Understanding Arc Length and Area of Segments in Radians

Comprehensive Definition, Description, Examples & Rules 

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Arc Length: The arc length and area of segments in geometry correspond to portions of a circle in geometry. The measurement of distance that lies along the curved line of a circular arc or a portion of a curve is called an arc length. You can calculate the length of an arc (s) of a circle with radius (r) and a subtending angle (θ) in radians, by making use of the following formula:

s = rθ

The concept of arc length is very important for various real-life applications in an assorted range of fields. For example:

  • Arc length calculations are made use of by engineers to design curved structures like bridges, roads, pipelines, etc. 
  • The length of the arc is used in today’s modern world for computer graphics and the development of several online games. 
  • Arc length can be used to decide on the exact length of curved pathways like trajectories in physics. 

Area of Segments: The region enclosed by a chord and the corresponding arc in a circle is called the area of a segment of a circle. The area of a segment of a circle (A) with radius (r) and a subtending angle (θ) can be calculated in radians with the help of the following formula:

A = (1/2) * r^2 * (θ – sin(θ)).

The concept of the area of a segment of a circle is also very crucial to everyday activities like:

  • Surveying to find out land areas and boundaries of properties.
  • Calculating the total area of irregular shapes.
  • Determining proportions in statistics and data visualisation. 

Arc Length

Arc length is what we call the total measure of the distance along the curved line of a circular arc or portion of a curve. A fundamental concept in geometry and trigonometry, arc length allows us to determine the amount of circumference of a circle or curved path covered by a given angle, which comes in handy in various fields like physics, navigation, engineering, etc. 

The arc length formula is written as s = rθ

Here,

  • ‘s’ corresponds to the length of the arc you’re supposed to find. 
  • ‘r’ corresponds to the radius of the given circle, i.e. the total distance from the centre of the circle to the starting and end points of the arc. 
  • θ corresponds to the subtending angle in radians, i.e. the angle formed at the centre of the circle by the arc. 

Here is an illustrative example to help you understand the arc length formula better:

Find the arc length of a circle with a radius of 10 m and a subtending angle of 1.5 radians. 

Solution: substituting the given values in the arc length formula, we get:

r= 10 m

θ=1.5 radians

Hence, arc length= rθ

s=10*1.5

s=15 m

Therefore, the arc length of a circle with a radius of 10 m and a subtending angle of 1.5 radians, is 1.5 metres. 

Comparison of Arc Length Formula in Degrees and Radians

We can also represent the arc length equation in degrees. The formula for finding the length of an arc in degrees is given as:

s= θ/360*2πr

Here,

‘s’ corresponds to the arc length

‘r’ corresponds to the radius of the circle

‘θ’ corresponds to the subtending angle in degrees

If we compare the two formulas, when θ is in radians, the arc lengths formula radians can help us find the arc length based on the angle in radians. Whereas when θ is in degrees, we first need to convert the angle into radians by multiplying it with π/180 and then calculate the arc length. 

Area of Segment Formula in Radians

In a circle, a segment refers to the region that is enclosed by a chord, i.e. a line segment that connects two points on the circumference of the circle, and the corresponding arc, i.e. the portion of the circumference of the circle, between those two points. 

To find the area of a segment in radians, we can use the area of segment formula radians:

A= 1/2r^2 [θ – sin (θ)]

Here, 

‘A’ corresponds to the area of the segment of the circle

‘r’  corresponds to the radius of the circle

‘θ’ corresponds to the subtending angle in radians

Sin (θ) is the sine of the subtending angle θ

Step-by-Step Process of Calculating the Area of Segment Radians

Here is how you can go about calculating the area of a segment of a circle in a few easy steps:

Find the area of a segment of a circle of radius 5 metres and the subtending angle of 1.2 radians. 

Solution:

  • Note the given values: r= 5 metres, θ= 1.2 radians
  • Put the above values in the formula to find the area of the segment in radians: A= 1/2r^2 [θ – sin (θ)]. Hence, we get, A= ½ (25) [1.2- sin (1.2)], which is equal to: A= ½ (25) (1.2- 0.9324), or A= ½ (25) (0.2676)
  • Therefore, the area of the segment is approximately 3.345. 

Areas Where Segment Area Calculations Are Useful

Segment area calculations have a lot of applications in the real world. Some of them are listed below:

  • Segment area calculations come in handy in architecture when trying to build circular elements like arches, domes, windows, etc. 
  • Segment area calculations are used by artists to create aesthetically pleasing designs in their work. 
  • Segment area calculations are made use of when trying to determine property boundaries by determining areas of land.

Length of Arc vs. Area of Segment

While finding an arc is crucial to both formulas, the arc length can be used to denote the measure of the total distance along the curved line of a circular arc or portion of a curve, and the area of a segment is used to refer to the region enclosed by a chord and the corresponding arc between those two points. 

Have a look at the following example to understand the difference better:

If we consider a circle of radius 8 metres and a subtending angle of 1.2 radians, the arc length would be the actual length along the curved path of the circle between two given points, A and B on the circle. On the other hand, the area of the segment would be the region that is covered by the chord AB and the curved arc.

Situations where you would use one calculation over the other

  • In scenarios where you have to find the total distance along a circular path, use the arc length formula. For example, calculating the distance travelled by an object along a circular path, deciding how much fencing you need to enclose a road that curves along a circular path, etc.  
  • In scenarios where you have to find the total area enclosed in a curved portion, use the segment area formula. For example, determining the area of a plot of land that is bounded by a circular fence. 

Quadrant Formula

One of the four equal parts into which a circle is divided by two perpendicular and intersecting lines is known as a quadrant of a circle. Each quadrant has an angle of 90 degrees or π/2 radians. 

The Quadrant Formula and its Significance

The quadrant formula can be used to find the coordinates of a point on the unit circle, i.e. a circle with a radius of 1 unit, and a subtending angle θ in radians, especially in geometry and trigonometry. The quadrant formula provides a straightforward way to compute trigonometric problems involving its fundamental functions like sine, cosine and tangent, and understanding the relationship between the unit circle and angles. 

Take a look at the following examples of using quadrant formula in radians:

  • What would be the coordinates of the point on the unit circle corresponding to an angle of π/4 radians or 45 degrees?

Solution: Using the quadrant formula, 

x= cos(π/4) = √2/2 ≈ 0.7071

y= sin(π/4) = √2/2 ≈ 0.7071 

Hence, the coordinates are 0.7071 and 0.7071 approximately. 

  • What would be the coordinates of the point on the unit circle corresponding to an angle of 3π/2 radians or 270 degrees?

Solution: Using the quadrant formula, 

x= cos(3π/2) = 0

y= sin(3π/2) = -1

Hence, the coordinates are 0 and -1 approximately. 

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Key Takeaways

  1. The measurement of distance that lies along the curved line of a circular arc or a portion of a curve is called an arc length.

  2. The region enclosed by a chord and the corresponding arc in a circle is called the area of a segment of a circle.

  3. The concepts of arc length and area of a segment of a circle are important in various everyday applications like navigation, surveying, architecture, etc.

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Frequently Asked Questions

You can use the formula s = rθ to find the arc length in a circle.

The arc length of a circle denotes a particular portion of the circumference of the circle whereas the circumference of a circle is the total distance around the circle.

You can use the formula A = (1/2) * r^2 * (θ – sin(θ)) to find out the arc length of a segment. 

some practical applications of the arc length formula are navigating, surveying, engineering, etc. 

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