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Mathematics-Circle

This is a mind map about circles, including understanding circles, circumference, area, etc. The introduction is detailed and the description is comprehensive. I hope it can help interested friends learn.

Edited at 2023-11-19 15:22:54

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Mathematics-Circle

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Outline

round
- 2
EllaGrace
Equations of Lines and Circles
- 1
21_ethanSawyer

round

Get to know circles

circle in life

wheel

cup

cake

introduction circle

center of circle

The point where the tip of the needle is located is called the center of the circle, usually represented by the letter o.

radius

The line segment connecting the center of the circle to any point on the circle is called the radius. Generally represented by the letter r.

diameter

The line segment that passes through the center of the circle and has both ends on the circle is called the diameter, generally represented by the letter d.

discovery of circle

Fold the circle in half along any diameter so that the two sides overlap.

A circle has an infinite number of radii and an infinite number of diameters.

In the same circle, all radii are equal, all diameters are equal, and the length of the diameter is twice the radius.

circumference of circle

example

The radius of the bicycle's wheels is approximately 33 centimeters. Approximately how far can this bicycle go if its wheels turn once? (Keep the whole meter in the result.) Xiao Ming’s home is 1 km away from the school. When he rides his bicycle from home to school, how many turns does the wheel make?

C＝2πr 2×33*3.14=207.24 (cm)≈2 (meter) 1 kilometer=1000 meters, 1000÷2=500 (circle)

Circumference formula of a circle

Using C to represent the circumference of a circle, we have: C=πd or C=2πr.

Someone has long studied the relationship between circumference and diameter and found that the ratio of the circumference of any circle to its diameter is a fixed number. We call it pi, represented by the letter π. It is an infinite non-repeating decimal, π = 3.14159265535... But in practical applications, only its approximate value is often taken, such as π ≈ 3.14.

sector

As shown in the picture on the left, the part between two points A and B on the circle is called an arc, pronounced "arc AB". The figure enclosed by an arc and two radii passing through both ends of the arc is called a sector. The colored part in the picture is the fan shape.

An angle like angle AOB with its vertex at the center of a circle is called the central angle.

It can be found that in the same circle, the size of the sector is related to the size of the central angle of the sector.

area of circle

If S is used to represent the area of a circle, then the calculation formula for the area of a circle is: S=πr².

example

Each square meter of turf is eight yuan. How many square meters does this circular lawn cover?

Draw a circle on the cardboard and divide the circle into several (even) equal parts. After cutting them, use these small pieces of paper that are similar to isosceles triangles to put them together. You will find that they look like a parallelogram when put together.

The more pieces you divide, the smaller each piece will be, and the closer the shape will be to a rectangle.

As can be seen from the figure above, the radius of the circle is r, the length of the rectangle is approximately C/2 (=πr), and the width is approximately r. Because the area of the rectangle = length × width, the area of the circle = π × r (2) = πr (2).

The diameter of the circular lawn is 20 meters, and the cost is 8 yuan per square meter of turf. How much does it cost to lay full sod?

20÷2=10 (meters), 3.14×10 (2)=314 (square meters), 314×8=2512 (yuan).

Answer: It costs 2512 yuan to cover the entire area with turf.

The silver part of the disc is a ring with an inner radius of 2 cm and an outer radius of 6 cm. What is the area of the ring?

3.14×6(2)-3.14×2(2)=113.04-12.56=100.48 (square centimeters)

3.14×(6(2)-2(2))=3.14×32=100.48 (square centimeters).

Answer: The area of the ring is 100.48 square centimeters.

In Chinese architecture, we often see outer squares, inner circles, and outer circles. The design of the inner square. The radius of the two circles in the picture above is one meter. Can you find the area between the square and the circle?

(1) 2×2=4 (square meters), 3.14×1 (2)=3.14 (square meters), 4-3.14=0.86 (square meters).

(2) (1/2×2×1)×2=2 (square meters), 3.14-2=1.14 (square meters).

Answer: The area between the square and the circle in the left picture is 0.86 square meters, and the area between the circle and the square in the right picture is 1.14 square meters.