Calculating Arc Length | Formulas and Methods for Different Curves

Arc Length Formula

The arc length formula is used to calculate the length of a curve or a portion of a curve

The arc length formula is used to calculate the length of a curve or a portion of a curve. It is particularly useful in geometry and trigonometry when dealing with circles or other curved shapes.

The formula for finding the arc length of a curve depends on the type of curve being considered. Here are the most common formulas for different types of curves:

1. Arc Length of a Circle:
For a circle with radius (r) and an angle (θ) measured in radians, the arc length (s) is given by the formula:
s = rθ

Here, θ is the central angle subtended by the arc, and the value of θ should be in radians for this formula to work. If you have the angle in degrees, you can convert it to radians by multiplying by π/180.

2. Arc Length of a Sector:
A sector is a portion of a circle enclosed by two radii and an arc. To find the arc length of a sector, you can use the formula:
s = (θ/360) × 2πr

Here, θ is the central angle of the sector measured in degrees, and r is the radius of the circle.

3. Arc Length of a Parametric Curve:
Parametric curves are a set of equations that represent a curve in terms of parameters. If the parametric equations for a curve are given by x = f(t) and y = g(t), where t belongs to an interval [a, b], the arc length (s) of the curve can be calculated using the following formula:
s = ∫(a to b) √[f'(t)^2 + g'(t)^2] dt

Here, f'(t) and g'(t) represent the derivatives of f(t) and g(t), respectively.

Note: The arc length formula for a parametric curve is based on the concept of integration, so it involves evaluating an integral to find the arc length.

It’s important to remember that these formulas provide an approximate arc length and may need to be adjusted based on the degree of accuracy required and the nature of the curve being considered. If a more precise value is needed, advanced mathematical techniques or numerical methods may be necessary.

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