First, the basic algebraic formula
1. Square difference formula: (a+b) × (a-b) = A2-B2.
2. Complete square formula: (AB) 2 = A2AB+B2
Complete cubic formula: (a b) 3 = (a b) (a2ab+B2)
3. Multiplication with the same base number: am× an = am+n (m both m and n are positive integers, a≠0).
Same base powers division: am ÷ an = am-n (m both m and n are positive integers, a≠0).
a0= 1(a≠0)
A-p = (a ≠ 0, p is a positive integer)
4. Arithmetic series:
( 1)sn = = na 1+n(n- 1)d;
(2)an = a 1+(n- 1)d;
(3)n =+ 1;
(4) If a, a and b are arithmetic progression, then: 2a = a+b;
(5) If m+n=k+i, then: AM+An = AK+AI;
(where: n is the number of terms, a 1 is the first term, an is the last term, d is the tolerance, and sn is the sum of the first n terms in arithmetic progression).
5. Geometric series:
( 1)an = a 1q- 1;
(2) Serial number = (q 1)
(3) If A, G and B are geometric series, then G2 = AB.
(4) If m+n=k+i, then: am? an=ak? ai;
(5)am-an=(m-n)d
(6) =q(m-n)
(where: n is the number of terms, a 1 is the first term, an is the last term, q is the common ratio, and sn is the sum of the first n terms in the geometric series).
6. The root formula of unary quadratic equation: ax2+bx+c=a(x-x 1)(x-x2).
Where: x1=; x2= (b2-4ac 0)
Relationship between root and coefficient: x 1+x2=-, x 1? x2=
Second, the basic geometric formula
1. triangle: three points that are not on a straight line can form a triangle; The sum of the internal angles of the triangle is equal to180; Any two triangles.
The sum of sides is greater than the third side, and the difference between any two sides is less than the third side;
(1) angle bisector: the bisector of an angle of a triangle intersects the opposite side of the angle, and the line segment between the intersection of the vertex and the angle is called the bisector of the angle of the triangle.
(2) The midline of a triangle: the line segment connecting a vertex of the triangle with the midpoint of its opposite side is called the midline of the triangle.
(3) Height of the triangle: The vertical line from the vertex of the triangle to its opposite side is called the height of the triangle.
(4) The midline of the triangle: the line segment connecting the midpoints of the two sides of the triangle is called the midline of the triangle.
(5) Inner heart: the intersection of bisectors of angles is called inner heart; The distance from the center to the three sides of the triangle is equal.
Center of gravity: the intersection of the center lines is called the center of gravity; The distance from the center of gravity to the midpoint of each side is equal to one third of the center line here.
Vertical line: the intersection of high lines is called vertical line; The vertex of a triangle must be perpendicular to the opposite side.
Exterior center: The intersection of the perpendicular lines of the three sides of a triangle is called the exterior center of the triangle. The distance from the outer center to the three vertices of the triangle is equal.
Right triangle: A triangle with an angle of 90 degrees is a right triangle.
Properties of right triangle:
(1) The two acute angles of a right triangle are complementary;
(2) The median line on the hypotenuse of the right triangle is equal to half of the hypotenuse;
(3) In a right triangle, if there is an acute angle equal to 30, then the right side it faces is equal to half of the hypotenuse;
(4) In a right triangle, if a right-angled side is equal to half of the hypotenuse, the acute angle of this right-angled side is 30;
(5) In a right triangle, C2 = A2+B2 (where: A and B are the lengths of two right angles and C is the length of the hypotenuse);
(6) The radius of the circumscribed circle of the right triangle is also the center line on the hypotenuse;
Determination of right triangle;
(1) has an angle of 90;
(2) The median line of one side is equal to half the length of the side;
(3) If C2 = A2+B2, a triangle with side lengths A, B and C is a right triangle;
2. Area formula:
Square = side length × side length;
Rectangular = length × width;
Triangle = x base x height;
Trapezoid =;
Circle = R2
Parallelogram = base × height
Department = R2
Cube = 6× side length× side length
Cuboid = 2× (length× width+width× height+length× height);
Cylinder = 2π R2+2π RH;
Surface area of ball = 4r2
3. Volume formula
Cube = side length × side length × side length;
Cuboid = length× width× height;
Cylinder = bottom area × height = sh = π r2h
Cone = π r2h
Ball =
4. Formulas related to circles
Let the radius of the circle be r and the distance from the point to the center of the circle be d, then there are:
(1)d¢r: the point is in the circle (that is, the inside of the circle is the set of points whose distance from the center of the circle is less than the radius);
(2) d = r: the point is on the circle (that is, the upper part of the circle is the set of points whose distance from the center of the circle is equal to the radius);
(3) d-r: the point is outside the circle (that is, outside the circle is the set of points whose distance from the center of the circle is greater than the radius);
The nature and judgment of the positional relationship between line and circle;
If the radius ⊙O is r and the distance from the center of o to the straight line is d, then:
(1) straight line intersection ⊙ o: dr;
(2) The line is tangent to ⊙O: d = r;;
(3) the line is separated from ⊙O: d r;
The nature and judgment of the positional relationship between circles;
Let the radii of two circles be r and r respectively and the center distance be d, then:
(1) Two circles are separated from each other:
(2) circumscribe two circles:
(3) Two circles intersect: ();
(4) Two circles are inscribed: ();
(5) Two circles contain: ().
The formula of the circumference of a circle: c = 2π r = π d (where r is the radius of the circle, d is the diameter of the circle, and π ≈ 3.1415926 ≈);
Calculation formula of arc length corresponding to central angle: =;
Sector area: (1)S sector = π R2; ②S fan = r;
If the radius of the cone bottom is R and the length of the generatrix is L, then its lateral area: S side = π r;
Volume of cone: v = sh = π r2h.