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Area Of A Sector and Segment



In this lesson, we will learn

  • the area of a sector in a circle
  • the formula for area of sector (in degrees)
  • the formula for area of sector (in radians)
  • how to calculate the area of a sector
  • how to calculate the central angle of a sector
  • how to calculate the radius of a sector
  • how to calculate the area of a segment

Related Topics: More Geometry Lessons

Area of Sector

A sector is like a "pizza slice" of the circle. It consists of a region bounded by two radii and an arc lying between the radii.

The area of a sector is a fraction of the area of the circle. This area is proportional to the central angle. In other words, the bigger the central angle, the larger is the area of the sector.

area sector



Formula for Area of Sector (in degrees)

We will now look at the formula for the area of a sector where the central angle is measured in degrees.

Recall that the angle of a full circle is 360˚ and that the formula for the area of a circle is πr2.

Comparing the area of sector and area of circle, we get the formula for the area of sector when the central angle is given in degrees.

area sector formula

where r is the radius of the circle

This formulas allow us to calculate any one of the values given the other two values.

Worksheet to calculate arc length and area of a sector (degrees)

Calculate The Area Of A Sector (using formula in degrees)

We can calculate the area of the sector, given the central angle and radius of circle.

Example:

Given that the radius of the circle is 5 cm, calculate the area of the shaded sector. (Take π = 3.142).

area of sector


Solution:

area of sector formula

Area of sector =

=13.09 cm2

Calculate Central Angle Of A Sector

We can calculate the central angle subtended by a sector, given the area of the sector and area of circle.

Example:

The area of a sector with a radius of 6 cm is 35.4 cm 2. Calculate the angle of the sector. (Take π = 3.142).

Solution:

area of sector formula

Central Angle =

= 112.67°



 



Videos

This video shows how to derive the formula to calculate the area of a sector in a circle.
It explains how to find the area of a sector of a circle. The formula for the area of a circle is given and the formula for the area of a sector of a circle is derived.

The following video shows how we can calculate the area of a sector using the formula in degrees.

This video gives some examples of how to work with sectors of a circle.  The formula used is in degrees.


 



Calculate the radius of a sector

This video shows how to find the missing radius after you are given the sector area and the size of the central angle.

Formula for Area of Sector (in radians)

Next, we will look at the formula for the area of a sector where the central angle is measured in radians. Recall that the angle of a full circle in radians is 2π.

Comparing the area of sector and area of circle, we get the formula for the area of sector when the central angle is given in radians.

formula area sector

where r is the radius of the circle

This formula allows us to calculate any one of the values given the other two values.

Worksheet to calculate arc length and area of sector (radians)

Calculate the area of sector (using formula in radians)

The following video shows how we can calculate the area of a sector using the formula in radians


 



This video provides an example of determining the area of a sector and the area bounded by a chord and an arc. The formula is given in radians.


Area of Segment (angle in degrees)

The segment of a circle is a region bounded by the arc of the circle and a chord.
The area of segment in a circle is equal to the area of sector minus the area of the triangle.

This video shows how to derive the area of a segment formula.
How do you find the area of a segment of a circle


This video gives three examples of how to calculate the area of segments of circles. It uses half the product of the base and the height to calculate the area of the triangle.

This video shows how to calculate the area of sector and the area of segment. It uses the sine rule to calculate the area of triangle.


 



Area of sectors & segments and arc length of circles (GCSE maths) in degrees
Definition of arc, segment and sector, Relationship to circle, Area of Sector, Area of Segment, Arc Length, Mixed problems with arcs

Area of segment (angle in radians)


Finding the area of a segment (angle given in radians)

Area of a segment from a circle (angle in radians)

Finding radius given Area of Segment


 



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