Extended knowledge
The derivation of circular area is only a simple discussion of the area of a basic geometric shape. However, in real life and mathematical research, the concept of circular area has been extended to a wider field. Here are some extended knowledge about circular areas.
1, area of circle: Sometimes, we need to consider the area of a circle, that is, the area where a circle cuts into another circle to form a ring. Assuming that the radius of the outer circle is (r) and the radius of the inner circle is (r), the area of the ring can be expressed as: [text {area} text {ring} = pi (R2R2)].
This formula is obtained by subtracting the area of the inner circle from the area of the outer circle. The concept of circular ring is widely used in engineering, physics and design.
2. Ellipse area: an ellipse is a generalized form of a circle, and its shape is determined by two axes, namely, the long axis and the short axis. The area formula of an ellipse is [text{ area }text{ ellipse }=piab], where (a) is the length of the major axis of the ellipse and (b) is the length of the minor axis. The area formula of ellipse is similar to that of circle, but the length of axis needs to be considered additionally.
3. Sector area: In practical problems, we may only care about a part of the circle, not the area of the whole circle. At this time, the calculation of sector area is involved. A sector is an area surrounded by the center of a circle, two points on a circle and an arc connecting these two points. The formula for calculating the sector area is: [text {area} text {sector} = frac {theta} {360circ} timespir2].
Where (θ) represents the degree of the central angle of the sector. This formula is calculated by the ratio of the central angle of the sector to the whole circle.
4. Solution of pi: In ancient times, people got a mysterious mathematical constant (pi) by studying the ratio of circumference to diameter of a circle. Although we usually take (pi) as 3, 14 159, there are many interesting stories and algorithms in the history of calculating (pi). For example, Archimedes used polygons to approximate circles to calculate (pi).
5. Area of curve: In calculus, a circle can also be described by integral. Using the method of calculus, we can find out the area surrounded by curves. For a circle, the area formula can be obtained by integration: [text {area} text {circle} = int {r} {r} sqrt {R2X2}, dx] This integration represents the area between the curve (y = sqrt {r 2x 2}) and the (x) axis.
In the continuous development of mathematics, the concept of circular area has been popularized and deepened, involving more mathematical principles and applications. These expanded knowledge not only plays an important role in the field of pure mathematics, but also plays a key role in the application fields such as physics and engineering.