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CIRCLE – LENGTH OF AN ARC
Circle – length of an arc of a circle. Knowledge of circles forms the basics of learning all about circles, circle geometry, etc. In this article, you will learn the definition of the whole terms related to circle – arc, sectors, segments chord, perimeter, radius, diameter, and lots more. You will also learn how to calculate the length of an arc of a circle.
A circle is all parts in the same plane that lies at an “equal distance” from a center point. The distance around this path is called the circumference of the circle. The straight line from the center of the circle to any point on the circumference is called the radius. The radius is half the diameter of any circle.
A chord is a line segment that has its endpoints on the circular border but does not pass through the mid-point. If the chord divides the circle into two equal parts, the chord is called a diameter. And each of the parts is called a semi-circle. See the figure below for a true representation of all these terms.
CIRCLE – DEFINITION OF TERMS
Having considered these few terms, let’s delve a little deeper into the breakdown of complex terms. We will now consider segments, sectors, arcs, etc.
When a chord divides a circle into two unequal parts, the smaller part is called the ‘minor segment‘ and the bigger part, the ‘major segment‘. The figure below shows a description of the above-stated segments in a circle.
An ‘arc’ is part of the circumference of a circle. In a circle, we have two arcs namely – the minor arc and the major arc.
Minor Arc: is an arc that is less than half of the circumference of a circle while,
Major Arc: is more than or greater than half of the circumference of a circle.
From the above diagram, APB is the ‘minor arc’ while ACB is the ‘major arc’. Note that the process of finding the major and minor arcs of a circle is the same. The only difference is the angle each subtends at the center of the circle.
THE SECTOR OF A CIRCLE
The sector of a circle is a plane figure bounded by two radii (plural of radius) and an arc. The yellow shaded part in the figure below is the minor sector of the circle and the blue shaded part is the major sector of the circle.
DIFFERENCE BETWEEN A SECTOR AND A SEGMENT OF A CIRCLE
The main difference between a sector and a segment is that: A sector is bounded by an arc and two radii while a segment of a circle is a region bounded by a chord and an arc.
The above explanations and diagrams have consolidated your knowledge of all the terms on the circle. I am now convinced that you will be able to tell which part of the circle you are been requested to find in any question on circle. Let’s consider some explanations and calculations on the length of an arc of a circle.
THE LENGTH OF AN ARC OF A CIRCLE
The length of an arc as show in the diagram below is written mathematically as,
Where θ = Angle subtended at the center of the circle (in degree) and The radius of the circle.
In cases where θ (angle) is in radian, then the Length of an arc will be
Where θ = radian measure of the angle subtended at the center and radius of the circle.
Let us consider three examples on the length of an arc of a circle. These examples, will show how to calculate the length of an arc in degree, radian and how to find the radius of a circle if given the length of an arc,
Example 1: Find the length of an arc of a circle of radius 7cm which subtends an angle 
at the center of the circle. {Take π 
}.
Given the parameters, θ
You See, this is just a piece of Cake! Let’s consider another example, this time around, the angle will be in radian.
Example 2: Find the length of an arc which subtends an angle of 
π radians at the center of the circle o and is of radius 
. {Take π}.
Given the parameters, , θ =
π
π
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Example 3: Find the radius of a circle which subtends an angle of 
at the centre of the circle and is of length 
. {Take π}.
Given the parameters, length of arc = 2.8cm, θ and
So
Cross multiply,
≅
Wow! Great solution! As you prepare for any examinations, expect a tricky questions like this.
Note: The sum of the lengths of the major and minor arc of a given circle is equal to the circumference of the circle. If the lengths of both the circumference and any arcs are known, the length of the other arc can be found.
Watch out for our next article, which will discuss the perimeter and area of sector and the segment of a circle. Meanwhile, click any of the following articles to learn more on these topics: Indices in Mathematics, Quadratic Equation, Logarithm laws and examples and the Standard form of a numbers.
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