the cyclic segment starts from the first digit of the decimal part, which is called pure cyclic decimal. For example: 3.111.5656 ...

Mixed cycle decimal: the cycle section does not start from the first digit of the decimal part. This is called mixed cyclic decimal. 3. 1222 …… 0.03333 ……

When writing a cyclic decimal, for simplicity, the cyclic part of the decimal only needs one cyclic segment, and a dot is added to the first and last digits of this cyclic segment. If there is only one number in the circle, just click a point on it. For example: 3.777 ... Jane writing 0.5302302 ... Jane writing.

(C) the significance of the score

Divide the unit "1" into several parts on average, and the number representing such a part or parts is called a fraction.

In the score, the middle horizontal line is called the dividing line; The number below the fractional line is called the denominator, indicating how many copies the unit "1" is divided into on average; The number below the fractional line is called the numerator, indicating how many copies there are.

Divide the unit "1" into several parts on average, and the number representing one part is called fractional unit.

Fractional classification

True fraction: The fraction with numerator less than denominator is called true fraction. The true score is less than 1.

False fraction: Fractions with numerator greater than denominator or numerator equal to denominator are called false fractions. False score is greater than or equal to 1.

With fraction: False fraction can be written as a number consisting of integer and true fraction, which is usually called with fraction.

Simplified points and general points

Changing a fraction into a fraction equal to it, but with smaller numerator and denominator, is called divisor.

The denominator of a molecule is a fraction of a prime number, which is called simplest fraction.

Dividing the scores of different denominators by the scores of the same denominator equals the original score, which is called the total score.

4) Percentage

A number indicating that one number is a percentage of another number is called a percentage, also called a percentage or a percentage. Percentages are usually expressed as "%". The percent sign is a symbol indicating percentage.