What is a fan? Do we often cut a cake when you celebrate your birthday? The piece cut off is a 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×πr? . If the vertex angle is in radians, it can be simplified as radius times arc length times 1/2 (arc length = radius times radians).
Sector area formula: s sector =(lR)/2(l is the arc length of the sector) =( 1/2)θR? (θ is the central angle in radians)
S fan =(n/360)πR？ ,
S fan = 1/2lr (when the arc length is known) (n is the degree of central angle, and r is the sector radius).
Note: π is pi, which is about 3. 14 15926535, generally 3. 14.
R is the sector radius, n is the degree of the central angle of the arc, π is π,
You can also divide the area of the circle where the sector is located by 360 and multiply it by the angle n of the central angle of the sector.
S=nπR？ /360,
S= 1/2LR .(l is arc length and r is 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.
Compared with the triangle, the sector is similar in overall shape. Three vertices, two sides and their included angles are the same, but one side of the triangle becomes an arc. Because "arc" is a curve, it is more difficult to calculate, understand and think about the sector area. Although there are difficulties, we believe that since the area formulas of sector and triangle are the same in mathematical structure, there must be an inevitable connection between them in essence.
As shown in figure 1, the area of 1 triangle is used to approximate the sector area, and the error is relatively large.
As shown in Figure 2, the sector is divided into two small sectors on average, and the sum of the two triangle areas is used to approximate the sector area, so the error becomes smaller.
We imagine that if the sector is divided into four small sectors on average, and the sum of the four triangle areas is used to approximate the sector area, the error will be smaller.
Let's imagine that if n is large enough to divide the sector into n small sectors on average, then approximate the sector area with the sum of n triangular areas, and the error can be small enough. Therefore:
Through the previous thinking and calculation process, we can see that when comparing the triangle area formula, the reason why the arc is equivalent to the bottom and the reason why the radius is equivalent to the height in the sector area formula can be seen, and we can see the subtleties of the process of arc transforming to the bottom, radius transforming to the height, curve transforming to the straight line and "replacing the curve with the straight line".
What I want to explain here is that when I study advanced mathematics in the future, I will understand that the full approximation mentioned above will become completely equal. It is necessary and important for us to cultivate some basic mathematical ideas and methods in junior high school.
The famous educator Suhomlinski said, "Think, don't memorize". When he was a principal, he asked teachers not only to teach students knowledge, but also to strengthen students' thinking training, and to pay attention to and try to solve the problem of "how to make students learn to think".
We should also pay attention to the two fan-shaped area formulas mentioned above and try to solve the problem of "how to make students learn to think". If you memorize by rote, you can't remember it well, and you can't use it flexibly. If we understand the various reasoning ideas and methods of the formula after some thinking, it will not only help us remember, but also fundamentally increase the accumulation of mathematical ability and literacy, so as to find the correct doorway to avoid mechanical sea tactics and effectively improve our mathematical achievements.