环形面积公式
它被定义为半径为 R 和 r 的两个圆的面积之差。 R是大圆的半径，r是小圆的半径。两个圆之间的公共区域称为环带区域，用于找到该区域的公式称为环带公式。它也可以被称为一个圆心在半径 r 上的孔。
环形公式
表示环形公式的阴影区域如下所示：
The radius of the inner circle = ‘r’
Area of the inner circle = πr2
The radius of the outer circle = ‘R’
Area of the outer circle = πR2
The difference is generally calculated by subtracting a small value from the larger one.
Annulus formula is given by:
π (R2 – r2)
Or
π (R – r)(R + r)
(a2 – b2 )= (a – b)(a + b))
The area of the annulus region can be calculated if the radius of the inner and outer circle is known. It is used in various applications, such as finding the area of the circular racing track.
For example,
If R = 5cm
R = 3cm
Annulus formula will be π (R2 – r2)
= π (5 – 3)
= π (25 – 9)
= π (16)
示例问题
问题1：求半径600m到300m的圆形场地之间形成的圆形赛道的面积？
解决方案：
The area of the circular track between the two tracks can be calculated using the annulus formula given by:
π (R2 – r2)
Where,
R = 600
r = 300
= π (6002 – 3002)
= π (360000 – 90000)
= π (270000)
= 848230 m2
Thus, the area of the circular track is 848230 square meter.
问题2：求半径为250m和400m的两个圆之间形成的圆形赛道的面积？
解决方案：
The area of the circular track between the two tracks can be calculated using the annulus formula given by:
π (R2 – r2)
Where,
R = 400
r = 250
R is the radius of outer circle and ‘r’ is the radius of inner circle.
= π (4002 – 2502)
= π (160000 – 62500)
= π (97500)
= 306296 m2
Thus, the area of the circular track is 306296 square meter.
问题 3： 找到直径为 300 厘米的草坪包围的 12 厘米宽的路径面积？
解决方案：
Given:
Width of the path = 12cm
Diameter of the circular lawn = 300cm
Radius of the inner lawn = diameter/2
= 300/2
= 150cm
Radius of outer circle = radius of inner circle + width of the path
= 150 + 12
= 162cm
The annulus formula given by:
π (R2 – r2)
Where,
R = 162
r = 150
A = π (R – r)(R + r)
A = 3.1415 (162 – 150)(162 + 150)
A = 3.1415 ×12 × 312
A =11761.776 cm2
Thus, the area of the path is 11761.776 square centimeter.
问题4：如果环形面积和圆场的宽度分别为3600m 2和10m，求内外圆的半径？
解决方案：
Given: A = 3600m2
Width = 10m
Width = R – r
Where,
R = radius of outer circle
r = radius of inner circle
R – r = 10
R = 10 + r
The annulus formula is given by:
A = π (R – r)(R + r)
Substituting the value of R in the annulus formula, we get:
A = π (10 + r – r)(10 + r + r)
A = π (10)(10 + 2r)
The area of the width or the annulus area is 3600m2. Substituting in the above equation, we get:
3600 = π (10)(10 + 2r)
3600 = π (100 + 20r)
100 + 20r = 1145.9
20r = 1145.9 – 100
20r = 1045.9
r = 52.3m
The radius of outer circle = r + 10
= 52.3 + 10
= 62.3m
Thus, the radii of the inner and outer circle are 52.3m and 62.3m.
问题 5：求一条 20m 宽的小路被直径 200m 的草坪包围的面积？
解决方案：
Given:
Width of the path = 20m
Diameter of the circular lawn = 200m
Radius of the inner lawn = diameter/2
= 200/2
= 100m
The radius of outer circle = radius of inner circle + width of the path
= 100 + 20
= 120m
The annulus formula given by:
π (R2 – r2)
Where,
R = 120
r = 100
A = π (R – r)(R + r)
A = 3.1415 (120 – 100)(120 + 100)
A = 3.1415 ×20 × 220
A =13822 m2
Thus, the area of the path is 13822 square meter.
问题6：如果环形面积和圆场的宽度分别为2400cm 2和15cm，求内外圆的半径？
解决方案：
Given: A = 2400cm2
Width = 15cm
Width = R – r
Where,
R = radius of outer circle
r = radius of inner circle
Difference in the outer and inner radius = width of the field
R – r = 15
R = 15 + r
The annulus formula given by:
A = π (R – r)(R + r)
Substituting the value of R in the annulus formula, we get:
A = π (15 + r – r)(15 + r + r)
A = π (15)(15 + 2r)
The area of the width or the annulus area is 2400cm2. Substituting in the above equation, we get:
2400 = π (15)(15 + 2r)
2400 = π (225 + 30r)
225 + 30r = 764
30r = 764 – 225
30r = 539
r = 18cm (approx)
The radius of outer circle = r + 15
= 18 + 15
= 33cm
Thus, the radii of inner and outer circle are 18cm and 33cm.