In the first step, we need to review the past three-dimensional figures, and we also know how to calculate the surface area and volume of each three-dimensional figure. We also think about the relationship between cone and cylinder, cuboid and cube, and then we know something new. We need to explore conical cylinders, and their surface areas and volumes are very clear, so the learning target begins to accurately calculate the surface areas and volumes of some cones and conical cones.
We can find the surface area of a thousand spheres. For example, here is a cube. How do we find the surface area of this cube? We can expand the cube, divide it into six squares, and then add up the areas of the squares, so the surface area of the cylinder should be the same. Our method is to expand the cylinder, which is two circles and a rectangle with the same size. We calculate the area of a circle and the area of a rectangle and add them to get the area of a cylinder. Now the radius of our members is R, and the area of the circle is r2 Uyghur (R squared times Uyghur). There are two identical circles, namely 2×r2. Now we have to calculate the area of a rectangle. If we want to calculate the area of a rectangle, it needs to be long and wide. The width is the height h in this original column. What is the length? You will find that the length is the perimeter of the bottom of the besieged city, so the length is the perimeter of the circle, the radius of the circle is R, and the perimeter is 2 r. When the length and width of the rectangle are all available, you will know how to calculate its area, that is, the area is 2 r×h, and the surface area of the cylinder is 2 rh+2r2. That is to say, if you want to know the surface area of the cylinder, you only need to know the radius and height. (manuscript)
We still use the previous method to find the surface area of the cone. The first step is to have a cone and then expand it. If you want to know what the shape of a cone is after expansion, you have to make a cone, which is also a rotating body. What is this? What about the geometry that forms the cone? We have two conjectures, the first is a triangle, and the second is a fan. We tried to conceive that it could be completely surrounded by a cone, so we had to operate it on the spot. I first drew a fan, then drew a triangle and cut it out. Finally, the fan surrounded a cone, so when the cone was unfolded, it was a fan and a circle, which was the bottom of the cone.
We calculate the area of the circle and the area of the sector, and then add up to the surface area of the cone. However, if you want to know the area of circle and sector, you must know the radius of circle and sector. Someone asked if the radius of a circle and a sector would be the same. We know that the line segment between two points is the shortest. We draw a line through the center of the circle, which is the diameter of the circle, but the cone hovers in the air from the beginning. Only at that moment does it mean that the diameter of goodness is not equal to the diameter of the circle, and the radius is certainly not equal, so now we understand that at least two pieces of information are needed, the radius of the circle and the radius of the fan.
Next, we need to calculate the surface area. The area of the circle is the radius of the bottom surface R, and the area is r2. Next, let's look at the area of the sector. The radius of the sector is r, and we use big r to represent it. Because the radius of the sector is different from the radius of the circle, we also need to know the angle of the central angle. Let's assume that the angle of the central angle is n, so the area of the sector is r2 xn/360. Then, add the area of the sector to the area of the circle, that is
However, this rule still has some shortcomings, because we can't directly know the angle of the sector after the cone is unfolded, so we'd better use another method to find the area of the sector, that is, 1/2lR, L, that is, the length of the arc AB. This law has been proved in this field, so it is feasible. Then we need to know the length of arc AB now. In fact, Hu is circling, because this is a city surrounded by foxes. Explain that the circumference of a circle is an arc with a radius of r, the radius of this sector is also called a generatrix, and the area of that sector is, so the area of the sector is 1/22 r×R, and the surface area of the cone is 1/22 r×R+r2.
This rule is obviously better than the first method I found. Now we need to find the volume of cone and cylinder step by step. (manuscript)
In fact, the original is very similar to the area we seek for a circle, except that the circle is two-dimensional and the original is three-dimensional, that is, the original adds a high condition. Now I divide a cylinder, as shown below.
First, we should have a cylinder, confirm its height and radius at the bottom, and then divide the cylinder. In fact, the smaller the score, the closer the picture is, just like a triangle, but we can't divide the auxiliary perimeter into the bottom of the triangle. Let's assume that it is a triangle, then expand it and make these triangles into a cuboid. We know how to calculate the volume of a cuboid, that is, length times width times height. What is the width of a cuboid? When we seek length and width, we think of the area of a circle. The method is exactly the same, except for one height. As shown in Figure 5, how do we form a rectangle? We separate the circumference of the circle and spell it out. The sum of two rectangles is the circumference of a circle. What is the length of this cuboid? , also round, half circumference, so it is 1/22 r, the width is the radius in the figure, r, so what is the height of the cuboid? The height of a cuboid is the height h of a cylinder, and all three conditions are met, so what is the volume? That is, length multiplied by width multiplied by height, that is, 1/22 Uyghur×r×h, let's solve it, that is, 1/22 Uyghur× r× r× h = r× r× h = R2 Uyghur h, and you will find that the bottom area of a cylinder multiplied by height is the volume law of a cylinder.
The volume of the cone, well, it's a little hard to say, because it is also a rotating body, but I think the volume of the cone is related to the volume of the cone with the same bottom area, so I made a pair of cones and cylinders with the same bottom area.
Then we go to the playground to get some soil to make it thinner, otherwise we will have the volume of a cone and a cylinder. Let's fill the cones first, and then fill the soil on the cones. Finally, we visited the original three times, almost full of cylinders, so the volume of the cone may be 65438+ 0/3 of the volume of the cylinder with the same bottom area as him. Actually, it is. We can only prove it through physical experiments.