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Basic Geometry : Quadrilaterals

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Example Questions

Example Question #401 : Quadrilaterals

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:

4.50

4.57

4.63

4.21

Correct answer:

4.57

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.

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

Simplify.

$\text{Radius}=2$

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

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

Simplify.

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

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.

$\text{Area of Square}=\frac{4^2}{2}$

Simplify.

$\text{Area of Square}=8$

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

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

$\text{Area of shaded region}=4\pi-8$

Solve.

$\text{Area of shaded region}=4.57$

Report an Error

Example Question #402 : Quadrilaterals

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:

7.06

7.13

7.79

7.51

Correct answer:

7.13

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, 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.

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

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

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

Simplify.

$\text{Area of circle}=\frac{25}{4}\pi$

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.

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

Simplify.

$\text{Area of Square}=\frac{25}{2}$

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

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

$\text{Area of shaded region}=\frac{25}{4}\pi-\frac{25}{2}$

Solve.

$\text{Area of shaded region}=7.13$

Report an Error

Example Question #403 : Quadrilaterals

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:

1.16

1.14

The area of the shaded region cannot be determined.

2.14

Correct answer:

1.14

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.

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

Simplify.

$\text{Radius}=1$

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

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

Simplify.

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

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.

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

Simplify.

$\text{Area of Square}=2$

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

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

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

Solve.

$\text{Area of shaded region}=1.14$

Report an Error

Example Question #404 : Quadrilaterals

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:

2.15

The area of the shaded region cannot be determined.

2.34

2.57

Correct answer:

2.57

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.

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

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

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

Simplify.

$\text{Area of circle}=\frac{9}{4}\pi$

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.

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

Simplify.

$\text{Area of Square}=\frac{9}{2}$

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

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

$\text{Area of shaded region}=\frac{9}{4}\pi-\frac{9}{2}$

Solve.

$\text{Area of shaded region}=2.57$

Report an Error

Example Question #405 : Quadrilaterals

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:

10.36

10.21

10.27

10.24

Correct answer:

10.27

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.

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

Simplify.

$\text{Radius}=3$

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

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

Simplify.

$\text{Area of circle}=9\pi$

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.

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

Simplify.

$\text{Area of Square}=18$

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

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

$\text{Area of shaded region}=9\pi-18$

Solve.

$\text{Area of shaded region}=10.27$

Report an Error

Example Question #406 : Quadrilaterals

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:

12.98

13.65

14.02

13.98

Correct answer:

13.98

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.

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

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

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

Simplify.

$\text{Area of circle}=\frac{49}{4}\pi$

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.

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

Simplify.

$\text{Area of Square}=\frac{49}{2}$

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

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

$\text{Area of shaded region}=\frac{49}{4}\pi-\frac{49}{2}$

Solve.

$\text{Area of shaded region}=13.98$

Report an Error

Example Question #407 : Quadrilaterals

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:

23.12

23.15

23.66

23.32

Correct answer:

23.12

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.

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

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

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

Simplify.

$\text{Area of circle}=\frac{81}{4}\pi$

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.

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

Simplify.

$\text{Area of Square}=\frac{81}{2}$

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

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

$\text{Area of shaded region}=\frac{81}{4}\pi-\frac{81}{2}$

Solve.

$\text{Area of shaded region}=23.12$

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:

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.

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

Report an Error

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

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Basic Geometry : Quadrilaterals

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #401 : Quadrilaterals

Example Question #402 : Quadrilaterals

Example Question #403 : Quadrilaterals

Example Question #404 : Quadrilaterals

Example Question #405 : Quadrilaterals

Example Question #406 : Quadrilaterals

Example Question #407 : Quadrilaterals

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

All Basic Geometry Resources

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