﻿ Area and Circumference of a Circle Worksheet | Problems & Solutions

# Area and Circumference of a Circle Worksheet

Area and Circumference of a Circle Worksheet
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1.
The diameter of a circular garden is 32 m. What is the new circumference of the garden, if its radius is decreased by 6 m? [ Use $\pi$ = 3.14]
 a. 66.80 m b. 125.60 m c. 64.80 m d. 62.80 m

#### Solution:

The diameter of the circular garden = 32 m.

So, the radius of the circular garden = diameter / 2 = 32 / 2 = 16 m.

If the radius is decreased by 6 m, then the new radius = 16 - 6 = 10 m.

The new circumference of the garden = 2 × π × new radius.
[Formula.]

= 2 × 3.14 × 10
[Substitute the values.]

= 62.80 m.
[Multiply.]

So, the new circumference of the garden = 62.80 m.

2.
A circular park of radius 300 yd has a circular pavement around it. What is the area of the pavement, if the width of the pavement is 4 yd? [Use $\pi$ = 3.14]
 a. 2426$\pi$ yd2 b. 7248 yd2 c. 2416$\pi$ yd2 d. None of the above

#### Solution:

Area of a circle with radius r = πr2
[Formula.]

Area of the circular park with radius 300 yd = π(3002).

Area of the park along with the pavement = π(r + 4)2 = π(300 + 4)2 = π(3042).

Area of the pavement = area of the park along with the pavement - area of the park.

Area of the pavement = π(304)2 - π(300)2 = π(3042 - 3002).

= π(304 - 300)(304 + 300)
[(a2 - b2) = (a +b)(a - b).]

= π(4)(604)
[Simplify.]

= 2416π

So, the area of the pavement is 2416π yd2.

3.
What is the area of the shaded region in the figure?[Use $\pi$ = 3.14]

 a. 177.7 cm2 b. 172.7 cm2 c. 127.7 cm2 d. None of the above

#### Solution:

Radius of the bigger circle = 8 cm and the radius of the smaller circle = 3 cm.

Area of the bigger circle = π x (radius of the bigger circle)2
[Formula.]

= 3.14 x 82
[Substitute the value of the radius.]

= 3.14 x 8 x 8
[Expand 82 as 8 x 8.]

= 200.96 cm2
[Multiply.]

Area of the bigger circle = 200.96 cm2.

Area of the smaller circle = π x (radius of the smaller circle)2
[Formula.]

= 3.14 x 32
[Substitute the radius of the smaller circle.]

= 3.14 x 3 x 3
[Expand 32 as 3 x 3.]

= 28.26 cm2
[Multiply.]

Area of the smaller circle = 28.26 cm2

Area of the shaded region = area of the bigger circle Ã¢â‚¬â€œ area of the smaller circle.

= 200.96 - 28.26
[Substitute the values.]

= 172.7
[Subtract.]

So, area of the shaded region is 172.7 cm2.

4.
The radius of a sector of a circle is 2 cm and its central angle is 90 degrees. Calculate the perimeter of the sector.
 a. 4 cm b. $\pi$ cm c. (4 + $\pi$) cm d. (2 + $\pi$) cm

#### Solution:

Perimeter of a sector of a circle = 2 × radius + arc length.
[Formula.]

= 2r + θ / 360 × 2πr
[Arc length of a circle = θ / 360 × 2πr]

Perimeter of a sector = 2 × 2 + 90 / 360(2π × 2)
[Substitute the value of r and θ].

= 4 + π.
[Simplify]

So, perimeter of the sector is (4 + π) cm.

5.
Find the area of the sector AOB.

 a. 3.25$\pi$ cm2 b. 3$\pi$ cm2 c. 6.75$\pi$ cm2 d. 4.5 cm2

#### Solution:

Area of the sector of a circle = 1 / 2r2θ
[Formula.]

θ = 120° = 120 × π / 180 = 2π3 radians

= 1 / 2 × (4.5)2 × 2π3
[Substitute the values.]

= 6.75π
[Simplify.]

So, the area of the sector of the circle is 6.75π cm2.

6.
Find the area of the sector POQ of the circle of radius 8 cm.

 a. 8 cm2 b. 4 cm2 c. 8$\pi$ cm2 d. 4$\pi$ cm2

#### Solution:

Area of the sector of a circle = 1 / 2r2θ
[Formula.]

45° = 45 × π / 180 = π / 4 radians

= 1 / 2 × (8)2 × π / 4
[Substitute the values.]

= 8π
[Simplify.]

So, the area of the sector of the circle is 8π cm2.

7.
The central angle of a sector is 30° and the radius of the circle is 6 cm. Find the area of the sector of the circle.
 a. $\frac{3\pi }{2}$ square cm b. $\frac{2\pi }{3}$ square cm c. 3$\pi$ square cm d. 6$\pi$ square cm

#### Solution:

Area of the sector of a circle = 1 / 2r2θ
[Formula.]

30° = 30 × π / 180 = π / 6 radians

= 1 / 2 × (6)2 × π / 6
[Substitute the values.]

= 3π
[Simplify.]

So, the area of the sector of the circle is 3π square cm.

8.
The angle of a sector is 90 degrees and the radius of the circle is 3 cm. Calculate the area of the sector.
 a. $\frac{3\pi }{2}$ square cm b. $\frac{9\pi }{2}$ square cm c. $\frac{9\pi }{4}$ square cm d. $\frac{3\pi }{4}$ square cm

#### Solution:

Area of the sector of a circle = 1 / 2r2θ
[Formula.]

90° = 90 × π / 180 = π / 2 radians

= 1 / 2 × (3)2 × π / 2
[Substitute the values.]

= 9π4

So, the area of the sector of the circle is 9π4 square cm.

9.
If the diameter of a circle is doubled, how is the circumference changed?
 a. multiplied by 2 b. divided by 2 c. divided by 4 d. multiplied by 4

#### Solution:

Let, the diameter of a circle be d units.

If diameter of a circle is d units, then circumference of the circle = πd units.

If the diameter of the circle is doubled, then diameter of the new circle will be 2d units.

Circumference of the new circle = π(2d) units = 2πd units = 2 × original circumference of the circle.

Therefore, circumference of the new circle will be two times the original circumference of the circle.

10.
If the radius of a circle is multiplied by 2, how is the area changed?
 a. multiplied by 4 b. divided by 4 c. divided by 2 d. multiplied by 2

#### Solution:

Let, the radius of a circle be r units.

If the radius of a circle is r units, then area of the circle = πr2 square units.

If the radius of the circle is multiplied by 2, then radius of the new circle will be 2r units.

Area of the new circle = π(2r)2 square units = 4πr2 square units = 4 × original area of the circle.

Therefore, area of the new circle will be four times the original area of the circle.