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Basic Geometry : Circles

Study concepts, example questions & explanations for Basic Geometry

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9 Diagnostic Tests 164 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

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Example Question #101 : Radius

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

Possible Answers:

$32\pi$

$40\pi$

$48\pi$

$24\pi$

Correct answer:

$32\pi$

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.

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

Simplify.

$\text{side}=8\sqrt2$

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

$\text{diameter}=\text{side}=8\sqrt2$

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.

$\text{radius}=\frac{1}{2}(8\sqrt2)$

Solve.

$\text{radius}=4\sqrt2$

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.

$\text{Area}=\pi\times (4\sqrt2)^2$

Solve.

$\text{Area}=32\pi$

Report an Error

Example Question #102 : Radius

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

Possible Answers:

$25\pi$

$75\pi$

$50\pi$

$100\pi$

Correct answer:

$50\pi$

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.

$\text{side}=\frac{20(\sqrt2)}{2}$

Simplify.

$\text{side}=10\sqrt2$

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

$\text{diameter}=\text{side}=10\sqrt2$

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.

$\text{radius}=\frac{1}{2}(10\sqrt2)$

Solve.

$\text{radius}=5\sqrt2$

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.

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

Solve.

$\text{Area}=50\pi$

Report an Error

Example Question #103 : Radius

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

Possible Answers:

$144\pi$

$72\pi$

$484\pi$

$12\pi$

Correct answer:

$72\pi$

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.

$\text{side}=\frac{24(\sqrt2)}{2}$

Simplify.

$\text{side}=12\sqrt2$

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

$\text{diameter}=\text{side}=12\sqrt2$

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.

$\text{radius}=\frac{1}{2}(12\sqrt2)$

Solve.

$\text{radius}=6\sqrt2$

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.

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

Solve.

$\text{Area}=72\pi$

Report an Error

Example Question #101 : How To Find The Area Of A Circle

Find the area of a circle given radius 4.

Possible Answers:

$32\pi$

$16\pi$

$8\pi$

$64\pi$

Correct answer:

$16\pi$

Explanation:

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

$A=\pi{r^2}=\pi*4^2=16\pi$

Report an Error

Example Question #105 : Radius

If a rectangle with a diagonal of is inscribed in a circle, what is the area of the circle?

Possible Answers:

$2\pi$

Cannot be determined.

$\pi$

$4\pi$

Correct answer:

$\pi$

Explanation:

Notice that the diagonal of the rectangle is the same as the diameter of the circle.

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

Now, recall the relationship between the diameter of a circle and its radius:

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

Substitute in the given diameter to find the radius of the circle.

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

Simplify.

$\text{radius}=1$

Finally, recall how 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.

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

Solve.

$\text{Area}=\pi$

Report an Error

Example Question #106 : Radius

If a rectangle with a diagonal of is inscribed in a circle, what is the area of the circle?

Possible Answers:

$4\pi$

$16\pi$

$8\pi$

$20\pi$

Correct answer:

$4\pi$

Explanation:

Notice that the diagonal of the rectangle is the same as the diameter of the circle.

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

Now, recall the relationship between the diameter of a circle and its radius:

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

Substitute in the given diameter to find the radius of the circle.

$\text{radius}=\frac{1}{2}(4)$

Simplify.

$\text{radius}=2$

Finally, recall how 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.

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

Solve.

$\text{Area}=4\pi$

Report an Error

Example Question #102 : Plane Geometry

If a rectangle with a diagonal of $2\sqrt2$ is inscribed in a circle, what is the area of the circle?

Possible Answers:

$16\pi$

$2\sqrt2\pi$

$4\sqrt2\pi$

$2\pi$

Correct answer:

$2\pi$

Explanation:

Notice that the diagonal of the rectangle is the same as the diameter of the circle.

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

Now, recall the relationship between the diameter of a circle and its radius:

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

Substitute in the given diameter to find the radius of the circle.

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

Simplify.

$\text{radius}=\sqrt2$

Finally, recall how 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.

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

Solve.

$\text{Area}=2\pi$

Report an Error

Example Question #108 : Radius

If a rectangle with a diagonal of $4\sqrt2$ is inscribed in a circle, what is the area of the circle?

Possible Answers:

$10\pi$

$6\pi$

$12\pi$

$8\pi$

Correct answer:

$8\pi$

Explanation:

Notice that the diagonal of the rectangle is the same as the diameter of the circle.

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

Now, recall the relationship between the diameter of a circle and its radius:

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

Substitute in the given diameter to find the radius of the circle.

$\text{radius}=\frac{1}{2}(4\sqrt2)$

Simplify.

$\text{radius}=2\sqrt2$

Finally, recall how 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.

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

Solve.

$\text{Area}=8\pi$

Report an Error

Example Question #109 : Radius

If a rectangle with a diagonal of is inscribed in a circle, what is the area of the circle?

Possible Answers:

$12\pi$

$36\pi$

$24\pi$

Correct answer:

$36\pi$

Explanation:

Notice that the diagonal of the rectangle is the same as the diameter of the circle.

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

Now, recall the relationship between the diameter of a circle and its radius:

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

Substitute in the given diameter to find the radius of the circle.

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

Simplify.

$\text{radius}=6$

Finally, recall how 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.

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

Solve.

$\text{Area}=36\pi$

Report an Error

Example Question #110 : Radius

If a rectangle with a diagonal of is inscribed in a circle, what is the area of the circle?

Possible Answers:

$196\pi$

$72\pi$

$80\pi$

$64\pi$

Correct answer:

$64\pi$

Explanation:

Notice that the diagonal of the rectangle is the same as the diameter of the circle.

$\text{Diameter}=\text{Diagonal}=16$

Now, recall the relationship between the diameter of a circle and its radius:

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

Substitute in the given diameter to find the radius of the circle.

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

Simplify.

$\text{radius}=8$

Finally, recall how 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.

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

Solve.

$\text{Area}=64\pi$

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 #101 : Radius

Example Question #102 : Radius

Example Question #103 : Radius

Example Question #101 : How To Find The Area Of A Circle

Example Question #105 : Radius

Example Question #106 : Radius

Example Question #102 : Plane Geometry

Example Question #108 : Radius

Example Question #109 : Radius

Example Question #110 : Radius

All Basic Geometry Resources

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