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Basic Geometry : Plane Geometry

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Example Questions

Example Question #116 : How To Find The Area Of A Square

In the figure, a square is inscribed in a circle. If the perimeter of the square is $\displaystyle 36$, then what is the area of the shaded region?

Possible Answers:

42.20 $\displaystyle 42.20$

48.69 $\displaystyle 48.69$

The area of the shaded region cannot be determined.

46.23 $\displaystyle 46.23$

Correct answer:

46.23 $\displaystyle 46.23$

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.

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

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

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

$\text{side}=\frac{36}{4}$ $\displaystyle \text{side}=\frac{36}{4}$

Simplify.

$\text{side}=9$ $\displaystyle \text{side}=9$

Now recall how to find the area of a square:

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

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

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

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

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

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

Simplify.

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

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

$\text{Diagonal}=9\sqrt2$ $\displaystyle \text{Diagonal}=9\sqrt2$

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

$\text{Diameter}=\text{Diagonal}=9\sqrt2$ $\displaystyle \text{Diameter}=\text{Diagonal}=9\sqrt2$

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

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

$\text{Radius}=\frac{9\sqrt2}{2}$ $\displaystyle \text{Radius}=\frac{9\sqrt2}{2}$

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

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

$\text{Area of circle}=\pi\times (\frac{9\sqrt2}{2})^2$ $\displaystyle \text{Area of circle}=\pi\times (\frac{9\sqrt2}{2})^2$

Simplify.

$\text{Area of circle}=\frac{81}{2}\pi$ $\displaystyle \text{Area of circle}=\frac{81}{2}\pi$

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

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

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

Solve.

$\text{Area of shaded region}=46.23$ $\displaystyle \text{Area of shaded region}=46.23$

Report an Error

Example Question #801 : Plane Geometry

In the figure, a square is inscribed in a circle. If the perimeter of the square is $\displaystyle 44$, then what is the area of the shaded region?

Possible Answers:

65.23 $\displaystyle 65.23$

71.05 $\displaystyle 71.05$

69.07 $\displaystyle 69.07$

70.02 $\displaystyle 70.02$

Correct answer:

69.07 $\displaystyle 69.07$

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.

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

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

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

$\text{side}=\frac{44}{4}$ $\displaystyle \text{side}=\frac{44}{4}$

Simplify.

$\text{side}=11$ $\displaystyle \text{side}=11$

Now recall how to find the area of a square:

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

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

$\text{Area of square}=11^2$ $\displaystyle \text{Area of square}=11^2$

Simplify.

$\text{Area of square}=121$ $\displaystyle \text{Area of square}=121$

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

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

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

Simplify.

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

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

$\text{Diagonal}=11\sqrt2$ $\displaystyle \text{Diagonal}=11\sqrt2$

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

$\text{Diameter}=\text{Diagonal}=11\sqrt2$ $\displaystyle \text{Diameter}=\text{Diagonal}=11\sqrt2$

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

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

$\text{Radius}=\frac{11\sqrt2}{2}$ $\displaystyle \text{Radius}=\frac{11\sqrt2}{2}$

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

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

$\text{Area of circle}=\pi\times (\frac{11\sqrt2}{2})^2$ $\displaystyle \text{Area of circle}=\pi\times (\frac{11\sqrt2}{2})^2$

Simplify.

$\text{Area of circle}=\frac{121}{2}\pi$ $\displaystyle \text{Area of circle}=\frac{121}{2}\pi$

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

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

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

Solve.

$\text{Area of shaded region}=69.07$ $\displaystyle \text{Area of shaded region}=69.07$

Report an Error

Example Question #118 : How To Find The Area Of A Square

In the figure, a square is inscribed in a circle. If the perimeter of the square is $\displaystyle 48$, then what is the area of the shaded region?

Possible Answers:

74.69 $\displaystyle 74.69$

82.19 $\displaystyle 82.19$

80.20 $\displaystyle 80.20$

85.33 $\displaystyle 85.33$

Correct answer:

82.19 $\displaystyle 82.19$

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.

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

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

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

$\text{side}=\frac{48}{4}$ $\displaystyle \text{side}=\frac{48}{4}$

Simplify.

$\text{side}=12$ $\displaystyle \text{side}=12$

Now recall how to find the area of a square:

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

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

$\text{Area of square}=12^2$ $\displaystyle \text{Area of square}=12^2$

Simplify.

$\text{Area of square}=144$ $\displaystyle \text{Area of square}=144$

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

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

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

Simplify.

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

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

$\text{Diagonal}=12\sqrt2$ $\displaystyle \text{Diagonal}=12\sqrt2$

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

$\text{Diameter}=\text{Diagonal}=12\sqrt2$ $\displaystyle \text{Diameter}=\text{Diagonal}=12\sqrt2$

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

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

$\text{Radius}=\frac{12\sqrt2}{2}$ $\displaystyle \text{Radius}=\frac{12\sqrt2}{2}$

Simplify.

$\text{Radius}=6\sqrt2$ $\displaystyle \text{Radius}=6\sqrt2$

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

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

$\text{Area of circle}=\pi\times (6\sqrt2)^2$ $\displaystyle \text{Area of circle}=\pi\times (6\sqrt2)^2$

Simplify.

$\text{Area of circle}=72\pi$ $\displaystyle \text{Area of circle}=72\pi$

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

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

$\text{Area of shaded region}=72\pi-144$ $\displaystyle \text{Area of shaded region}=72\pi-144$

Solve.

$\text{Area of shaded region}=82.19$ $\displaystyle \text{Area of shaded region}=82.19$

Report an Error

Example Question #119 : How To Find The Area Of A Square

In the figure, a square is inscribed in a circle. If the perimeter of the square is $\displaystyle 8$, then what is the area of the shaded region?

Possible Answers:

2.31 $\displaystyle 2.31$

2.28 $\displaystyle 2.28$

2.98 $\displaystyle 2.98$

2.14 $\displaystyle 2.14$

Correct answer:

2.28 $\displaystyle 2.28$

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.

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

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

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

$\text{side}=\frac{8}{4}$ $\displaystyle \text{side}=\frac{8}{4}$

Simplify.

$\text{side}=2$ $\displaystyle \text{side}=2$

Now recall how to find the area of a square:

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

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

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

Simplify.

$\text{Area of square}=4$ $\displaystyle \text{Area of square}=4$

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

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

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

Simplify.

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

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

$\text{Diagonal}=2\sqrt2$ $\displaystyle \text{Diagonal}=2\sqrt2$

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

$\text{Diameter}=\text{Diagonal}=2\sqrt2$ $\displaystyle \text{Diameter}=\text{Diagonal}=2\sqrt2$

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

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

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

Simplify.

$\text{Radius}=\sqrt2$ $\displaystyle \text{Radius}=\sqrt2$

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

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

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

Simplify.

$\text{Area of circle}=2\pi$ $\displaystyle \text{Area of circle}=2\pi$

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

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

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

Solve.

$\text{Area of shaded region}=2.28$ $\displaystyle \text{Area of shaded region}=2.28$

Report an Error

Example Question #801 : Basic Geometry

In the figure, a square is inscribed in a circle. If the perimeter of the square is $\displaystyle 4$, then what is the area of the shaded region?

Possible Answers:

0.69 $\displaystyle 0.69$

0.57 $\displaystyle 0.57$

0.32 $\displaystyle 0.32$

0.49 $\displaystyle 0.49$

Correct answer:

0.57 $\displaystyle 0.57$

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, let's find the area of the square.

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

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

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

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

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

Simplify.

$\text{side}=1$ $\displaystyle \text{side}=1$

Now recall how to find the area of a square:

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

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

$\text{Area of square}=1^2$ $\displaystyle \text{Area of square}=1^2$

Simplify.

$\text{Area of square}=1$ $\displaystyle \text{Area of square}=1$

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

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

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

Simplify.

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

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

$\text{Diagonal}=1\sqrt2$ $\displaystyle \text{Diagonal}=1\sqrt2$

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

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

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

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

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

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

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

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

Simplify.

$\text{Area of circle}=\frac{1}{2}\pi$ $\displaystyle \text{Area of circle}=\frac{1}{2}\pi$

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

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

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

Solve.

$\text{Area of shaded region}=0.57$ $\displaystyle \text{Area of shaded region}=0.57$

Report an Error

Example Question #121 : How To Find The Area Of A Square

In the figure, a square is inscribed in a circle. If the perimeter of the square is $\displaystyle 12$, then what is the area of the shaded region?

Possible Answers:

The area of the shaded region cannot be determined.

5.14 $\displaystyle 5.14$

4.66 $\displaystyle 4.66$

6.98 $\displaystyle 6.98$

Correct answer:

5.14 $\displaystyle 5.14$

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, let's find the area of the square.

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

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

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

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

$\text{side}=\frac{12}{4}$ $\displaystyle \text{side}=\frac{12}{4}$

Simplify.

$\text{side}=3$ $\displaystyle \text{side}=3$

Now recall how to find the area of a square:

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

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

$\text{Area of square}=3^2$ $\displaystyle \text{Area of square}=3^2$

Simplify.

$\text{Area of square}=9$ $\displaystyle \text{Area of square}=9$

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

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

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

Simplify.

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

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

$\text{Diagonal}=3\sqrt2$ $\displaystyle \text{Diagonal}=3\sqrt2$

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

$\text{Diameter}=\text{Diagonal}=3\sqrt2$ $\displaystyle \text{Diameter}=\text{Diagonal}=3\sqrt2$

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

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

$\text{Radius}=\frac{3\sqrt2}{2}$ $\displaystyle \text{Radius}=\frac{3\sqrt2}{2}$

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

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

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

Simplify.

$\text{Area of circle}=\frac{9}{2}\pi$ $\displaystyle \text{Area of circle}=\frac{9}{2}\pi$

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

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

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

Solve.

$\text{Area of shaded region}=5.14$ $\displaystyle \text{Area of shaded region}=5.14$

Report an Error

Example Question #122 : How To Find The Area Of A Square

In the figure, a square is inscribed in a circle. If the perimeter of the square is $\displaystyle 40$, then what is the area of the shaded region?

Possible Answers:

57.08 $\displaystyle 57.08$

58.15 $\displaystyle 58.15$

60.98 $\displaystyle 60.98$

52.07 $\displaystyle 52.07$

Correct answer:

57.08 $\displaystyle 57.08$

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.

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

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

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

$\text{side}=\frac{40}{4}$ $\displaystyle \text{side}=\frac{40}{4}$

Simplify.

$\text{side}=10$ $\displaystyle \text{side}=10$

Now recall how to find the area of a square:

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

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

$\text{Area of square}=10^2$ $\displaystyle \text{Area of square}=10^2$

Simplify.

$\text{Area of square}=100$ $\displaystyle \text{Area of square}=100$

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

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

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

Simplify.

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

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

$\text{Diagonal}=10\sqrt2$ $\displaystyle \text{Diagonal}=10\sqrt2$

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

$\text{Diameter}=\text{Diagonal}=10\sqrt2$ $\displaystyle \text{Diameter}=\text{Diagonal}=10\sqrt2$

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

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

$\text{Radius}=\frac{10\sqrt2}{2}$ $\displaystyle \text{Radius}=\frac{10\sqrt2}{2}$

Simplify.

$\text{Radius}=5\sqrt2$ $\displaystyle \text{Radius}=5\sqrt2$

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

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

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

Simplify.

$\text{Area of circle}=50\pi$ $\displaystyle \text{Area of circle}=50\pi$

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

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

$\text{Area of shaded region}=50\pi-100$ $\displaystyle \text{Area of shaded region}=50\pi-100$

Solve.

$\text{Area of shaded region}=57.08$ $\displaystyle \text{Area of shaded region}=57.08$

Report an Error

Example Question #123 : 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 $\displaystyle 12$, then what is the area of the shaded region?

Possible Answers:

The area of the shaded region cannot be determined.

41.10 $\displaystyle 41.10$

42.15 $\displaystyle 42.15$

40.98 $\displaystyle 40.98$

Correct answer:

41.10 $\displaystyle 41.10$

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.

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

Now, use the diameter to find the radius.

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

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

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

Simplify.

$\text{Radius}=6$ $\displaystyle \text{Radius}=6$

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

$\text{Area of circle}=\pi\times(6)^2$ $\displaystyle \text{Area of circle}=\pi\times(6)^2$

Simplify.

$\text{Area of circle}=36\pi$ $\displaystyle \text{Area of circle}=36\pi$

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.

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

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

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

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

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

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:

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

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

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

Simplify.

$\text{Area of Square}=72$ $\displaystyle \text{Area of Square}=72$

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

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

$\text{Area of shaded region}=36\pi-72$ $\displaystyle \text{Area of shaded region}=36\pi-72$

Solve.

$\text{Area of shaded region}=41.10$ $\displaystyle \text{Area of shaded region}=41.10$

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Example Question #124 : 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 $\displaystyle 4$, then what is the area of the shaded region?

Possible Answers:

4.57 $\displaystyle 4.57$

4.63 $\displaystyle 4.63$

4.50 $\displaystyle 4.50$

4.21 $\displaystyle 4.21$

Correct answer:

4.57 $\displaystyle 4.57$

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.

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

Now, use the diameter to find the radius.

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

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

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

Simplify.

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

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

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

Simplify.

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

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.

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

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

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

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

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

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:

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

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

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

Simplify.

$\text{Area of Square}=8$ $\displaystyle \text{Area of Square}=8$

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

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

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

Solve.

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

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Example Question #125 : 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 $\displaystyle 5$, then what is the area of the shaded region?

Possible Answers:

7.13 $\displaystyle 7.13$

7.79 $\displaystyle 7.79$

7.51 $\displaystyle 7.51$

7.06 $\displaystyle 7.06$

Correct answer:

7.13 $\displaystyle 7.13$

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.

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

Now, use the diameter to find the radius.

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

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

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

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

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

Simplify.

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

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.

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

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

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

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

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

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:

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

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

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

Simplify.

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

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

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

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

Solve.

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

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