AREAS OF CIRCLES AND SECTORS WORKSHEET

About "Areas of Circles and Sectors Worksheet"

Areas of Circles and Sectors Worksheet :

Worksheet given in this section is much useful to the students who would like to practice problems on areas of circles and sectors. 

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Areas of Circles and Sectors Worksheet - Problems

Problem 1 :

Find the area of the circle shown below. 

Problem 2 :

If the area of a circle is 96 square centimeters, find its diameter. 

Problem 3 :

Find the area of the sector shown at the right.

Problem 4 :

A and B are two points on a P with radius 9 inches and APB = 60°. Find the areas of the sectors formed by APB.

Problem 5 : 

Find the area of the shaded region shown below. 

Problem 6 :

You are cutting the front face of a clock out of wood, as shown in the diagram. What is the area of the front of the case ?

Areas of Circles and Sectors Worksheet - Solutions

Problem 1 :

Find the area of the circle shown below. 

Solution : 

Formula area of a circle is given by 

A  =  πr2

Plug r  =  8.

A  =  π(8)2

A  =  64π

Use calculator. 

A  ≈  201.06

So, the area is 64π, or about 201.06, square inches.

Problem 2 :

If the area of a circle is 96 square centimeters, find its diameter. 

Solution :

Formula area of a circle is given by 

A  =  πr2

Plug A  =  96.

96  =  πr2

Divide each side π.

96/π  =  πr2/π

96/π  =  r2

Use calculator.

30.56    r2

Take square root on each side. 

5.53    r

So, the diameter of the circle is about 2(5.53), or about 11.06, centimeters.

Problem 3 :

Find the area of the sector shown at the right.

Solution : 

Sector CPD intercepts an arc whose measure is 80°. The radius is 4 feet. 

Formula for area of a sector is given by

A  =  [m∠arc CD / 360°] ⋅ πr2

Plug m∠arc CD  =  80°  and r  =  4.

A  =  [80° / 360°] ⋅ π(4)2

A  =  (2 / 9) ⋅ 16π

Use calculator.

A  ≈  11.17

So, the area of the sector is about 11.17 square feet.

Problem 4 :

A and B are two points on a P with radius 9 inches and APB = 60°. Find the areas of the sectors formed by APB.

Solution : 

Draw a diagram of ⊙P and APB. Shade the sectors. Label a point Q on the major arc. 

Find the measures of the minor and major arcs.

Because m∠APB  =  60°, we have

m∠arc AB  =  60°

and 

m∠AQB  =  360° - 60°  =  300°

Use the formula for the area of a sector. 

A  =  [m∠arc CD / 360°] ⋅ πr2

Plug m∠arc CD  =  80°  and r  =  4.

A  =  [80° / 360°] ⋅ π(4)2

A  =  (2 / 9) ⋅ 16π

Use calculator.

A  ≈  11.17

So, the area of the sector is about 11.17 square feet.

Area of Smaller Sector

A  =  60°/360° ⋅ π(9)2

A  =  1/⋅ π ⋅ 81

A  ≈  42.41 square inches

Area of Larger Sector

A  =  300°/360° ⋅ π(9)2

A  =  5/⋅ π ⋅ 81

A  ≈  212.06 square inches

Problem 5 : 

Find the area of the shaded region shown below. 

Solution : 

The diagram shows a regular hexagon inscribed in a circle with radius 5 meters. The shaded region is the part of the circle that is outside of the hexagon.

Area of shaded region  = Area of circle - Area of hexagon

Area of shaded region  =  πr2 - 1/2 ⋅ a ⋅ p

Radius of the circle is 5 and the apothem of a hexagon is 

=  1/2 ⋅ side length ⋅ √3

=  1/2 ⋅ 5 ⋅ √3

5√3/2

So, the area of the shaded region is

=  [π ⋅ 52]  -  [1/2 ⋅ (5√3/2) ⋅ (6 ⋅ 5)]

=  25π  -  75√3/2

Use calculator. 

≈  13.59

So, the area of the shaded region is about 13.59 square meters.

Problem 6 :

You are cutting the front face of a clock out of wood, as shown in the diagram. What is the area of the front of the case ?

Solution : 

The front of the case is formed by a rectangle and a sector, with a circle removed. Note that the intercepted arc of the sector is a semicircle.

So, the required area is 

=  Area of rectangle + Area of sector - Area of circle

=  [6 ⋅ 11/2] + [180°/360° ⋅ π ⋅ 32] -  [π ⋅ (1/2  4)2]

=  33 + 9/2 ⋅ π -  4π

Use calculator. 

≈  34.57

The area of the front of the case is about 34.57 square inches.

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