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Basic Geometry : How to find the area of a square

Study concepts, example questions & explanations for Basic Geometry

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Example Questions

Example Question #131 : How To Find The Area Of A Square

In the figure, a square is inscribed in a circle. If the diameter of the circle is , then what is the area of the shaded region?

Possible Answers:

21.23

28.54

29.33

31.57

Correct answer:

28.54

Explanation:

From the figure, you should notice that the diameter of the circle is also the diagonal of the square.

In order to find the area of the shaded region, we will first need to find the area of the circle and the area of the square.

First, recall how to find the area of a circle.

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

Now, use the diameter to find the radius.

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

Substitute in the given value of the diameter to find the length of the radius.

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

Simplify.

$\text{Radius}=5$

Now, substitute in the value of the radius into the equation to find the area of the circle.

$\text{Area of circle}=\pi\times(5)^2$

Simplify.

$\text{Area of circle}=25\pi$

Now, we will need to find the area of the square.

Notice that the diagonal of the square is also the hypotenuse of a right isosceles triangle that has the sides of the square as the legs of the triangle. We can then use the Pythagorean theorem.

$\text{Diagonal}^2=\text{side}^2+\text{side}^2$

$\text{Diagonal}^2=2(\text{side})^2$

$\text{side}^2=\frac{\text{Diagonal}^2}{2}$

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

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

Notice that the area of the square is the same as the equation we found through using the Pythagorean theorem.

So now, we can write the following equation:

$\text{Area of Square}=\frac{\text{Diagonal}^2}{2}$

Substitute in the value of the diagonal to find the area of the square.

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

Simplify.

$\text{Area of Square}=50$

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

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

$\text{Area of shaded region}=25\pi-50$

Solve.

$\text{Area of shaded region}=28.54$

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Example Question #132 : How To Find The Area Of A Square

In the figure, a square is inscribed in a circle. If the diameter of the circle is , then what is the area of the shaded region?

Possible Answers:

33.36

36.22

30.96

34.53

Correct answer:

34.53

Explanation:

From the figure, you should notice that the diameter of the circle is also the diagonal of the square.

In order to find the area of the shaded region, we will first need to find the area of the circle and the area of the square.

First, recall how to find the area of a circle.

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

Now, use the diameter to find the radius.

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

Substitute in the given value of the diameter to find the length of the radius.

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

Now, substitute in the value of the radius into the equation to find the area of the circle.

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

Simplify.

$\text{Area of circle}=\frac{121}{4}\pi$

Now, we will need to find the area of the square.

Notice that the diagonal of the square is also the hypotenuse of a right isosceles triangle that has the sides of the square as the legs of the triangle. We can then use the Pythagorean theorem.

$\text{Diagonal}^2=\text{side}^2+\text{side}^2$

$\text{Diagonal}^2=2(\text{side})^2$

$\text{side}^2=\frac{\text{Diagonal}^2}{2}$

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

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

Notice that the area of the square is the same as the equation we found through using the Pythagorean theorem.

So now, we can write the following equation:

$\text{Area of Square}=\frac{\text{Diagonal}^2}{2}$

Substitute in the value of the diagonal to find the area of the square.

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

Simplify.

$\text{Area of Square}=\frac{121}{2}$

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

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

$\text{Area of shaded region}=\frac{121}{4}\pi-\frac{121}{2}$

Solve.

$\text{Area of shaded region}=34.53$

Report an Error

Example Question #131 : How To Find The Area Of A Square

In the figure, a square is inscribed in a circle. If the diameter of the circle is , then what is the area of the shaded region?

Possible Answers:

0.32

0.36

0.15

0.29

Correct answer:

0.29

Explanation:

From the figure, you should notice that the diameter of the circle is also the diagonal of the square.

In order to find the area of the shaded region, we will first need to find the area of the circle and the area of the square.

First, recall how to find the area of a circle.

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

Now, use the diameter to find the radius.

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

Substitute in the given value of the diameter to find the length of the radius.

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

Now, substitute in the value of the radius into the equation to find the area of the circle.

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

Simplify.

$\text{Area of circle}=\frac{1}{4}\pi$

Now, we will need to find the area of the square.

Notice that the diagonal of the square is also the hypotenuse of a right isosceles triangle that has the sides of the square as the legs of the triangle. We can then use the Pythagorean theorem.

$\text{Diagonal}^2=\text{side}^2+\text{side}^2$

$\text{Diagonal}^2=2(\text{side})^2$

$\text{side}^2=\frac{\text{Diagonal}^2}{2}$

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

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

Notice that the area of the square is the same as the equation we found through using the Pythagorean theorem.

So now, we can write the following equation:

$\text{Area of Square}=\frac{\text{Diagonal}^2}{2}$

Substitute in the value of the diagonal to find the area of the square.

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

Simplify.

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

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

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

$\text{Area of shaded region}=\frac{1}{4}\pi-\frac{1}{2}$

Solve.

$\text{Area of shaded region}=0.29$

Report an Error

Example Question #133 : How To Find The Area Of A Square

In the figure, a square is inscribed in a circle. If the diameter of the circle is , then what is the area of the shaded region?

Possible Answers:

18.66

18.27

18.14

18.02

Correct answer:

18.27

Explanation:

From the figure, you should notice that the diameter of the circle is also the diagonal of the square.

In order to find the area of the shaded region, we will first need to find the area of the circle and the area of the square.

First, recall how to find the area of a circle.

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

Now, use the diameter to find the radius.

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

Substitute in the given value of the diameter to find the length of the radius.

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

Simplify.

$\text{Radius}=4$

Now, substitute in the value of the radius into the equation to find the area of the circle.

$\text{Area of circle}=\pi\times(4)^2$

Simplify.

$\text{Area of circle}=16\pi$

Now, we will need to find the area of the square.

Notice that the diagonal of the square is also the hypotenuse of a right isosceles triangle that has the sides of the square as the legs of the triangle. We can then use the Pythagorean theorem.

$\text{Diagonal}^2=\text{side}^2+\text{side}^2$

$\text{Diagonal}^2=2(\text{side})^2$

$\text{side}^2=\frac{\text{Diagonal}^2}{2}$

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

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

Notice that the area of the square is the same as the equation we found through using the Pythagorean theorem.

So now, we can write the following equation:

$\text{Area of Square}=\frac{\text{Diagonal}^2}{2}$

Substitute in the value of the diagonal to find the area of the square.

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

Simplify.

$\text{Area of Square}=32$

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

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

$\text{Area of shaded region}=16\pi-32$

Solve.

$\text{Area of shaded region}=18.27$

Report an Error

Example Question #131 : How To Find The Area Of A Square

Find the area of a square with side length 2.

Possible Answers:

Correct answer:

Explanation:

To solve, simply use the formula for the area of a square. Thus,

A=s^2=2^2=4

If the formula escapes you, draw a picture and imagine you have 2 rows of objects with two in them each. If you add them together, you will get a total of 4!

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Example Question #162 : Squares

Square A has an area of 81 cm^2 . The sides of Square B are triple the size of the sides of Square A. What is the area of Square B?

Possible Answers:

1008cm^2

551cm^2

729cm^2

243 cm^2

27cm^2

Correct answer:

729cm^2

Explanation:

Because the area of Square A is 81 cm^2 , and the formula for the area of a square is A=s^2 , we simply plug it in, as follows:

81=s^2

$\sqrt{81}=\sqrt{s^2}$

s=9

So the length of Square A's sides is 9 cm . In order to get the length of Square B's sides, we must multiply this by , which gives us 27cm .

We then plug this into the formula for area:

A=27^2

A=729cm^2

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Example Question #163 : Squares

What is the area of a square that has side lengths of 4 inches?

Possible Answers:

4in^2

8in^2

12in^2

20in^2

16in^2

Correct answer:

16in^2

Explanation:

To find the area of a square you must know that a square has four sides and all sides are the same length. The formula for finding the area of a square is the length multiplied by the width, or l * w .

The length times width in this case would be $4 \cdot 4 = 16$ .

When answering an area question the answer must be put in units squared, or units^2 . In this case it would be inches squared, or in^2 .

Therefore the answer is 16in^2

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Example Question #132 : How To Find The Area Of A Square

What is the area of a square that has side lengths of 10 inches?

Possible Answers:

75in^2

20in^2

100in^2

10in^2

50in^2

Correct answer:

100in^2

Explanation:

The length times width in this case would be $10 \cdot 10 = 100$ .

When answering an area question the answer must be put in units squared, or units^2 . In this case it would be inches squared, or in^2 .

Therefore the answer is 100in^2

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Example Question #133 : How To Find The Area Of A Square

One of the sides of a square has a length of 8cm . The perimeter of the square is 4 times larger than one side of the square. What is the area of the square?

Possible Answers:

1024 cm^2

64 cm^2

32 cm^2

16cm^2

There is not enough information given.

Correct answer:

64 cm^2

Explanation:

The formula for the area of a square is A=s^2 , with being one side of the square. Since we know one side is 8 cm , we simply plug this in to get the area, as follows:

A=8^2

A=64 cm^2 .

*Note that while the statement about perimeter is true (and always will be, since the formula for the perimeter of a square is P=4s ), this statement is not needed to solve the problem.

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Example Question #132 : How To Find The Area Of A Square

A cube has a side length of 5 inches. What is the total surface area of all its faces?

Possible Answers:

25in^2

175in^2

None of these.

25in^2

150in^2

Correct answer:

150in^2

Explanation:

A cube has 6 faces with congruent squares. Since the length of one edge is 5 inches, the area of that face is A=s*s=5*5=25in^2 .

Multiply this by the 6 faces present to get $\boldsymbol{150in^2}$ .

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Basic Geometry : How to find the area of a square

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #131 : How To Find The Area Of A Square

Example Question #132 : How To Find The Area Of A Square

Example Question #131 : How To Find The Area Of A Square

Example Question #133 : How To Find The Area Of A Square

Example Question #131 : How To Find The Area Of A Square

Example Question #162 : Squares

Example Question #163 : Squares

Example Question #132 : How To Find The Area Of A Square

Example Question #133 : How To Find The Area Of A Square

Example Question #132 : How To Find The Area Of A Square

All Basic Geometry Resources

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