S=πRL

The side area of the cone =n/360×π×R? = 1/2LR(n refers to the degree of the sector vertex angle, r refers to the radius of the cone bottom surface, and l refers to the generatrix)

The lateral area derivative of a cone needs to expand the cone;

(2) Mathematically, the line from the apex of the cone to any point on the circumference of the bottom of the cone is called the generatrix of the cone;

(3) Cut along any generatrix of the cone and spread it into a plane figure, that is, a sector;

(4) The radius of the expanded sector is the generatrix of the cone,

The arc length of the expanded sector is the circumference of the cone bottom;

⑤ By extension, the side area of three-dimensional graphics is transformed into the area of plane graphics.

Let the length of the generatrix of the cone be l and the radius of the bottom surface of the cone be r,

Then the fan-shaped radius is L, and the arc length is the circumference of the cone bottom (2πR).

The formula of sector area is: S=( 1/2)× sector radius× sector arc length.

=( 1/2)×L×(2πR)

=πRL

That is, the lateral area of the cone is π times the product of the radius of the cone bottom and the length of the cone generatrix.

Knowledge points of arc length and sector area

Arc length formula: n is the degree of central angle, r is the radius, and α is the radian of central angle.

L=nπr÷ 180 or l=n/ 180? πr or l=|α|r

In a circle with radius r, because the arc length corresponding to the central angle of 360 is equal to the circumference of the circle C=2πR, the arc length corresponding to the central angle n is l = n π r ÷ 180.

In the arc system, if the central angle of the arc is θ, there is a formula L=Rθ. The formula of sector area is S=LR/2, and the corresponding formula is S=RRθ/2.

S fan =LR/2(L is the arc length of the fan and R is the radius) or π (r 2) * n/360 (that is, the degree of the fan).

Sector is an important figure related to the circle, and its area is related to the central angle (vertex angle) and radius of the circle. The area of a sector with a central angle of n and a radius of r is n/360 * π r 2. If the vertex angle is in radians, it can be simplified as 1/2× arc length× (radius).

The sector is also similar to a triangle, and the simplified area formula above can also be regarded as: 1/2× arc length× (radius), which is similar to the triangle area: 1/2× bottom× height.

Arc length (L)=n/360? 2πr=nπr/ 180, and one side of the arc is similar to a fan-shaped triangle.

Knowledge points of regular polygons and circles

1, regular polygons are closely related to circles;

1) Divide the circumference of a circle into n equal parts, and connect the points in turn to get the graph, which is the inscribed regular N-polygon of the circle. This circle is called the circumscribed circle of this regular N-polygon.

2) Related concepts of regular polygon: the center of regular polygon is the center of circumscribed circle of regular polygon. Regular Polygon Radius-The radius of the inscribed circle of a regular polygon. (rn) The central angle of a regular polygon-it is the central angle of a circumscribed circle opposite to the side of a regular polygon. (αn)

Vertex of a regular polygon-is the distance from the edge to the center of a regular polygon. (Registered Nurse)

3) Calculation of regular N polygons: the relationship among edge an, radius r N and apogee rn: rn2-rn2 = () 2 (Pythagorean theorem).

Area of a regular n-polygon: sn = lnrn (ln- perimeter of a regular n-polygon) (the difference in the number of sides only reflects the difference in the central angle αn).

2. When all sides are equal, the polygon inscribed in the circle is a regular polygon; When the angles are equal, the polygon circumscribed by the circle is a regular polygon.

3. When the angles of a polygon inscribed in a circle are equal and the number of sides is odd, the inscribed polygon is a regular polygon;

When the number of sides of a circular circumscribed polygon is equal and odd, the circumscribed polygon is a regular polygon.

4. The inscribed regular N polygon of a circle is similar to its circumscribed regular N polygon, and the similarity ratio is equal to COS (180/n);

5. Compared with a circle, a regular polygon with the same circumference has a larger area, and the more sides a polygon has, the closer it is to a circle.

Compared with a circle, a regular polygon with the same area has a smaller perimeter, and the more sides a polygon has, the closer it is to a circle.

6. The circle is an axisymmetric figure with numerous symmetry axes; Regular polygons are also axisymmetric figures, and the number of symmetry axes is equal to the number of sides.

7. The circle is also a central symmetrical figure; A regular polygon is centrosymmetric only when the number of sides is even.

Knowledge points of positional relationship between upper straight line and circle

(1) A straight line and a circle have nothing in common, which is called separation. AB is separated from circle O, d>r.

② A straight line and a circle have two common points, which are called intersections. This straight line is called the secant of a circle. AB intersects with ⊙O and d.

③ A straight line and a circle have only one common point, which is called tangency. This straight line is called the tangent of the circle, and this common point is called the tangent point. AB is tangent to ⊙O, and d = r. (d is the distance from the center of the circle to the straight line)

In the plane, the general method to judge the positional relationship between the straight line Ax+By+C=0 and the circle X 2+Y 2+DX+EY+F = 0 is:

1. You can get y=(-C-Ax)/B from Ax+By+C=0 (where b is not equal to 0), and substitute it into x 2+y 2+dx+ey+f = 0, and the equation about x becomes.

If b 2-4ac > 0, the circle and the straight line have two intersections, that is, the circle and the straight line intersect.

If b 2-4ac = 0, the circle and the straight line have 1 intersections, that is, the circle is tangent to the straight line.

If b 2-4ac

2. If B=0 indicates that the straight line is Ax+C=0, that is, x=-C/A, parallel to the Y axis (or perpendicular to the X axis), change X 2+Y 2+DX+EY+F = 0 to (X-A) 2+(Y-B) 2 = R, and let Y =

When x=-C/Ax2, the straight line deviates from the circle;

Circular angle knowledge point

Theorem of circumferential angle: In the same circle or equal circle, the circumferential angle of the same arc or equal arc is equal to half the central angle of the arc.

Proof (classification idea, 3 kinds, equal radius)

The degree theorem of the circle angle: the degree of the circle angle is equal to half of the degree of the arc it faces.

(2) In the same circle or equal circle, the circumferential angles of the same arc or equal arc are equal, and so are the arcs with equal circumferential angles. (Not in the same circle, not in the same circle, actually equal. Note: This article only. [2])

③ The circumference angle (or diameter) of a semicircle is a right angle, and the chord with a circumference angle of 90 is a diameter.

(4) Diagonal lines of the inscribed quadrilateral of a circle are complementary, and any external angle is equal to its internal angle.

⑤ In the same circle or in the same circle, the circumferential angle, arc, chord and chord center distance are equal.

Proposition 1: Make a chord MN in the circle, take points A, B and C on the same side of the straight line MN as points A, B and C in the circle, on the circle and outside the circle respectively, and connect points A, B and C with points M and N respectively, and there will be ∠ A >; ∠B& gt； ∠C .

Proposition 2: the angle of the vertex outside the circle (the two sides intersect the circle) is equal to half of the difference between the two radians it cuts; The angle of the vertex in a circle (two sides intersect the circle) is equal to half of the sum of the radians cut by the vertex and the vertex.

Determine the conditional knowledge points of a circle

Through the exploration of circle determination by three points not on a straight line, we can know the circle determination by three points not on a straight line, master the method of circle determination by three points not on a straight line, understand the concepts of triangle circumscribed circle, triangle outer center and circle inscribed triangle, and further understand the problem-solving strategy.

Key points:

1. Theorem: Three points that are not on a straight line determine a circle. The condition of "not in a straight line" in the theorem can not be ignored, and the word "sure" should be understood as "there and only"

2. The circle passing through the vertex of the triangle is called the circumscribed circle of the triangle, and the center of the circumscribed circle is the outer center of the triangle. This triangle is called the inscribed triangle of a circle. As long as the triangle is determined, the outer center and radius of its circumscribed circle are also determined.

Difficulties:

The essence of analyzing the circle making method is to try to find the center of the circle.

Symmetry knowledge point of circle

In the algorithm of generating a circle, the computational overhead can be reduced to 1/8.

Symmetry principle:

The (1) circle is X-axis symmetric, so it is only necessary to calculate the position of the original 1/2 point.

(2) The circle satisfies the Y-axis symmetry, so only the position of the origin 1/2 needs to be calculated;

(3) The circle is axisymmetrical, y=xory=-x, so it is only necessary to calculate the position of the origin 1/2;

Through the analysis of the above three properties, it can be known that the calculation element only needs to analyze 1/8 points.

For example, the target point (x, y) for analysis must exist.

(x，-y)，(-x，y)，(-x，-y)，(y，x)，(y，-x)，(-y，x)，(-y，-x)。