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# 5 Examples of Arc Length

October 18, 2023
written by Rida Mirza

Arc length is a fundamental concept in geometry and trigonometry, representing the distance along the curved line of a circle or any other curved shape.

In this article, we will explore ten solved examples of arc length in mathematics.

## Examples of Arc Length

These are 5 examples of arc length.

### 1: Finding the Arc Length of a Quarter Circle

#### Example

Given a quarter circle with a radius of 4 units, find the arc length from the starting point to the end.

#### Solution

The formula for calculating the arc length of a quarter circle is,

L = (Ï€ * r) / 2, where “r” is the radius

L = (Ï€ * 4) / 2 L = (4Ï€) / 2 L = 2Ï€

So, the arc length of the quarter circle is 2Ï€ units.

### 2: Calculating the Arc Length of a Half Circle

#### Example

Given a semicircle with a radius of 5 cm, find the arc length from the starting point to the end.

#### Solution

The formula for calculating the arc length of a semicircle is,

L = (Ï€ * r), where “r” is the radius

L = Ï€ * 5 L = 5Ï€

So, the arc length of the semicircle is 5Ï€ cm.

### 3: Arc Length of a Circle Sector

#### Example

Find the arc length of a circle sector with a central angle of 60 degrees and a radius of 7 inches.

#### Solution

The formula for calculating the arc length of a circle sector is,

L = (Î¸/360) * 2Ï€r, where “Î¸” is the central angle, and “r” is the radius

L = (60/360) * 2Ï€ * 7 L = (1/6) * 14Ï€ L = (7/3)Ï€

So, the arc length of the circle sector is (7/3)Ï€ inches.

### 4: Calculating Arc Length Using Trigonometry

#### Example

Given a circle with a radius of 10 units and a central angle of 45 degrees, find the arc length between the two points.

#### Solution

Use the formula L = rÎ¸,

where “r” is the radius and “Î¸” is the central angle in radians

First, convert 45 degrees to radians: Î¸ = (45/180) * Ï€ Î¸ = (1/4) * Ï€

Now,

L = 10 * (1/4) * Ï€ L = (10/4)Ï€ L = (5/2)Ï€

So, the arc length is (5/2)Ï€ units.

### 5: Finding the Arc Length of a Curved Road

#### Example

Given a curved road with a radius of 100 meters and a central angle of 30 degrees, find the arc length of the road between two points.

#### Solution

Use the formula L = rÎ¸,

where “r” is the radius and “Î¸” is the central angle in radians

First, convert 30 degrees to radians: Î¸ = (30/180) * Ï€ Î¸ = (1/6) * Ï€

Now,

L = 100 * (1/6) * Ï€ L = (100/6)Ï€ L = (50/3)Ï€

So, the arc length of the curved road is (50/3)Ï€ meters

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