Arc Formula - How to Find Length of an Arc?

Arc Length Formula: Have you ever wondered how to measure just the curved part of a circle? That’s where the arc length formula works. The length of an arc helps us find out how long that curved section is. By using this formula, which involves the circle's radius and the angle of the arc, one can easily calculate the distance along the curve.

Arc length formula works like a handy tool for solving problems related to circles in real life. To explore how the arc length formula works and how you can apply it in different problems, keep reading.

Read More: Fun Math Activities for Kids to Develop Calculation Speed

What is an Arc?

An arc is a curved part of the edge of a circle. It’s like a small section of the circle that connects two points along its circumference. The length of the arc depends on how large the angle is at the center of the circle.

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Definition of an Arc Length

Arc length is the distance along the curved part of a circle or any curved line. It’s like the length of the curved edge of a circle that connects two points. Unlike a straight line that connects the two points directly, the arc length follows the curve, so it’s longer than the straight-line distance between the points, which is called the chord.

To find the arc length of a circle, we use a formula that includes the circle’s radius and the angle of the arc. The formula changes depending on whether the angle is measured in degrees or radians, helping us calculate the exact length of the curved part.

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What is an Arc Length Formula?

The arc length formula helps us calculate the distance along a curved part of a circle, known as an arc. Imagine the length of an arc as the portion of the circle's circumference that connects two points. The full circumference of a circle is given by the formula 2πr (where r is the radius), and the arc length is just a part of that full distance.

When the angle of the arc is measured in degrees, we use the length of arc formula to find the distance along the curve. The formula is based on the fraction of the circle that the arc represents, which is θ/360 (because there are 360° in a full circle).

• To find the arc length, we multiply that fraction by the total circumference: Arc length = (θ/360) × 2πr or Arc length = rθ × π/180.

• If the angle is in radians, the formula becomes simpler: Arc length = rθ.

To sum it up:

• When the angle is in degrees, we use Arc length = rθ × (π/180) to find the length of an arc.

• When the angle is in radians, it’s just Arc length = rθ.

The arc formula helps us determine the length of the arc by using the radius and the angle. By applying this formula of length of arc, we can measure the curved distance in various situations, whether the angle is in degrees or radians.

Arc Formula if θ is in Radians

To find the arc length when the angle is measured in radians, we use a simple formula: 

Arc Length = θ × r

Here, θ is the central angle of the arc in radians, and r is the radius of the circle. This arc formula directly relates the angle and the radius to calculate the arc length, making it easy to find the distance along the curve when the angle is given in radians.

Arc Length Formula if θ is in Degrees

When the angle θ is measured in degrees, the arc length can be found using the formula: 

Arc Length = 2πr × (θ/360°)

Here, r is the radius of the circle, and θ is the angle in degrees. This formula helps calculate the length of an arc by taking the fraction of the full circle that the arc represents.

Methods to Find Arc Length of a Curve

The arc length is the distance along the curved part of a circle. There are different methods to find the arc length depending on the details given, such as the radius, central angle, or sector area. Here are the methods to find an arc length of a curve with examples:

1. Calculating Arc Length with Radius and Central Angle

When the radius of the circle and the central angle of the arc are given in any question, you can directly apply the arc length formula.

• Arc Formula (when the angle is in radians): Arc Length = θ × r,
  where θ is the central angle in radians, and r is the radius.

• Arc Formula (when the angle is in degrees): Arc Length = θ × (π/180) × r,
  where θ is the central angle in degrees, and r is the radius.

 Example: Find arc length if the radius is 5 units and the central angle is 60°.

Solution: Arc Length = θ × (π/180) × r = 60 × (π/180) × 5 = 5.24 units.

Example: Find the arc length if the radius is 6 units and the central angle is 2 radians.

Solution: Arc Length = θ × r = 2 × 6 = 12 units

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2. Calculating Arc Length Without the Radius

In some cases, you can calculate the arc length without directly knowing the radius. This can be done using either the sector area or chord length.

1. Using Sector Area and Central Angle

If the sector area and the central angle are given in the question, you can solve for the radius and then use the arc length formula.

• Sector Area Formula (when θ is in degrees) = (θ/360) × πr²

• Sector Area Formula (when θ is in radians) = (1/2)r²θ

Example: Find the length of an arc when the sector area is 25 square units and the central angle is 2 radians.

Solution:

Step 1: Use the sector area formula: (1/2)r²θ = Sector Area
(1/2)r²(2) = 25
r² = 25
r = 5 units

Step 2: Use the arc length formula: r × θ = 5 × 2 = 10 units.

Example: Find the length of an arc when the sector area is 40 square units and the central angle is 90°.

Solution:

Step 1: Use the sector area formula = (θ/360) × πr²
40 = (90/360) × πr²
40 = (1/4) × πr²
r² ≈ 40 × 4 / π
r² ≈ 50.93
r ≈ 7.14 units

Step 2: Use the arc length formula = r × (π/180) × θ

= 7.14 × (π/180) × 90 ≈ 11.28 units.

2.  Using Chord Length and Central Angle

You can also calculate the arc length if you know the chord length and central angle by solving for the radius and then applying the arc length formula.

• Chord Length Formula = 2r × sin(θ/2)

Example: Find arc length when chord length is 5 units and central angle is 2 radians.

Solution:

• Chord Length = 5 units

• Central Angle = 2 radians

Step 1: Use the chord length formula: 2r × sin(θ/2) = Chord Length
2r × sin(1) = 5
r = 5 / (2 × sin(1)) ≈ 2.97 units

Step 2: Use the arc length formula: r × θ = 2.97 × 2 ≈ 5.94 units.

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3. How to Find Arc Length Without the Central Angle?

If the details about the central angle are not provided but the radius or sector area is given, you can calculate the angle and then use the arc length formula.

a) Calculating the length of an arc using radius and sector area:

If you have the radius and sector area, you can solve for the central angle and then find the arc length.

For example: If the sector area is 25 square units with a radius of 5 units, what will be the arc length?

Solution:

• Sector Area = 25 square units

• Radius = 5 units

Step 1: Use the sector area formula: (1/2)r²θ = Sector Area
(1/2)(5)²θ = 25
12.5θ = 25
θ = 2 radians

Step 2: Use the arc length formula: r × θ = 5 × 2 = 10 units 

Arc Length = 10 units.

b) Calculating Arc Length Using Radius and Chord Length

If the radius and chord length are given in the question, you can solve for the central angle using the chord formula and then calculate the arc length.

For example: If the chord length is 3 units and the radius is 2 units, what will be the length of an arc?

Solution:

• Chord Length = 3 units

• Radius = 2 units

Arc Length Formula: Have you ever wondered how to measure just the curved part of a circle? That’s where the arc length formula works. The length of an arc helps us find out how long that curved section is. By using this formula, which involves the circle's radius and the angle of the arc, one can easily calculate the distance along the curve.

Arc length formula works like a handy tool for solving problems related to circles in real life. To explore how the arc length formula works and how you can apply it in different problems, keep reading.

Read More: Fun Math Activities for Kids to Develop Calculation Speed

Practice Questions on How to Find Length of an Arc?

Now that you’ve learned about the arc length formula and seen examples of how to apply it, it’s time to assess your understanding. Try solving the practice questions provided below to check how well you’ve understood the concept of finding the length of an arc:

Practice Question 1: The radius of a circle is 8 units, and the central angle of the arc is 45°. Find the arc length. (Hint: Use the arc length formula for when the angle is in degrees.).

Practice Question 2: The central angle of an arc is 3 radians, and the radius of the circle is 10 units. Calculate the arc length. (Hint: Use the arc formula for when the angle is in radians.)

Practice Question 3: A sector area is 36 square units, and the central angle is 60°. Find the arc length of the circle, assuming the radius is unknown. (Hint: Use the sector area formula to first find the radius, then calculate the length with the arc formula.)

Practice Question 4: The chord length of a circle is 6 units, and the radius is 4 units. The central angle is unknown. Find the arc length. (Hint: Use the chord length formula to first find the central angle, then calculate the length with the arc formula.)

Practice Question 5: The sector area of a circle is 50 square units, and the radius is 7 units. Find the arc length of the circle.

Answers:

• Final Answer 1: 7.85 units

• Final Answer 2: 30 units

• Final Answer 3: 8.72 units

• Final Answer: 6.78 units

• Final Answer: 14.28 units

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