**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.

Before look at the worksheet, if you would like to know the stuff related to areas of circles and worksheet,

**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 ?

**Problem 1 :**

Find the area of the circle shown below.

**Solution : **

Formula area of a circle is given by

A = πr^{2}

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 = πr^{2}

Plug A = 96.

96 = πr^{2}

Divide each side π.

96/π = πr^{2}/π

96/π = r^{2}

Use calculator.

30.56 ≈ r^{2}

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°] ⋅ πr^{2}

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°] ⋅ πr^{2}

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.

A = 60°/360° ⋅ π(9) A = 1/6 ⋅ π ⋅ 81 A ≈ 42.41 square inches |
A = 300°/360° ⋅ π(9) A = 5/6 ⋅ 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 = πr^{2 }- 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 ⋅ 5^{2}] - [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 ⋅ 3^{2}] - [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.

After having gone through the stuff given above, we hope that the students would have understood, "Areas of Circles and Sectors Worksheet".

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