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Basic Geometry : Quadrilaterals

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Example Questions

Example Question #381 : Quadrilaterals

Find the area of a square inscribed in a circle that has a diameter of .

Possible Answers:

$6\sqrt2$

144

$24\sqrt2$

Correct answer:

Explanation:

Notice that when a squre is inscribed in a circle, the diameter of the circle is also the diagonal of the square.

$\text{Diameter}=\text{Diagonal}$

Thus, we can figure out the diagonal of the square.

$\text{Diagonal}=4$

Recall that the diagonal of a square is also the hypotenuse of a right isosceles triangle that has the sides of the square as its legs. We can then use the Pythagorean Theorem to find the length of the sides of the square.

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

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

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

Now, recall the formula for the area of a square.

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

Thus, we can also write the following formula to find the area of the square:'

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

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

$\text{Area}=\frac{12^2}{2}=72$

Report an Error

Example Question #790 : Basic Geometry

Find the area of a square inscribed in a circle that has a diameter of .

Possible Answers:

196

Correct answer:

Explanation:

Notice that when a squre is inscribed in a circle, the diameter of the circle is also the diagonal of the square.

$\text{Diameter}=\text{Diagonal}$

Thus, we can figure out the diagonal of the square.

$\text{Diagonal}=4$

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

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

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

Now, recall the formula for the area of a square.

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

Thus, we can also write the following formula to find the area of the square:'

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

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

$\text{Area}=\frac{14^2}{2}=98$

Report an Error

Example Question #104 : How To Find The Area Of A Square

Find the area of a square inscribed in a circle that has a diameter of .

Possible Answers:

128

112

256

362

Correct answer:

128

Explanation:

Notice that when a squre is inscribed in a circle, the diameter of the circle is also the diagonal of the square.

$\text{Diameter}=\text{Diagonal}$

Thus, we can figure out the diagonal of the square.

$\text{Diagonal}=4$

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

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

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

Now, recall the formula for the area of a square.

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

Thus, we can also write the following formula to find the area of the square:'

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

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

$\text{Area}=\frac{16^2}{2}=128$

Report an Error

Example Question #105 : How To Find The Area Of A Square

Find the area of a square inscribed in a circle that has a diameter of .

Possible Answers:

162

810

324

486

Correct answer:

162

Explanation:

Notice that when a squre is inscribed in a circle, the diameter of the circle is also the diagonal of the square.

$\text{Diameter}=\text{Diagonal}$

Thus, we can figure out the diagonal of the square.

$\text{Diagonal}=4$

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

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

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

Now, recall the formula for the area of a square.

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

Thus, we can also write the following formula to find the area of the square:'

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

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

$\text{Area}=\frac{18^2}{2}=162$

Report an Error

Example Question #382 : Quadrilaterals

Find the area of a square inscribed in a circle with a diameter of 100 .

Possible Answers:

500

10000

1000

5000

Correct answer:

5000

Explanation:

Notice that when a squre is inscribed in a circle, the diameter of the circle is also the diagonal of the square.

$\text{Diameter}=\text{Diagonal}$

Thus, we can figure out the diagonal of the square.

$\text{Diagonal}=4$

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

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

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

Now, recall the formula for the area of a square.

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

Thus, we can also write the following formula to find the area of the square:'

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

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

$\text{Area}=\frac{100^2}{2}=5000$

Report an Error

Example Question #383 : Quadrilaterals

Find the area of a square given side length 3x^2 .

Possible Answers:

9x^4

9x^2

6x^4

3x^4

Correct answer:

9x^4

Explanation:

To solve, simply use the formula for the area of a square and remember to distribute the square to both the constant and the variable. Thus,

A=s^2=(3x^2)^2=3^2*(x^2)^2=9x^4

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Example Question #384 : Quadrilaterals

Find the area of a square given side length of 4.

Possible Answers:

Correct answer:

Explanation:

To find the area of a square multiply the width with the length. In the case of a square, all sides are the same length thus, the area is simply the side length squarred.

To solve, simply use the formula for the area of a square and let,

s=4 .

Thus,

A=s^2=4^2=16 .

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Example Question #385 : Quadrilaterals

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

Possible Answers:

8.87

10.38

The area of the shaded region cannot be determined.

9.13

Correct answer:

9.13

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, let's find the area of the square.

From the given information, we can find the length of a side of the square.

$\text{Perimeter}=4(\text{side})$

$\text{side}=\frac{\text{Perimeter}}{4}$

Substitute in the value of the perimeter to find the length of a side of the square.

$\text{side}=\frac{16}{4}$

Simplify.

$\text{side}=4$

Now recall how to find the area of a square:

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

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

$\text{Area of square}=4^2$

Simplify.

$\text{Area of square}=16$

Now, use the Pythagorean theorem to find the length of the diagonal of the square.

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

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

Simplify.

$\text{Diagonal}=\text{side}\sqrt2$

Substitute in the value of the side of the square to find the length of the diagonal.

$\text{Diagonal}=4\sqrt2$

Recall that the diagonal of the square is the same as the diameter of the circle.

$\text{Diameter}=\text{Diagonal}=4\sqrt2$

From the diameter, we can then find the radius of the circle:

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

$\text{Radius}=\frac{4\sqrt2}{2}$

Simplify.

$\text{Radius}=2\sqrt2$

Now, use the radius to find the area of the circle.

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

$\text{Area of circle}=\pi\times (2\sqrt2)^2$

Simplify.

$\text{Area of circle}=8\pi$

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

$\text{Area of Shaded Region}=\text{Area of circle}-\text{Area of square}$

$\text{Area of shaded region}=8\pi-16$

Solve.

$\text{Area of shaded region}=9.13$

Report an Error

Example Question #795 : Plane Geometry

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

Possible Answers:

14.27

The area of the shaded region cannot be determined.

12.02

13.62

Correct answer:

14.27

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, let's find the area of the square.

From the given information, we can find the length of a side of the square.

$\text{Perimeter}=4(\text{side})$

$\text{side}=\frac{\text{Perimeter}}{4}$

Substitute in the value of the perimeter to find the length of a side of the square.

$\text{side}=\frac{20}{4}$

Simplify.

$\text{side}=5$

Now recall how to find the area of a square:

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

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

$\text{Area of square}=5^2$

Simplify.

$\text{Area of square}=25$

Now, use the Pythagorean theorem to find the length of the diagonal of the square.

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

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

Simplify.

$\text{Diagonal}=\text{side}\sqrt2$

Substitute in the value of the side of the square to find the length of the diagonal.

$\text{Diagonal}=5\sqrt2$

Recall that the diagonal of the square is the same as the diameter of the circle.

$\text{Diameter}=\text{Diagonal}=5\sqrt2$

From the diameter, we can then find the radius of the circle:

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

$\text{Radius}=\frac{5\sqrt2}{2}$

Now, use the radius to find the area of the circle.

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

$\text{Area of circle}=\pi\times (\frac{5\sqrt2}{2})^2$

Simplify.

$\text{Area of circle}=\frac{25}{2}\pi$

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

$\text{Area of Shaded Region}=\text{Area of circle}-\text{Area of square}$

$\text{Area of shaded region}=\frac{25}{2}\pi-25$

Solve.

$\text{Area of shaded region}=14.27$

Report an Error

Example Question #141 : Squares

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

Possible Answers:

20.55

19.87

18.60

21.63

Correct answer:

20.55

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, let's find the area of the square.

From the given information, we can find the length of a side of the square.

$\text{Perimeter}=4(\text{side})$

$\text{side}=\frac{\text{Perimeter}}{4}$

Substitute in the value of the perimeter to find the length of a side of the square.

$\text{side}=\frac{24}{4}$

Simplify.

$\text{side}=6$

Now recall how to find the area of a square:

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

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

$\text{Area of square}=6^2$

Simplify.

$\text{Area of square}=36$

Now, use the Pythagorean theorem to find the length of the diagonal of the square.

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

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

Simplify.

$\text{Diagonal}=\text{side}\sqrt2$

Substitute in the value of the side of the square to find the length of the diagonal.

$\text{Diagonal}=6\sqrt2$

Recall that the diagonal of the square is the same as the diameter of the circle.

$\text{Diameter}=\text{Diagonal}=6\sqrt2$

From the diameter, we can then find the radius of the circle:

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

$\text{Radius}=\frac{6\sqrt2}{2}$

Simplify.

$\text{Radius}=3\sqrt2$

Now, use the radius to find the area of the circle.

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

$\text{Area of circle}=\pi\times (3\sqrt2)^2$

Simplify.

$\text{Area of circle}=18\pi$

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

$\text{Area of Shaded Region}=\text{Area of circle}-\text{Area of square}$

$\text{Area of shaded region}=18\pi-36$

Solve.

$\text{Area of shaded region}=20.55$

Report an Error

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Basic Geometry : Quadrilaterals

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #381 : Quadrilaterals

Example Question #790 : Basic Geometry

Example Question #104 : How To Find The Area Of A Square

Example Question #105 : How To Find The Area Of A Square

Example Question #382 : Quadrilaterals

Example Question #383 : Quadrilaterals

Example Question #384 : Quadrilaterals

Example Question #385 : Quadrilaterals

Example Question #795 : Plane Geometry

Example Question #141 : Squares

All Basic Geometry Resources

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