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Basic Geometry : Circles

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Example Question #91 : Plane Geometry

Find the area of a circle that is inscribed in a square with side lengths of $\frac{1}{2}$ .

Possible Answers:

$\frac{\pi}{8}$

$\frac{\pi}{4}$

$\frac{\pi}{16}$

$\frac{\pi}{12}$

Correct answer:

$\frac{\pi}{16}$

Explanation:

Notice that when a circle is inscribed in a square, the length of the side of the square is the same as the diameter of a circle.

Recall how to find the area of a circle:

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

Now, recall the relationship between the radius and the diameter.

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

Use the given information to find the radius.

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

Simplify. Dividing by a number is the same as multiplying by its reciprocal.

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

$\text{Radius}=\frac{1}{4}$

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

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

Solve.

$\text{Area}=\frac{\pi}{16}$

Report an Error

Example Question #91 : Basic Geometry

Find the area of a circle that is inscribed in a square that has a diagonal of $2\sqrt2$ .

Possible Answers:

$\frac{1}{4}\pi$

$\pi$

$2\pi$

$\sqrt2\pi$

Correct answer:

$\pi$

Explanation:

When you draw out the circle that is inscribed in a square, you should notice two things. The first thing you should notice is that the diagonal of the square is also the hypotenuse of a right isosceles triangle that has the side lengths of the square as its legs. The second thing you should notice is that the diameter of the circle has the same length as the length of one side of the square.

First, use the Pythagorean theorem to find the length of a side of the square.

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

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

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

$\text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}$

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

$\text{side}=\frac{2\sqrt2(\sqrt2)}{2}$

Simplify.

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

Solve.

$\text{side}=2$

Now, recall the relationship between the diameter of the circle and the side of the square.

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

Now, recall that the radius is half the length of the diameter.

$\text{radius}=\frac{1}{2}(\text{diameter})$

Substitute in the value o fthe diameter to find the length of the radius.

$\text{radius}=\frac{1}{2}(2)=1$

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

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

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

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

Solve.

$\text{Area}=\pi$

Report an Error

Example Question #91 : How To Find The Area Of A Circle

Find the area of a circle that is inscribed in a square with a diagonal of $10\sqrt2$ .

Possible Answers:

$25\pi$

$100\pi$

$50\pi$

$200\pi$

Correct answer:

$25\pi$

Explanation:

First, use the Pythagorean theorem to find the length of a side of the square.

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

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

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

$\text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}$

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

$\text{side}=\frac{10\sqrt2(\sqrt2)}{2}$

Simplify.

$\text{side}=\frac{20}{2}$

Solve.

$\text{side}=10$

Now, recall the relationship between the diameter of the circle and the side of the square.

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

Now, recall that the radius is half the length of the diameter.

$\text{radius}=\frac{1}{2}(\text{diameter})$

Substitute in the value o fthe diameter to find the length of the radius.

$\text{radius}=\frac{1}{2}(10)=5$

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

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

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

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

Solve.

$\text{Area}=25\pi$

Report an Error

Example Question #91 : How To Find The Area Of A Circle

Find the area of a circle that is inscribed in a square that has a diagonal of $12\sqrt2$ .

Possible Answers:

$288\pi$

$24\pi$

$36\pi$

$144\pi$

Correct answer:

$36\pi$

Explanation:

First, use the Pythagorean theorem to find the length of a side of the square.

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

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

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

$\text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}$

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

$\text{side}=\frac{12\sqrt2(\sqrt2)}{2}$

Simplify.

$\text{side}=\frac{24}{2}$

Solve.

$\text{side}=12$

Now, recall the relationship between the diameter of the circle and the side of the square.

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

Now, recall that the radius is half the length of the diameter.

$\text{radius}=\frac{1}{2}(\text{diameter})$

Substitute in the value o fthe diameter to find the length of the radius.

$\text{radius}=\frac{1}{2}(12)=6$

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

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

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

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

Solve.

$\text{Area}=36\pi$

Report an Error

Example Question #93 : Plane Geometry

Find the area of a circle inscribed in a square that has a diagonal of $14\sqrt2$ .

Possible Answers:

$49\pi$

$42\pi$

$14\pi$

$28\pi$

Correct answer:

$49\pi$

Explanation:

First, use the Pythagorean theorem to find the length of a side of the square.

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

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

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

$\text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}$

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

$\text{side}=\frac{14\sqrt2(\sqrt2)}{2}$

Simplify.

$\text{side}=\frac{28}{2}$

Solve.

$\text{side}=14$

Now, recall the relationship between the diameter of the circle and the side of the square.

$\text{diameter}=\text{side}=14$

Now, recall that the radius is half the length of the diameter.

$\text{radius}=\frac{1}{2}(\text{diameter})$

Substitute in the value o fthe diameter to find the length of the radius.

$\text{radius}=\frac{1}{2}(14)=7$

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

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

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

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

Solve.

$\text{Area}=49\pi$

Report an Error

Example Question #96 : How To Find The Area Of A Circle

Find the area of a circle that is inscribed in a square that has a diagonal of $16\sqrt2$ .

Possible Answers:

$16\pi$

$64\pi$

$256\pi$

$32\pi$

Correct answer:

$64\pi$

Explanation:

First, use the Pythagorean theorem to find the length of a side of the square.

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

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

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

$\text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}$

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

$\text{side}=\frac{16\sqrt2(\sqrt2)}{2}$

Simplify.

$\text{side}=\frac{32}{2}$

Solve.

$\text{side}=16$

Now, recall the relationship between the diameter of the circle and the side of the square.

$\text{diameter}=\text{side}=16$

Now, recall that the radius is half the length of the diameter.

$\text{radius}=\frac{1}{2}(\text{diameter})$

Substitute in the value o fthe diameter to find the length of the radius.

$\text{radius}=\frac{1}{2}(16)=8$

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

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

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

$\text{Area}=\pi\times (8^2)$

Solve.

$\text{Area}=64\pi$

Report an Error

Example Question #94 : Plane Geometry

Find the area of a circle inscribed in a square that has a diagonal of $18\sqrt2$ .

Possible Answers:

$72\pi$

$81\pi$

$36\pi$

$18\pi$

Correct answer:

$81\pi$

Explanation:

First, use the Pythagorean theorem to find the length of a side of the square.

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

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

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

$\text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}$

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

$\text{side}=\frac{18\sqrt2(\sqrt2)}{2}$

Simplify.

$\text{side}=\frac{36}{2}$

Solve.

$\text{side}=18$

Now, recall the relationship between the diameter of the circle and the side of the square.

$\text{diameter}=\text{side}=18$

Now, recall that the radius is half the length of the diameter.

$\text{radius}=\frac{1}{2}(\text{diameter})$

Substitute in the value o fthe diameter to find the length of the radius.

$\text{radius}=\frac{1}{2}(18)=9$

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

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

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

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

Solve.

$\text{Area}=81\pi$

Report an Error

Example Question #95 : Plane Geometry

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

Possible Answers:

$36\pi$

$24\pi$

$18\pi$

$15\pi$

Correct answer:

$18\pi$

Explanation:

First, use the Pythagorean theorem to find the length of a side of the square.

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

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

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

$\text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}$

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

$\text{side}=\frac{12(\sqrt2)}{2}$

Simplify.

$\text{side}=6\sqrt2$

Now, recall the relationship between the diameter of the circle and the side of the square.

$\text{diameter}=\text{side}=6\sqrt2$

Now, recall that the radius is half the length of the diameter.

$\text{radius}=\frac{1}{2}(\text{diameter})$

Substitute in the value o fthe diameter to find the length of the radius.

$\text{radius}=\frac{1}{2}(6\sqrt2)$

Solve.

$\text{radius}=3\sqrt2$

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

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

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

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

Solve.

$\text{Area}=18\pi$

Report an Error

Example Question #96 : Plane Geometry

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

Possible Answers:

$\pi$

$2\pi$

$4\pi$

$6\pi$

Correct answer:

$2\pi$

Explanation:

First, use the Pythagorean theorem to find the length of a side of the square.

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

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

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

$\text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}$

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

$\text{side}=\frac{4(\sqrt2)}{2}$

Simplify.

$\text{side}=2\sqrt2$

Now, recall the relationship between the diameter of the circle and the side of the square.

$\text{diameter}=\text{side}=2\sqrt2$

Now, recall that the radius is half the length of the diameter.

$\text{radius}=\frac{1}{2}(\text{diameter})$

Substitute in the value o fthe diameter to find the length of the radius.

$\text{radius}=\frac{1}{2}(2\sqrt2)$

Solve.

$\text{radius}=\sqrt2$

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

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

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

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

Solve.

$\text{Area}=2\pi$

Report an Error

Example Question #100 : How To Find The Area Of A Circle

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

Possible Answers:

$\frac{25}{2}\pi$

$\frac{50}{3}\pi$

$\frac{15}{2}\pi$

$25\pi$

Correct answer:

$\frac{25}{2}\pi$

Explanation:

First, use the Pythagorean theorem to find the length of a side of the square.

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

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

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

$\text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}$

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

$\text{side}=\frac{10(\sqrt2)}{2}$

Simplify.

$\text{side}=5\sqrt2$

Now, recall the relationship between the diameter of the circle and the side of the square.

$\text{diameter}=\text{side}=5\sqrt2$

Now, recall that the radius is half the length of the diameter.

$\text{radius}=\frac{1}{2}(\text{diameter})$

Substitute in the value o fthe diameter to find the length of the radius.

$\text{radius}=\frac{1}{2}(5\sqrt2)$

Solve.

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

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

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

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

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

Solve.

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

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 #91 : Plane Geometry

Example Question #91 : Basic Geometry

Example Question #91 : How To Find The Area Of A Circle

Example Question #91 : How To Find The Area Of A Circle

Example Question #93 : Plane Geometry

Example Question #96 : How To Find The Area Of A Circle

Example Question #94 : Plane Geometry

Example Question #95 : Plane Geometry

Example Question #96 : Plane Geometry

Example Question #100 : How To Find The Area Of A Circle

All Basic Geometry Resources

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