The size of a circle is called its area.
Divide the circle into several equal parts, and after cutting, these approximate isosceles triangles can be put together into an approximate parallelogram.
As shown below.
Because the area of parallelogram = base × height, and the area of circle ISR × r×r= ?r2.
The diameter or circumference of a circle is known, and the area of the circle is required. The radius must be found first before the area of the circle can be found.
Formula law
Area of circle = ×
S= ?
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■ Example 1: perimeter of central island garden12.56m.. What is the area of this garden?
■ Idea: First, the radius of the garden is 12.56÷3. 14÷2=2 (meters) through the perimeter, and then the area of the garden is 2 (square) × 3.14 =12 by using the circular area formula.
■ Solution: (12.56÷3. 14÷2)(2 square) ×3. 14.
=4×3. 14
= 12.56 m2
■ Answer: The garden area is 12.56 square meters.
■ Note: It is required to calculate the radius of the circle before calculating the area with the formula.
Example 2: A rope is 62.8 meters long. Use it to form a rectangle, square or circle. Calculate which graph has the largest area.
■ Idea: Use rope to form a figure, and the length of the rope is the perimeter of the figure, that is, the area of three figures is obtained through the perimeter.
When enclosed in a rectangle, we can know that length+width =62.8÷2=3 1.4 (m).
Maximize the area of a rectangle. The closer the values of length and width are, the larger its area will be, so the length is 15.8 cm and the width is15.6cm.. Its area is 15.8× 15.6=246.48 (square centimeter); When a square is enclosed, the side length of the square is 62.8÷4= 15.7 (cm).
Its area is 15.7× 15.7=246.49 (square centimeter); Circle a circle, the radius of the circle is 62.8 ÷ 3.14 ÷ 2 =10 (cm), and the area of the circle is 10(2 square) × 3.14 = 3/kloc-0.
■ Solution: Surround a rectangle:
Length+width =62.8÷2=3 1.4 (cm)
Area: 15.8× 15.6=246.48 cm2.
Form a square:
Side length: 62.8÷4= 15.7 (cm)
Area: 15.7× 15.7=246.49 (square centimeter)
Form a circle:
Radius: 62.8÷3. 14÷2= 10 (cm)
Area: 10(2 m2) ×3. 14=3 14 (cm2)
■ A: The enclosed circle area is the largest.
■ Note: In the plane figure with equal perimeter, the area of the circle is the largest.
■ Example 3: It is known that the diameters of the four circles in the figure below are all 10 cm. Find the area of the shaded part.
■ Thinking and inspiration: directly using the formula, the area of the shadow part in the middle of the square is not easy to calculate. You can see that the blank part in the square is four quarter circles, and the following figure can be obtained by digging and filling:
The shadow area of the original image is equal to the sum of the square area with a side length of 10 cm and the areas of four semicircles (i.e. two circles) with a radius of 5 cm, as shown in the following figure.
■ Solution: 10(2 square) +5(2 square) ×3. 14×2.
= 100+ 157
=257 (square centimeter).
■ A: The shadow area is 257 square centimeters.
■ Note: When answering questions about graphics, it is necessary to cut and supplement them into regular graphics according to the characteristics of graphics to facilitate the answer.
ring
The part between two concentric circles with different sizes is a ring. As shown on the right, the shaded part is a ring.
The symmetry of a ring is very strong. It is a central symmetric figure with the center of the circle as the center of symmetry, and it is also an axisymmetric figure with numerous symmetry axes, and the symmetry axis is the diameter of a great circle.
Annular area
Usually, the area of a circle can be obtained by subtracting the area of a small circle from the area of a big circle, and some can also be obtained by multiplying the result of (R(2 square) -r(2 square)) according to the characteristics of the topic.
Formula law
Annular area
= large circle area-small circle area
S= ?-? =? (-)
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■ Example 1: The cross section of the steel pipe is shown on the right. Its inner circle radius is 2 cm and its outer circle radius is 4 cm. What is its cross-sectional area?
■ Idea: The cross section of the steel pipe is a ring, and the cross section area of the steel pipe is 50.24 (square centimeter) minus the area of the small circle 2 (square centimeter) ×3. 14= 12.56 (square centimeter).
■ Solution: 4(2 square) ×3. 14-2(2 square) ×3. 14=37.68 (square centimeter).
A: Its cross-sectional area is 37.68 square centimeters.
■ Example 2: There is a circular garden with a circumference of 37.68 meters, and a path with a width of 1.5 meters is opened around the garden, as shown below.
Find the area of this road.
Road width is1.5m.
■ Inspiration: The area of this path is the area of the circle. The perimeter of the garden is 37.68m, and it can be found that the garden radius is 37.68÷3. 14÷2=6 (m), and the excircle radius is 6+ 1.5=7.5 (m).
Using the area formula of the ring, we can find that the area of this path is 7.5(2 square) ×3. 14-6(2 square) ×3. 14=63.585 (square meter).
■ Solution:
Garden radius: 37.68÷3. 14÷2=6 (m).
Great circle radius: 6+ 1.5=7.5 (m)
Area of this path:
7.5(2 square) ×3. 14-6(2 square) ×3. 14
=(7.5(2 square) -6(2 square)) ×3. 14
=63.585 square meters
■ A: The area of this path is 63.585 square meters.
■ Note: To find the area of a circular ring, it is generally to find the area of a large circle and a small circle according to known conditions, and the difference between them is the area of the circular ring. The problem is to find the radius from the circumference, and then solve it with the formula of circular area.
■ Example 3: The shadow area below is 20 square centimeters. Find the area of the ring.
■ Thinking: As can be seen from the figure, the two right-angled sides of a large right-angled triangle are the radius r of a big circle, and the two right-angled sides of a small right-angled triangle are the radius r of a small circle. The shadow area is the difference between the area of a large right triangle 1/2r(2 square) and the area of a small right triangle 1/2R(2 square), that is, 1/2R(2 square).
According to the formula S = R (2 square) -r(2 square), we can get that the area of this ring is 40×3. 14= 125.6 (square centimeter).
■ solution: 20 ÷1/2× 3.14 =125.6 (square centimeter)
■ Answer: The area of the ring is 125.6 square centimeters.
■ Note: According to the characteristics of the topic, you can skillfully calculate the area of the circle by using the area formula of the circle.
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