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Chapter 9: Area of Plane Figures

    Introduction

    Area of a Trapezium

    Area of a Quadrilateral

    Area of a Rhombus

    Surveying the Field

    Area of a Polygon

    Area of a Circle

    Area of a Circular Path or Area of a Ring

    Length of the Arc

    Area of Sector

    Exercise 9.1

    Exercise 9.2

    What We Have Discussed?

    Easy Level Worksheet

    Moderate Level Worksheet

    Hard Level Worksheet

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Chapter 9: Area of Plane Figures > Length of the Arc

Length of the Arc

Let's look at Length of the Arc!

As you can see in the diagram, an arc is a part of the of a circle, and a sector is a part of the of a circle.

The arc between two points A and B is often written as AB⌒. This definition is slightly ambiguous: there is a second arc that connects A and B but goes the other way around the circle.

The smaller of the two arcs is called the minor arc, and the larger one is called the major arc. If points A and B are exactly opposite each other, both arcs have the same length and are .

To find the length of an arc or the area of a sector, we need to know about the corresponding angle at the center of the circle: this is called the central angle.

Notice how the arc, sector and angle all take up the same proportion of a full circle. For example, if the central angle is , it takes up of a full circle.

This means that the length of the arc is also 14 of the whole circumference of the circle, and the area of the sector is 14 of the whole area of the circle.

We can express this relationship in an equation:

arc lengthcircumference=circle area=central angle

Now we can rearrange these equations to find whichever variable we’re interested in. For example,

arc length = circumference×c360 = 2πr×c360

where r is the radius of the circle, and c is the size of the central angle.

If the central angle is measured in radians rather than degrees, we can use the same equations, but have to replace 360° with :

arc length = 2πr×c2π = r×c

Notice how the equations become much simpler, and π cancels out everywhere. This is because, as you might recall, the definition of radians, is basically the length of an arc in a circle with radius 1.