In geometry, an arc is a segment of a curve that lies within a circle. It is defined as the distance between two points along the curved line. The length of an arc is an important measure in geometry and is used in various calculations and applications. Understanding how to find the length of an arc is essential in solving problems involving circles and curved shapes.
What is an Arc?
Before we delve into the concept of arc length, let’s first understand what an arc is. An arc is a portion of a curve that is formed by two points on the curve and the section of the curve that connects them. In other words, it is a segment of a curve that lies within a circle. When we think of a circle, we can imagine an infinite number of arcs that can be formed by connecting different pairs of points on the circumference of the circle.
What is Arc Length?
Arc length is the measure of the distance along the curved line that makes up the arc. It represents the length of the arc and is typically denoted by the letter “s”. It is important to note that the arc length is longer than the straight line distance between its endpoints. This is because the arc follows the curve of the circle, resulting in a longer path.
Arc Length Formula
To calculate the length of an arc, we use the arc length formula. The formula for arc length depends on whether the central angle of the arc is given in degrees or radians. Let’s explore both cases:
Arc Length Formula in Radians
If the central angle of the arc is given in radians, we can use the following formula to calculate the arc length:
s = θ * r
Where:
- s is the arc length
- θ is the central angle of the arc in radians
- r is the radius of the circle
Arc Length Formula in Degrees
If the central angle of the arc is given in degrees, we can use the following formula to calculate the arc length:
s = (θ/360) * 2π * r
Where:
- s is the arc length
- θ is the central angle of the arc in degrees
- r is the radius of the circle
- π is a mathematical constant approximately equal to 3.14159
Arc Length Formula in Integral Form
Another way to express the arc length formula is in integral form. This form is useful when dealing with curves that are not easily represented by simple equations. The integral form of the arc length formula is:
∫[a,b] √(1 + (dy/dx)^2) dx
Where:
- [a, b] represents the interval over which the arc length is being calculated
- dy/dx represents the derivative of the function that defines the curve
Arc Length Equation
The arc length equation is an expression that relates the length of an arc to the radius of the circle and the central angle of the arc. It allows us to calculate the length of an arc without explicitly knowing the angle or the radius. The equation is derived from the arc length formula and is given by:
L = r * θ
Where:
- L is the length of the arc
- r is the radius of the circle
- θ is the central angle of the arc
The arc length equation is a simplified version of the arc length formula and is commonly used in calculations involving arcs.
How To Find Arc Length
Now that we understand the concept of arc length and have explored the formulas and equations associated with it, let’s learn how to find the arc length of a curve. There are various methods depending on the information given. We’ll cover three common scenarios: finding arc length with the radius and central angle, finding arc length without the radius, and finding arc length without the central angle.
How to Find Arc Length With the Radius and Central Angle
To find the arc length when given the radius and central angle of the arc, you can use the arc length formula. Here’s a step-by-step guide:
Step 1: Determine the values of the radius (r) and the central angle (θ) of the arc. Step 2: Plug the values into the arc length formula: s = θ * r for radians or s = (θ/360) * 2π * r for degrees. Step 3: Calculate the arc length using the formula.
Let’s work through an example to illustrate this method:
Example: Find the length of an arc if the radius of the arc is 8 cm and the central angle is 40°.
Solution: Step 1: Radius, r = 8 cm Central angle, θ = 40°
Step 2: Using the arc length formula for degrees, we have: s = (θ/360) * 2π * r s = (40/360) * 2π * 8
Step 3: Calculate the arc length: s = (40/360) * 2π * 8 s ≈ 5.582 cm
So, the length of the arc is approximately 5.582 cm.
How to Find Arc Length Without the Radius
In some cases, you may not be given the radius of the arc. However, you can still find the arc length using the central angle and another given parameter, such as the sector area or the chord length. Here’s how:
Finding Arc Length with Sector Area
To find the arc length when given the sector area, you can use the following steps:
Step 1: Determine the sector area (A) and the central angle (θ) of the arc.
Step 2: Multiply the sector area by 2 and divide the result by the central angle in radians.
Step 3: Take the square root of the division result.
Step 4: Multiply the square root by the central angle to get the arc length.
Let’s work through an example:
Example: Find the length of an arc if the sector area is 12 cm² and the central angle is 60°.
Solution:
Step 1: Sector area, A = 12 cm² Central angle, θ = 60°
Step 2: Calculate the division result: (A * 2) / θ = (12 * 2) / 60
Step 3: Take the square root of the division result: √(24/60)
Step 4: Multiply the square root by the central angle: √(24/60) * 60
So, the length of the arc is approximately 8.37 cm.
Finding Arc Length with Chord Length
To find the arc length when given the chord length, follow these steps:
Step 1: Determine the chord length (c) and the central angle (θ) of the arc. Step 2: Divide the chord length by twice the sine of half the central angle in radians. Step 3: Multiply the division result by the central angle to get the arc length.
Let’s work through an example:
Example: Find the length of an arc if the chord length is 10 cm and the central angle is 30°.
Solution:
Step 1: Chord length, c = 10 cm Central angle, θ = 30°
Step 2: Calculate the division result: (c / (2 * sin(θ/2))) = (10 / (2 * sin(30°/2)))
Step 3: Multiply the division result by the central angle: (10 / (2 * sin(30°/2))) * 30°
So, the length of the arc is approximately 17.32 cm.
How to Find Arc Length Without the Central Angle
In some cases, you may not be given the central angle of the arc. However, you can still find the arc length using the radius and another given parameter, such as the chord length or the sector area. Here’s how:
Finding Arc Length with Chord Length
To find the arc length when given the chord length, follow these steps:
Step 1: Determine the chord length (c) and the radius (r) of the arc.
Step 2: Divide the chord length by the radius.
Step 3: Take the inverse sine of the division result to find the central angle in radians.
Step 4: Multiply the central angle by the radius to get the arc length.
Let’s work through an example:
Example: Find the length of an arc if the chord length is 12 cm and the radius is 6 cm.
Solution:
Step 1: Chord length, c = 12 cm Radius, r = 6 cm
Step 2: Calculate the division result: c / r = 12 / 6
Step 3: Take the inverse sine of the division result: sin^(-1)(12/6)
Step 4: Multiply the central angle by the radius: sin^(-1)(12/6) * 6
So, the length of the arc is approximately 12 cm.
Finding Arc Length with Sector Area
To find the arc length when given the sector area, follow these steps:
Step 1: Determine the sector area (A) and the radius (r) of the arc.
Step 2: Divide the sector area by twice the radius squared.
Step 3: Take the square root of the division result.
Step 4: Multiply the square root by the radius to get the arc length.
Let’s work through an example:
Example: Find the length of an arc if the sector area is 16 cm² and the radius is 4 cm.
Solution:
Step 1: Sector area, A = 16 cm² Radius, r = 4 cm
Step 2: Calculate the division result: √(A / (2 * r^2))
Step 3: Take the square root of the division result: √(16 / (2 * 4^2))
Step 4: Multiply the square root by the radius: √(16 / (2 * 4^2)) * 4
So, the length of the arc is approximately 4 cm.
Solved Examples on Arc Length
Let’s work through a few more examples to solidify our understanding of finding arc length.
Example 1: Find the length of an arc if the radius is 10 cm and the central angle is 45°.
Solution:
Step 1: Radius, r = 10 cm Central angle, θ = 45°
Step 2: Using the arc length formula for degrees, we have: s = (θ/360) * 2π * r s = (45/360) * 2π * 10
Step 3: Calculate the arc length: s = (45/360) * 2π * 10 s ≈ 7.854 cm
So, the length of the arc is approximately 7.854 cm.
Example 2: Find the length of an arc if the sector area is 20 cm² and the central angle is 60°.
Solution:
Step 1: Sector area, A = 20 cm² Central angle, θ = 60°
Step 2: Calculate the division result: √(A / (2 * r^2))
Step 3: Take the square root of the division result: √(20 / (2 * 4^2))
Step 4: Multiply the square root by the radius: √(20 / (2 * 4^2)) * 4
So, the length of the arc is approximately 4.472 cm.
Example 3: Find the length of an arc if the chord length is 8 cm and the radius is 6 cm.
Solution:
Step 1: Chord length, c = 8 cm Radius, r = 6 cm
Step 2: Calculate the division result: c / r = 8 / 6
Step 3: Take the inverse sine of the division result: sin^(-1)(8/6)
Step 4: Multiply the central angle by the radius: sin^(-1)(8/6) * 6
So, the length of the arc is approximately 8.377 cm.
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