Area of a sector is the region bounded by the bounding radii and the arc of the sector.

There are two methods to find the area of a sector.

(i) when the central angle and the radius are given

(ii) when length of arc and the radius are given

Area of a sector :

= (θ/360) ⋅ πr^{2} square units

(or)

= (lr/2) square units

θ - central angle formed by the sector

L - length of arc

r - radius of the sector

We can use the first formula if the central angle(θ) formed by the sector and radius are given. If the length of arc(L) are given, we have to use the second formula.

**Example 1 :**

Find the area of the sector whose radius and central angle are 42 cm and 60° respectively.

**Solution :**

**Area of the sector :**

**= (θ/360**°**) **⋅** **π**r ^{2}**

**Substitute r = 42 cm , ****θ = **60°.

= (60°/360°) ⋅ π ⋅ 42^{2}

= (60°/360°) ⋅ π ⋅ 42 ⋅ 42

= (1/6) ⋅ π ⋅ 42 ⋅ 42

= 294π cm^{2}

**Example 2 :**

Find the area of the sector whose radius and central angle are 21 cm and 60° respectively.

**Solution:**

**Area of the sector :**

** = (θ/360**°**) **⋅** **π**r ^{2}**

**Substitute r = 21 cm , ****θ = **60°.

= (60°/360°) ⋅ π ⋅ 21^{2}

= (1/6) ⋅ π ⋅ 21 ⋅ 21

= 73.5π cm^{2}

**Example 3 :**

Find the area of the sector whose radius and central angle are 4.9 cm and 30° respectively.

**Solution :**

**Area of the sector : **

**= (θ/360**°**) **⋅** **π**r ^{2}**

**Substitute r = 4.9 cm , ****θ = 3**0°.

= (30°/360°) ⋅ π ⋅ 4.9^{2}

= (1/12) ⋅ π ⋅ 4.9 ⋅ 4.9

≈ 73.5π cm^{2}

**Example 4 :**

Find the area of the sector and also find the central angle formed by the sector whose radius is 21 cm and length of arc is 66 cm (Use π ≈ 22/7).

**Solution :**

Area of the sector :

= (lr/2) square units

Substitute l = 66 and r = 21.

= (66 ⋅ 21)/2

= 693 cm^{2}

Central angle formed by the sector :

Area of sector = 693 cm^{2}

(θ/360)⋅** **π**r ^{2 }**= 693

Substitute the known values and solve for θ.

(θ/360)⋅** (22/7)**(21)** ^{2 }**= 693

1386θ/360** ^{ }**= 693

77θ/20 = 693

Multiply each side by 20/77.

θ = 180°

**Example 5 :**

Find the area of the sector whose radius and length of arc are 6 cm and 20 cm.

**Solution :**

Area of the sector :

= (lr/2) square units

Substitute l = 6 and r = 20.

= (6 ⋅ 20)/2

= 60 cm^{2}

**Example 6 :**

Find the area of the sector whose diameter and length of arc are 10 cm and 40 cm respectively.

**Solution :**

Radius = Diameter /2

= 10/2

= 5 cm

Area of the sector :

= (lr/2) square units

Substitute l = 40 and r = 5.

= (40 ⋅ 5)/2

= 100 cm^{2}

**Example 7 :**

Find the area of the sector whose radius is 35 cm and perimeter is 147 cm.

**Solution :**

Perimeter of sector = 147 cm

l + 2r = 147

l + 2(35) = 147

l + 70 = 147

l = 77 cm

Area of the sector :

= (lr)/2 square units

Substitute l = 77 and r = 35.

= (77 ⋅ 35)/2

= 1347.5 cm^{2}

**Example 8 :**

Find the area of the sector whose radius is 20 cm and perimeter is 110 cm.

**Solution :**

Perimeter of sector = 110 cm

l + 2r = 110

l + 2(20) = 110

l + 40 = 110

l = 70 cm

Area of the sector :

= (lr)/2 square units

Substitute l = 70 and r = 20.

= (70 ⋅ 20)/2

= 700 cm^{2}

Apart from the stuff given above, please use our google custom search here.

If you have any feedback about our math content, please mail us :

**v4formath@gmail.com**

We always appreciate your feedback.

You can also visit the following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**