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A Sector of a Circle is referred to as the pie-shaped portion of a circle consisting of an arc and two radii. A circle is defined as a two-dimensional figure that consists of all points in a plane.
- A sector of a circle is also popular with the name circle sector or disk sector.
- There are two sectors, minor and major sectors.
- The major sector encloses the bigger part of the circle with a larger angle.
- The minor sector encloses the smaller part of the circle with a smaller angle.
- The sector of a circle comprises the perimeter of the circle in the form of the arc it forms with the diameter, i.e. the two radii.
- A sector forming an angle of 180 degrees is called a half disk.
Read More: Geometry Formula
Key Terms: Sector of a Circle, Circle, Major Sector, Minor Sector, Area, Perimeter, Arc Length, Diameter, Radii, Two-dimensional Figure
Sector of a Circle
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A sector of a circle is a part of the circle that is produced when a section of the circumference and two radii meet at both extremities of the arc. It can be compared to shape of a slice of a pizza. A sector of a circle is divided into two parts which are as follows:
Major Sector
Major Sector is also known as bigger area. It is formed when larger angle is enclosed by two radii and an arc. The angle formed by the major sector is greater than 180 degrees. It is the part of the boundary.
Read More: Surface Area and Volume Force
Minor Sector
Minor Sector is also known as smaller area. It is formed when smaller angle is enclosed by two radii and an arc. The angle formed by the major sector is less than 180 degrees.
Area of Sector of a Circle
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Area of a sector of a circle is → \(\Pi\)r2 where r is radius and \(\Pi\) is a constant whole value is 22/7. The whole circular angle will always be 360° in a circle. Therefore, when the angle at the centre of the circle is 360°, then the area of the sector= \(\Pi\)r2
- Applying the unitary method,
- When the angle at the centre is 1°, then the area of the sector will be = \(\Pi\)r2 X 1/360
- Therefore, the area of the sector with the angle at the centre θ = \(\Pi\)r2 X θ/360
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Length of an Arc of Sector of a Circle
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The length of a sector of a circle is →2\(\Pi\)r where r is radius and \(\Pi\) is a constant whole value is 22/7. The angle of the sector is θ° in the circle. Therefore, when the angle at the centre of the circle is 360°, then the perimeter of the circle= 2\(\Pi\)r
- When the angle at the centre is θ °, then the length of the arc XMY will be = 2\(\Pi\)r* (θ/360)
- Applying the unitary method,
- Therefore, the area of the sector with the angle at the centre θ = 2\(\Pi\)r* θ/360
Read More: Constructions
Perimeter of the Sector of a Circle
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There is no need to memorize the formula here but to think logically that the perimeter of the sector should be the length of its constituents i.e the 2 radii ( 2r) and the arc [2\(\Pi\)r X (θ/360)]. Therefore, the perimeter of the sector of the circle = 2r + 2\(\Pi\)r * (θ/360)
Perimeter of the Sector of a Circle
Read More: Areas Related to Circles Revision Notes
Solved Examples of a Sector of a CircleGiven below are the examples of a sector of a circle are as follows: Example 1.The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes. [ 2013 ] Ans. The minute hand of a clock completes a full circle degree in 60 minutes. Swept in 1 hour (60 mints) = 360? Swept in 1 minute = 360/60 = 6? Swept by minutes hand in 5 minutes = 5*6? = 30? Hence \(\theta\)= 300, r = 14cm Area swept by minutes hand = Area of sector Area swept by minutes hand = \(\theta\)/3600 * πr² Area swept by minutes hand = 30/3600 * 22/7 * (14)² Area swept by minutes hand = 1/12 * 22/7 * 14*14 Area swept by minutes hand = 1/12 * 22/1 * 2*14 Area swept by minutes hand in 5 minutes = 154/3 cm² Example 2. The circumference of a circle is 22 cm. Calculate the area of its quadrant. [ 2011 ] Ans. The circumference of a circle is →2 r 2 x 22/7 x r = 22 r = 72 The area of the quadrant is 1/4th of the area of a circle. Therefore, the area swept can be calculated as \(\Pi\)r2 X1 4 22/7 x 72 x 72 x 1 4= 9.625 cm2. Read More: Statistics Example 3. In Figure, find the area of the shaded region. [ 2011 ] Ans. Area of the region can be found as: 30/360( \(\Pi\)52 - \(\Pi\)42 ) = 1/12x\(\Pi\) (25- 16) =33/14 =2.36 cm2 |
Things to Remember
- The sector of a Circle is an enclosed figure of two radii and an arc.
- The larger area has the bigger angle and is the major sector.
- The smaller area has a lesser angle and is called the minor sector.
- The area of the minor sector is lesser than the area of the major sector.
- While finding the perimeter of the circle, make sure to add to the length of the two radii.
Sample Problems
Ques. Find the area of the sector of a circle where the radius is 8 cm and the angle is 60°. Also, find the area of the corresponding major sector.(Use π = 3.14).(3 Marks)
Ans. The formula for the area of the sector is \(\Pi\)r2 * (θ/360) where θ is 60°.
3.14 x 8 x 8 x (60/360) = 33.49 cm2.
The area of the major sector is = \(\Pi\)r2 - area of minor sector
= (3.14 x 64)-33.49
= 167.47 cm2.
Ques. Find the circumference of the sector with a radius of 6 cm and an angle of 72°. π = 3.14. (2 Marks)
Ans. The formula for finding the length of a sector is 2r + 2\(\Pi\)r X θ /360
2 x 6 + 2 x 3.14 x 6 x 72 /360
= 12 + 7.54
= 19.54 cm.
Ques. Find the length of the arc of the circle with a radius of 7 cm and an angle of 36°. Use π = 22/7.(2 Marks)
Ans. The formula to find the length of an arc of a circle is (2\(\Pi\)r) X θ/360
= (2 x 22/7x 7 )x 36/360
= 2 x 22 x 110
= 4.4 cm
Ques. Find the area of the sector of a circle where the diameter is 42 cm and the angle is 45°. (2 Marks)
Ans. The diameter is 42 cm. The radius of the circle = 21 cm.
The formula for the area of the sector is (\(\Pi\)r2 )Xθ/360 where θ is 45°.
22/7 x 21 x 21 x 60 x (45 /360)= 173.25 cm2.
Ques. Find the length of the arc of the circle with a diameter of 7 cm and angle of 72°. (2 Marks)
Ans. The diameter is 7 cm. The radius of the circle = 3.5 cm.
The formula to find the length of an arc of a circle is (2\(\Pi\)r)Xθ/360
= 2 x 22/7 x 3.5 x 72/360
= 4.4 cm
Ques. In Figure, find the area of the shaded region. (3 Marks)
Ans. Area of the region can be found as:
360°-θ/360° x π(R² - r²)
=320°/360° x π[(14)²-(7)²]
=8/9 x 22/7(196-49)
=1232/3
=410.67 cm²
Ques. In the given figure, the shape is given as a sector of a circle with centre O and ∠AOB = 90°. If AO = OB = 42 cm, then find the circumference of the top of the table.(2 Marks)
Ans. Circumference = Length of the arc + 2 radii
= (2\(\Pi\)r )X θ/360+ 2r
= 2 x 22/7x 42 x 270/360 + 2 x 42
= 282 cm
Ques. Find the area of the sector of a circle where the radius is 5 cm and the angle is 60°. Also, find the area of the corresponding major sector.(Use π = 3.14). (3 Marks)
Ans. The formula for the area of the sector is \(\Pi\)r2 * (θ/360) where θ is 60°.
3.14 x 5 x 5 x (60/360) = 13.08cm2.
The area of the major sector is = \(\Pi\)r2 - area of minor sector
= (3.14 x 25) – 13.08
= 65.42 cm2.
Ques. The circumference of a circle is 44 cm. Calculate the area of its quadrant. (2 Marks)
Ans. The circumference of a circle is →2 r
= 2 x 22/7 x r = 44
r = 7
The area of the quadrant is 1/4th of the area of a circle.
Therefore, the area swept can be calculated as \(\Pi\)r2 X1 /4
22/7 x 7 x 7 x 1 / 4= 38.5 cm2.
Ques. Find the area of the sector of a circle where the diameter is 42 cm and the angle is 45°. (2 Marks)
Ans. The diameter is 36 cm. The radius of the circle = 12 cm.
The formula for the area of the sector is (\(\Pi\)r2 )Xθ/360 where θ is 45°.
22/7 x 12 x 12 x (45 /360)= 56.57cm2.
Ques. Find the circumference of the sector with a radius of 8 cm and an angle of 72°. π = 3.14 (2 Marks)
Ans. The formula for finding the length of a sector is 2r + 2\(\Pi\)r X θ /360
2 x 8 + 2 x 3.14 x 8 x 72 /360
= 16 + 10.048
= 26.048 cm.
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