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Basic Geometry : Circles

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Example Questions

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Example Question #111 : Circles

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

Possible Answers:

$100\pi$

$400\pi$

$40\pi$

$20\pi$

Correct answer:

$100\pi$

Explanation:

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

$\text{Diameter}=\text{Diagonal}=20$

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.

$\text{radius}=\frac{1}{2}(20)$

Simplify.

$\text{radius}=10$

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.

$\text{Area}=\pi\times(10)^2$

Solve.

$\text{Area}=100\pi$

Report an Error

Example Question #111 : Plane Geometry

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

Possible Answers:

$28\pi$

$49\pi$

$241\pi$

$196\pi$

Correct answer:

$49\pi$

Explanation:

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

$\text{Diameter}=\text{Diagonal}=14$

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.

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

Simplify.

$\text{radius}=7$

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.

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

Solve.

$\text{Area}=49\pi$

Report an Error

Example Question #113 : Circles

If a rectangle with a diagonal of $6\sqrt3$ is inscribed in a circle, what is the area of the circle?

Possible Answers:

$27\pi$

$6\sqrt3\pi$

$54\pi$

Correct answer:

$27\pi$

Explanation:

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

$\text{Diameter}=\text{Diagonal}=6\sqrt3$

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.

$\text{radius}=\frac{1}{2}(6\sqrt3)$

Simplify.

$\text{radius}=3\sqrt3$

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.

$\text{Area}=\pi\times(3\sqrt3)^2$

Solve.

$\text{Area}=27\pi$

Report an Error

Example Question #111 : Basic Geometry

If a rectangle with a diagonal of $18\sqrt2$ is inscribed in a circle, what is the area of the circle?

Possible Answers:

$18\sqrt2\pi$

$72\pi$

$162\pi$

$81\pi$

Correct answer:

$162\pi$

Explanation:

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

$\text{Diameter}=\text{Diagonal}=18\sqrt2$

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.

$\text{radius}=\frac{1}{2}(18\sqrt2)$

Simplify.

$\text{radius}=9\sqrt2$

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.

$\text{Area}=\pi\times(9\sqrt2)^2$

Solve.

$\text{Area}=162\pi$

Report an Error

Example Question #115 : Circles

If a rectangle with a diagonal of $16\sqrt3$ is inscribed in a circle, what is the area of the circle?

Possible Answers:

$81\pi$

$16\sqrt3\pi$

$32\sqrt2\pi$

$192\pi$

Correct answer:

$192\pi$

Explanation:

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

$\text{Diameter}=\text{Diagonal}=16\sqrt3$

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.

$\text{radius}=\frac{1}{2}(16\sqrt3)$

Simplify.

$\text{radius}=8\sqrt3$

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.

$\text{Area}=\pi\times(8\sqrt3)^2$

Solve.

$\text{Area}=192\pi$

Report an Error

Example Question #116 : Circles

A rectangle with a diagonal of $2\sqrt{13}$ is incribed in a circle, what is the area of the circle?

Possible Answers:

$39\pi$

$52\pi$

$26\pi$

$13\pi$

Correct answer:

$13\pi$

Explanation:

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

$\text{Diameter}=\text{Diagonal}=2\sqrt{13}$

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.

$\text{radius}=\frac{1}{2}(2\sqrt{13})$

Simplify.

$\text{radius}=\sqrt{13}$

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.

$\text{Area}=\pi\times(\sqrt{13})^2$

Solve.

$\text{Area}=13\pi$

Report an Error

Example Question #111 : How To Find The Area Of A Circle

Find the area of a circle given radius is x^3 .

Possible Answers:

$x^3\pi$

$x^6\pi$

$x^5\pi$

$x^2\pi$

Correct answer:

$x^6\pi$

Explanation:

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

Given the radius is x^3 , substitute it in for r in the below formula.

Also recall the rules of exponents which states that when an exponent is raised to another exponent, they are multiplied together.

$(x^n)^m=x^{m\cdot n}$

Thus,

$A=\pi{r^2}=\pi*(x^3)^2=\pi*x^{3*2}=x^6\pi$

Report an Error

Example Question #111 : Basic Geometry

Find the area of a circle given a radius of 3.

Possible Answers:

$27\pi$

$6\pi$

$9\pi$

$\frac{9}{2}\pi$

Correct answer:

$9\pi$

Explanation:

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

Thus,

given, r=3

$A=\pi{r^2}=\pi*3^2=9\pi$ .

Report an Error

Example Question #114 : Plane Geometry

In the figure below, a circle is inscribed in a square. If the side length of the square is , what is the area of the shaded region?

Possible Answers:

0.86

0.88

1.04

0.96

Correct answer:

0.86

Explanation:

From the figure, you should notice that the side length of the square is also the same as the length of the diameter of the circle.

To find the area of the shaded region, we will need to subtract the area of the circle from the area of the square.

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

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

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

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

Now, find the area of the circle.

We will need to first find the length of the radius of the circle.

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

Plug in the value of the diameter to find the length of the radius.

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

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

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

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

$\text{Area of Circle}=\pi\times 1^2=\pi$

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

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

$\text{Area of Shaded Region}=4-\pi=0.86$

Round to two places after the decimal.

Report an Error

Example Question #111 : How To Find The Area Of A Circle

In the figure below, a circle is inscribed in a square. If the length of a side of the square is , what is the area of the shaded region?

Possible Answers:

3.43

3.19

3.67

3.34

Correct answer:

3.43

Explanation:

From the figure, you should notice that the side length of the square is also the same as the length of the diameter of the circle.

To find the area of the shaded region, we will need to subtract the area of the circle from the area of the square.

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

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

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

$\text{Area of Square}=4^2=16$

Now, find the area of the circle.

We will need to first find the length of the radius of the circle.

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

Plug in the value of the diameter to find the length of the radius.

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

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

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

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

$\text{Area of Circle}=\pi\times 2^2=4\pi$

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

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

$\text{Area of Shaded Region}=16-4\pi=3.43$

Round to two places after the decimal.

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 #111 : Circles

Example Question #111 : Plane Geometry

Example Question #113 : Circles

Example Question #111 : Basic Geometry

Example Question #115 : Circles

Example Question #116 : Circles

Example Question #111 : How To Find The Area Of A Circle

Example Question #111 : Basic Geometry

Example Question #114 : Plane Geometry

Example Question #111 : How To Find The Area Of A Circle

All Basic Geometry Resources

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