如何找到弧的长度？
弧是沿曲线边界的两个半径所对的两点之间的距离。如果圆上的两点正好相反，则形成一个 180 度的内角。否则，形成的角度始终小于 180 度或 π 弧度。圆弧 AB 的示意图如下：
弧长
圆弧 AB 的距离大于点 A 和 B 之间直线的距离。圆弧是圆的圆周的一部分。圆周是圆的另一个边界。周长也可以称为圆的周长。
案例1：给定半径和角度时
The formula to calculate the length of an arc is given by:
L = 2πr × (θ ÷ 360) ⇢ (1)
Where
r is the radius of the circle
θ is the angle in degrees. It is the angle between the two radii forming the arc or the central angle of the arc.
L = Arc length
Arc length when the angle is represented in radians
1 radian = π/180°
Or
1 radian = 2π/360°
Substituting the value of radian in equation (1):
r × (θ× (2π ÷ 360))
L = r θ ⇢ (2)
Where,
r is the radius of the circle
θ is the angle in radians.
案例2：给定面积和角度时
The formula to calculate the length of an arc is given by:
L = 2πr × (θ ÷ 360)
Where,
r is the radius of the circle
θ is the angle in degrees
Or
L = r θ
Where,
r is the radius of the circle
θ is the angle in radians.
We need to find the radius of the circle from the given area. After finding the radius, we will substitute the value of radius in the formula.
Area of the circle = πr2
For example,
If area = 314 m2
πr2 = 314 m2
r2 = 314/π
(π = 3.14)
r2 = 314/3.14
r2 = 100
r = √100
r = 10 m
The length of the arc with angle π radians will be:
L = r θ
L = 10 × π
L = 10 × 3.1415
L = 31.415 m
The value of r can be used in the same formula, as discussed above.
案例3 ：积分形式的弧长
The arc length in integral form is given by:
L = ∫√(1 + (dy/dx)2)dx
The limit of integral is [a, b]
Where,
Y is the f(x) function
示例问题
问题1：求半径为2m、角度为π/2弧度的圆弧的长度。
解决方案：
The formula to calculate the length of the arc is given by:
L = r θ
Where,
L is the length of the arc
Given: r = 2m and θ = π/2 radians
Length of arc = 2 × π/2
Length of arc = π
(π = 3.1415)
Length of arc = 3.1415 m
Thus, the length of the arc is 3.1415 m.
问题 2：求函数f(x) = 8 在 x =2 和 x = 4 之间的弧长。
解决方案：
The formula to calculate the arc length for the function is given by:
L = ∫√(1 + (dy/dx)2)dx
The limit of integral is [a, b]
Substituting the values a = 2, b = 4, and y = 6 or dy/dx = 0 in the above formula,
L = ∫√(1 + (0)2)dx
L = ∫√1 dx
L = ∫1 dx
L = x
(Integral of 1 is x)
The limit of integral is [2, 4]
L = (4 – 2)
L = 2
Thus, the length of the arc of function f(x) = 8 between x = 2 and x = 4 is 2.
问题3：求半径为5cm，角度为60°的圆弧的长度。
解决方案：
The formula to calculate the length of the arc is given by:
L = 2πr × (θ ÷ 360)
Where,
L is the length of the arc
Given: r = 5cm and θ = 60°
Length of arc = 2πr × (60 ÷ 360)
Length of arc = 2πr × 1/6
Length of arc = 2 × 3.1415 × 5/6
(π = 3.1415)
Length of arc = 5.235cm
Thus, the length of the arc is 5.235cm
问题4：求半径为0.5m，角度为π/4弧度的弧的长度。
解决方案：
The formula to calculate the length of the arc is given by:
L = r θ
Where,
L is the length of the arc
Given: r = 0.5m and θ = π/4 radians
Length of arc = 0.5 × π/4
Length of arc = 0.392 m
(π = 3.1415)
Thus, the length of the arc is 0.392 m
问题5：求半径为10cm，角度为135°的圆弧的长度。
解决方案：
The formula to calculate the length of the arc is given by:
L = 2πr × (θ ÷ 360)
Where,
L is the length of the arc
Given: r = 10cm and θ = 135°
Length of arc = 2πr × (135÷360)
Length of arc = (2 × 3.1415 × 10 × 135)/360°
(π = 3.1415)
Length of arc = 23.56cm
Thus, the length of the arc is 23.56cm.
问题 6：求半径为 20mm、角度为 π/6 弧度的圆弧的长度。
解决方案：
The formula to calculate the length of the arc is given by:
L = r θ
Where,
L is the length of the arc
Given: r = 20mm and θ = π/6 radians
Length of arc = 20 × π/6
Length of arc = 10.47 mm
(π = 3.1415)
Thus, the length of the arc is 10.47 mm
问题 7：求半径为 2 厘米，角度为 90°的圆弧的长度。
解决方案：
The formula to calculate the length of the arc is given by:
L = 2πr × (θ ÷ 360)
Where,
L is the length of the arc
Given: r = 2cm and θ = 90°
Length of arc = 2πr × (90 ÷ 360)
Length of arc = 2πr × 1/4
Length of arc = 2 ×3.1415 × 2 × 1/4
(π = 3.1415)
Length of arc = 3.1415 cm
Thus, the length of the arc is 3.1415 cm.