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Basic Geometry : Circles

Study concepts, example questions & explanations for Basic Geometry

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9 Diagnostic Tests 164 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #121 : Basic Geometry

In the figure below, a circle is inscribed in a square. If the length of a side of the square is , what is the area of the shaded region?

Possible Answers:

7.90

7.73

7.66

The area of the shaded region cannot be determined.

Correct answer:

7.73

Explanation:

From the figure, you should notice that the side length of the square is also the same as the length of the diameter of the circle.

To find the area of the shaded region, we will need to subtract the area of the circle from the area of the square.

First, recall how to find the area of a square:

$\text{Area of square}=\text{side}^2$

Plug in the given value to find the area of the square.

$\text{Area of Square}=6^2=36$

Now, find the area of the circle.

We will need to first find the length of the radius of the circle.

$\text{Radius}=\frac{\text{Diameter}}{2}$

Plug in the value of the diameter to find the length of the radius.

$\text{Radius}=\frac{6}{2}=3$

Now, recall the formula to find the area of a circle:

$\text{Area of Circle}=\pi\times\text{radius}^2$

Plug in the value of the radius to find the area of the circle.

$\text{Area of Circle}=\pi\times 3^2=9\pi$

Now, to find the area of the shaded region, subtract the area of the circle from the area of the square.

$\text{Area of Shaded Region}=\text{Area of Square}-\text{Area of Circle}$

$\text{Area of Shaded Region}=36-9\pi=7.73$

Round to two places after the decimal.

Report an Error

Example Question #121 : Plane Geometry

In the figure below, a circle is inscribed in a square. If a side length of the square is , what is the area of the shaded region?

Possible Answers:

2.89

1.09

1.93

1.87

Correct answer:

1.93

Explanation:

From the figure, you should notice that the side length of the square is also the same as the length of the diameter of the circle.

To find the area of the shaded region, we will need to subtract the area of the circle from the area of the square.

First, recall how to find the area of a square:

$\text{Area of square}=\text{side}^2$

Plug in the given value to find the area of the square.

$\text{Area of Square}=3^2=9$

Now, find the area of the circle.

We will need to first find the length of the radius of the circle.

$\text{Radius}=\frac{\text{Diameter}}{2}$

Plug in the value of the diameter to find the length of the radius.

$\text{Radius}=\frac{3}{2}$

Now, recall the formula to find the area of a circle:

$\text{Area of Circle}=\pi\times\text{radius}^2$

Plug in the value of the radius to find the area of the circle.

$\text{Area of Circle}=\pi\times (\frac{3}{2})^2=\frac{9}{4}\pi$

Now, to find the area of the shaded region, subtract the area of the circle from the area of the square.

$\text{Area of Shaded Region}=\text{Area of Square}-\text{Area of Circle}$

$\text{Area of Shaded Region}=9-\frac{9}{4}\pi=1.93$

Round to two places after the decimal.

Report an Error

Example Question #123 : Plane Geometry

In the figure below, a circle is inscribed in a square. If a length of the side of the square is , what is the area of the shaded region?

Possible Answers:

5.29

5.51

5.37

5.82

Correct answer:

5.37

Explanation:

From the figure, you should notice that the side length of the square is also the same as the length of the diameter of the circle.

To find the area of the shaded region, we will need to subtract the area of the circle from the area of the square.

First, recall how to find the area of a square:

$\text{Area of square}=\text{side}^2$

Plug in the given value to find the area of the square.

$\text{Area of Square}=5^2=25$

Now, find the area of the circle.

We will need to first find the length of the radius of the circle.

$\text{Radius}=\frac{\text{Diameter}}{2}$

Plug in the value of the diameter to find the length of the radius.

$\text{Radius}=\frac{5}{2}$

Now, recall the formula to find the area of a circle:

$\text{Area of Circle}=\pi\times\text{radius}^2$

Plug in the value of the radius to find the area of the circle.

$\text{Area of Circle}=\pi\times (\frac{5}{2})^2=\frac{25}{4}\pi$

Now, to find the area of the shaded region, subtract the area of the circle from the area of the square.

$\text{Area of Shaded Region}=\text{Area of Square}-\text{Area of Circle}$

$\text{Area of Shaded Region}=25-\frac{25}{4}\pi=5.37$

Round to two places after the decimal.

Report an Error

Example Question #124 : Plane Geometry

In the figure below, a circle is inscribed in a square. If a length of the side of the square is , what is the area of the shaded region?

Possible Answers:

0.28

0.11

0.17

0.21

Correct answer:

0.21

Explanation:

From the figure, you should notice that the side length of the square is also the same as the length of the diameter of the circle.

To find the area of the shaded region, we will need to subtract the area of the circle from the area of the square.

First, recall how to find the area of a square:

$\text{Area of square}=\text{side}^2$

Plug in the given value to find the area of the square.

$\text{Area of Square}=1^2=1$

Now, find the area of the circle.

We will need to first find the length of the radius of the circle.

$\text{Radius}=\frac{\text{Diameter}}{2}$

Plug in the value of the diameter to find the length of the radius.

$\text{Radius}=\frac{1}{2}$

Now, recall the formula to find the area of a circle:

$\text{Area of Circle}=\pi\times\text{radius}^2$

Plug in the value of the radius to find the area of the circle.

$\text{Area of Circle}=\pi\times( \frac{1}{2})^2=\frac{1}{4}\pi$

Now, to find the area of the shaded region, subtract the area of the circle from the area of the square.

$\text{Area of Shaded Region}=\text{Area of Square}-\text{Area of Circle}$

$\text{Area of Shaded Region}=1-\frac{1}{4}\pi=0.21$

Round to two places after the decimal.

Report an Error

Example Question #125 : Plane Geometry

In the figure below, a circle is inscribed in a square. If the length of a side of the square is , what is the area of the shaded region?

Possible Answers:

10.52

10.47

10.98

10.55

Correct answer:

10.52

Explanation:

From the figure, you should notice that the side length of the square is also the same as the length of the diameter of the circle.

To find the area of the shaded region, we will need to subtract the area of the circle from the area of the square.

First, recall how to find the area of a square:

$\text{Area of square}=\text{side}^2$

Plug in the given value to find the area of the square.

$\text{Area of Square}=7^2=49$

Now, find the area of the circle.

We will need to first find the length of the radius of the circle.

$\text{Radius}=\frac{\text{Diameter}}{2}$

Plug in the value of the diameter to find the length of the radius.

$\text{Radius}=\frac{7}{2}$

Now, recall the formula to find the area of a circle:

$\text{Area of Circle}=\pi\times\text{radius}^2$

Plug in the value of the radius to find the area of the circle.

$\text{Area of Circle}=\pi\times (\frac{7}{2})^2=\frac{49}{4}\pi$

Now, to find the area of the shaded region, subtract the area of the circle from the area of the square.

$\text{Area of Shaded Region}=\text{Area of Square}-\text{Area of Circle}$

$\text{Area of Shaded Region}=49-\frac{49}{4}\pi=10.52$

Round to two places after the decimal.

Report an Error

Example Question #126 : Plane Geometry

In the figure below, a circle is inscribed in a square. If the length of a side of the square is , what is the area of the shaded region?

Possible Answers:

13.73

13.71

13.77

13.75

Correct answer:

13.73

Explanation:

From the figure, you should notice that the side length of the square is also the same as the length of the diameter of the circle.

To find the area of the shaded region, we will need to subtract the area of the circle from the area of the square.

First, recall how to find the area of a square:

$\text{Area of square}=\text{side}^2$

Plug in the given value to find the area of the square.

$\text{Area of Square}=8^2=64$

Now, find the area of the circle.

We will need to first find the length of the radius of the circle.

$\text{Radius}=\frac{\text{Diameter}}{2}$

Plug in the value of the diameter to find the length of the radius.

$\text{Radius}=\frac{8}{2}=4$

Now, recall the formula to find the area of a circle:

$\text{Area of Circle}=\pi\times\text{radius}^2$

Plug in the value of the radius to find the area of the circle.

$\text{Area of Circle}=\pi\times 4^2=16\pi$

Now, to find the area of the shaded region, subtract the area of the circle from the area of the square.

$\text{Area of Shaded Region}=\text{Area of Square}-\text{Area of Circle}$

$\text{Area of Shaded Region}=64-16\pi=13.73$

Round to two places after the decimal.

Report an Error

Example Question #127 : Plane Geometry

In the figure below, a circle is inscribed in a square. If the length of a side of the square is , what is the area of the shaded region?

Possible Answers:

17.33

17.49

17.20

17.38

Correct answer:

17.38

Explanation:

From the figure, you should notice that the side length of the square is also the same as the length of the diameter of the circle.

To find the area of the shaded region, we will need to subtract the area of the circle from the area of the square.

First, recall how to find the area of a square:

$\text{Area of square}=\text{side}^2$

Plug in the given value to find the area of the square.

$\text{Area of Square}=9^2=81$

Now, find the area of the circle.

We will need to first find the length of the radius of the circle.

$\text{Radius}=\frac{\text{Diameter}}{2}$

Plug in the value of the diameter to find the length of the radius.

$\text{Radius}=\frac{9}{2}$

Now, recall the formula to find the area of a circle:

$\text{Area of Circle}=\pi\times\text{radius}^2$

Plug in the value of the radius to find the area of the circle.

$\text{Area of Circle}=\pi\times (\frac{9}{2})^2=\frac{81}{4}\pi$

Now, to find the area of the shaded region, subtract the area of the circle from the area of the square.

$\text{Area of Shaded Region}=\text{Area of Square}-\text{Area of Circle}$

$\text{Area of Shaded Region}=81-\frac{81}{4}\pi=17.38$

Round to two places after the decimal.

Report an Error

Example Question #128 : Plane Geometry

In the figure below, a circle is inscribed in a square. If the length of a side of the square is , what is the area of the shaded region?

Possible Answers:

21.46

20.99

22.90

21.91

Correct answer:

21.46

Explanation:

From the figure, you should notice that the side length of the square is also the same as the length of the diameter of the circle.

To find the area of the shaded region, we will need to subtract the area of the circle from the area of the square.

First, recall how to find the area of a square:

$\text{Area of square}=\text{side}^2$

Plug in the given value to find the area of the square.

$\text{Area of Square}=10^2=100$

Now, find the area of the circle.

We will need to first find the length of the radius of the circle.

$\text{Radius}=\frac{\text{Diameter}}{2}$

Plug in the value of the diameter to find the length of the radius.

$\text{Radius}=\frac{10}{2}=5$

Now, recall the formula to find the area of a circle:

$\text{Area of Circle}=\pi\times\text{radius}^2$

Plug in the value of the radius to find the area of the circle.

$\text{Area of Circle}=\pi\times 5^2=25\pi$

Now, to find the area of the shaded region, subtract the area of the circle from the area of the square.

$\text{Area of Shaded Region}=\text{Area of Square}-\text{Area of Circle}$

$\text{Area of Shaded Region}=100-25\pi=21.46$

Round to two places after the decimal.

Report an Error

Example Question #129 : Plane Geometry

In the figure below, a circle is inscribed in a square. If the length of a side of the square is , find the area of the shaded region.

Possible Answers:

25.97

26.71

25.89

25.11

Correct answer:

25.97

Explanation:

From the figure, you should notice that the side length of the square is also the same as the length of the diameter of the circle.

To find the area of the shaded region, we will need to subtract the area of the circle from the area of the square.

First, recall how to find the area of a square:

$\text{Area of square}=\text{side}^2$

Plug in the given value to find the area of the square.

$\text{Area of Square}=11^2=121$

Now, find the area of the circle.

We will need to first find the length of the radius of the circle.

$\text{Radius}=\frac{\text{Diameter}}{2}$

Plug in the value of the diameter to find the length of the radius.

$\text{Radius}=\frac{11}{2}$

Now, recall the formula to find the area of a circle:

$\text{Area of Circle}=\pi\times\text{radius}^2$

Plug in the value of the radius to find the area of the circle.

$\text{Area of Circle}=\pi\times (\frac{11}{2})^2=\frac{121}{4}\pi$

Now, to find the area of the shaded region, subtract the area of the circle from the area of the square.

$\text{Area of Shaded Region}=\text{Area of Square}-\text{Area of Circle}$

$\text{Area of Shaded Region}=121-\frac{121}{4}\pi=25.97$

Round to two places after the decimal.

Report an Error

Example Question #130 : Plane Geometry

In the figure below, a circle is inscribed in a square. If the length of a side of the square is , what is the area of the shaded region?

Possible Answers:

30.90

29.14

33.10

30.95

Correct answer:

30.90

Explanation:

From the figure, you should notice that the side length of the square is also the same as the length of the diameter of the circle.

To find the area of the shaded region, we will need to subtract the area of the circle from the area of the square.

First, recall how to find the area of a square:

$\text{Area of square}=\text{side}^2$

Plug in the given value to find the area of the square.

$\text{Area of Square}=12^2=144$

Now, find the area of the circle.

We will need to first find the length of the radius of the circle.

$\text{Radius}=\frac{\text{Diameter}}{2}$

Plug in the value of the diameter to find the length of the radius.

$\text{Radius}=\frac{12}{2}=6$

Now, recall the formula to find the area of a circle:

$\text{Area of Circle}=\pi\times\text{radius}^2$

Plug in the value of the radius to find the area of the circle.

$\text{Area of Circle}=\pi\times 6^2=36\pi$

Now, to find the area of the shaded region, subtract the area of the circle from the area of the square.

$\text{Area of Shaded Region}=\text{Area of Square}-\text{Area of Circle}$

$\text{Area of Shaded Region}=144-36\pi=30.90$

Round to two places after the decimal.

Report an Error

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Basic Geometry : Circles

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #121 : Basic Geometry

Example Question #121 : Plane Geometry

Example Question #123 : Plane Geometry

Example Question #124 : Plane Geometry

Example Question #125 : Plane Geometry

Example Question #126 : Plane Geometry

Example Question #127 : Plane Geometry

Example Question #128 : Plane Geometry

Example Question #129 : Plane Geometry

Example Question #130 : Plane Geometry

All Basic Geometry Resources

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