5 Examples of Arc Length

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 = (π * r) / 2, where “r” is the radius

L = (π * 4) / 2 L = (4π) / 2 L = 2π

So, the arc length of the quarter circle is 2π 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 = (π * r), where “r” is the radius

L = π * 5 L = 5π

So, the arc length of the semicircle is 5π 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 = (θ/360) * 2πr, where “θ” is the central angle, and “r” is the radius

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

So, the arc length of the circle sector is (7/3)π 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 = rθ,

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

First, convert 45 degrees to radians: θ = (45/180) * π θ = (1/4) * π

Now,

L = 10 * (1/4) * π L = (10/4)π L = (5/2)π

So, the arc length is (5/2)π 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 = rθ,

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

First, convert 30 degrees to radians: θ = (30/180) * π θ = (1/6) * π

Now,

L = 100 * (1/6) * π L = (100/6)π L = (50/3)π

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