What is the formula for area of a sector?

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A sector is a section of a circle bounded by its two radii and the arc that connects them. A semi-circle, which represents half of a circle, is the most frequent sector of a circle. A sector of a circle is a pie-shaped section of a circle formed by the arc and its two radii. A sector is produced when a section of the circle’s circumference (also known as an arc) and two radii meet at both extremities of the arc.

In the above diagram, the minor sector is the section of the circle OAXB, whereas the large sector is the portion of the circle OAYB.

The area of a sector of a circle is the amount of space occupied inside a sector of a circle’s border. A sector always begins at the circle’s centre. The semi-circle is likewise a sector of a circle; in this case, a circle has two equal-sized sectors.

Formula

A = (θ/360°) × πr2 

where, 

θ is the sector angle subtended by the arcs at the center (in degrees),

r is the radius of the circle.

If the subtended angle θ is in radians, the area is given by, A = 1/2 × r2 × θ.

Derivation

Consider a circle with centre O and radius r, suppose OAPB is its sector and θ (in degrees) be the angle subtended by the arcs at the centre.

We know, area of the whole circular region is given by, πr².

If the subtended angle is of 360°, the area of the sector is equal to that of whole circle, that is, πr².

Apply unitary method to find the area of sector for any angle θ.

If the subtended angle is of 1°, the area of the sector is given by, πr²/360°.

Hence, when the angle is θ, the area of sector, OAPB = (θ/360°) × πr2 

This derives the formula for area of a sector of a circle.

Sample Problems

Problem 1. Find the area of the sector for a given circle of radius 5 cm if the angle of its sector is 30°.

Solution:

We have, r = 5 and θ = 30°.

Use the formula A = (θ/360°) × πr2 to find the area.

A = (30/360) × (22/7) × 52

= 550/840

= 0.65 sq. cm

Problem 2. Find the area of the sector for a given circle of radius 9 cm if the angle of its sector is 45°.

Solution:

We have, r = 9 and θ = 45°.

Use the formula A = (θ/360°) × πr2 to find the area.

A = (45/360) × (22/7) × 92

= 1782/56

= 31.82 sq. cm

Problem 3. Find the area of the sector for a given circle of radius 15 cm if the angle of its sector is π/2 radians.

Solution:

We have, r = 15 and θ = π/2.

Use the formula A = 1/2 × r2 × θ to find the area.

A = 1/2 × 152 × π/2

= 1/2 × 225 × 11/7

= 2475/14

= 176.78 sq. cm

Problem 4. Find the angle subtended at the centre of the circle if the area of its sector is 770 sq. cm and its radius is 7 cm.

Solution:

We have, r = 7 and A = 770.

Use the formula A = (θ/360°) × πr2 to find the value of θ.

=> 770 = (θ/360) × (22/7) × 72

=> 770 = (θ/360) × 154

=> θ/360 = 5

=> θ = 1800°

Problem 5. Find the area of a circle if the area of its sector is 132 sq. cm and the angle subtended at the centre of the circle is 60°.

Solution:

We have, θ = 60° and A = 132.

Use the formula A = (θ/360°) × πr2 to find the value of θ.

=> 132 = (60/360) × (22/7) × r2

=> 132 = (1/6) × (22/7) × r2

=> r2 = 252

=> r = 15.87 cm

Now, Area of circle = πr2 

= (22/7) × 15.87 ×15.87

= 5540.85/7

= 791.55 sq. cm

Related Pages
Circles, Sectors, Segments
Area Of Circles
More Geometry Lessons

The following table gives the formulas for the area of sector and area of segment for angles in degrees or radians. Scroll down the page for more explanations, examples and worksheets for the area of sectors and segments.

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.

The following diagrams give the formulas for the area of circle and the area of sector. Scroll down the page for more examples and solutions.

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 derive the formula for the area of sector when the central angle is given in degrees.

This formula allows 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).

Solution:

Area of sector = 60°/360° × 25π
    = 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 cm2. Calculate the angle of the sector. (Take π = 3.142).

Solution:

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.

Example:
Janice needs to find the area of the red section of the circular table top in order to buy the right amount of paint. What is the area of the red section of the circular table top?

Solution:
Step 1: Find the area of the entire circle using the area formula A = πr2.
Step 2: Find the fraction of the circle by putting the angle measurement of the sector over 360°, the total number of degrees in a circle.
Step 3: Multiply the fraction by the area of the circle. Leave your answer in terms of π.

• Show Video Lesson

How To Calculate The Area Of A Sector Using The Formula In Degrees And The Missing Radius Given The Sector Area And The Size Of The Central Angle?

Example 1: Find the area of the shaded region.

Example 2: Find the radius of the circle if the area of the shaded region is 50π

• Show Video Lesson

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.

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)

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

Example:
A lawn sprinkler located at the corner of a yard rotates through 90° and sprays water 30ft. What is the area of the sector watered?

• Show Video Lesson

How To Determine The Area Of A Sector?

The formula is given in radians.
How to determine the area of a segment? (the area bounded by a chord and an arc).

Example 1: Find the area of the sector of a circle with radius 8 feet formed by a central angle of 110°

Example 2: Find the area of the shaded region in the circle with radius 12cm and a central angle of 80°.

• Show Video Lesson

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.

How To Derive The Area Of A Segment Formula?

How do you find the area of a segment of a circle?

• Show Video Lesson

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.

• Show Video Lesson

How To Calculate The Area Of Sector And The Area Of Segment?

It uses the sine rule to calculate the area of triangle.

• Show Video Lesson

Area Of Segment (Angle In Radians)

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

• Show Video Lesson

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