I. Concept definition
Cylinder 1, definition of rotation: a rectangle rotates clockwise or counterclockwise on one side, and the space it passes through is called
Make a cylinder.
2. Translation definition: A circle whose bottom moves up or down a certain distance, and the space it passes through is called a cylinder.
Second, the concept.
1. The two circular surfaces of a cylinder are called the bottom surface, and the surrounding surfaces are called the side surfaces. A cylinder consists of two bottom surfaces and one side surface.
2. The two bottom surfaces of a cylinder are two identical circular surfaces. The distance between the two bottom surfaces is the height of the cylinder.
3. The side surface of the cylinder is a curved surface, and the development diagram of the side surface of the cylinder is a rectangle, a square or a parallelogram (diagonal cutting). Lateral area of a cylinder = the height of the X base perimeter, that is, S lateral area =Ch=2πrh base perimeter C = 2π r = π d.
Surface area of cylinder = lateral area+bottom area x2 = ch+2π r 2 = 2π r (r+h).
4. Volume of cylinder = base area x height, that is, V=S base area x h = (π× r× r) h.
The volume of a cylinder with equal base and equal height is three times that of a cone.
6. A parallelogram can enclose a cylinder.
7. Surface area of cylinder = side area+bottom area x2.
8. Divide the cylinder into two identical parts along the diameter of the bottom surface, and each part is called a semi-cylinder. Compared with the original cylinder,
Surface area =πr(r+h)+2rh, and the volume is half of the original.
9. The axial section of a cylinder is a rectangle with a diameter of x and the section is a circle with the same bottom.
Third, the basic introduction
There is a fixed line and a moving line in the same plane. When a plane rotates once around a fixed line, the surface formed by the moving line is called the rotating surface, the fixed line is called the axis of the rotating surface, and the moving line is called the generatrix of the rotating surface. If the generatrix is a straight line parallel to the axis, then the generated surface of revolution is called a cylindrical surface. If two planes perpendicular to the axis are used to cut the cylindrical surface, then the geometry surrounded by these two sections and the cylindrical surface is called a straight cylinder, which is called cylinder for short. A cylinder can also be regarded as a rectangle that rotates around one side.
Fourth, characteristics:
A cylinder consists of two bottom surfaces and one side surface.
The two bottom surfaces of a cylinder are exactly the same two circles.
The distance between the two bottom surfaces is the height of the cylinder.
Cylinders have countless heights and axes of symmetry.
The side of a cylinder is a curved surface.
It is the same thickness from top to bottom.
There are countless heights.
The lateral spread is rectangular or parallelogram.
Verb (abbreviation of verb) area formula
Side area of cylinder = bottom circumference x height =Ch.
Surface area of cylinder = side area+bottom area x2.
S=2πr^2+Ch
Volume of cylinder = bottom area x height V=πrh/V=Sh
Relationship between cylinder and cone: the volume of a cylinder with equal bottom and equal height is 3 times that of a cone, and the volume of a cone with equal bottom and equal height is 1/3 of that of a cylinder.
Six, application examples
After the cylinder was cut off by 5cm, the surface area of the cylinder decreased by 31.4cm2. What is the surface area of the original cylinder in square centimeters?
Original cylinder: 20 cm high.
Answer:
The radius of the cylinder is: 31.4 ÷ 5 ÷ 3.14 ÷ 2 =1(cm).
The original surface area of the cylinder is/kloc-0 /×/kloc-0 /× 3.14× 2+1× 2.14× 20 =131.88 (square
Cut the 48 cm long cylinder into two small cylinders according to the ratio of 5:3. After cutting, the surface area increased by 7 square centimeters. Find the volume of a long cylinder.
Answer:
Cut into two small cylinders, then the cross section is round, there are two, one is 7 ÷ 2 = 3.5 square centimeters. A large cylinder accounts for 5/8 of the whole cylinder, that is, the height of 48×5/8=30 cm. The bottom area of 3.5 is 105 cubic centimeter, which is the volume of a large cylinder.
You can also use two circles plus four rectangles, which is equal to the volume of a cylinder.