Circles

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Geometry › Circles

Questions 1 - 10

A square with a side length of 4 inches is inscribed in a circle, as shown below. What is the area of the unshaded region inside of the circle, in square inches?

Act_math_01

8π - 16

4π-4

8π-4

2π-4

8π-8

Explanation

Using the Pythagorean Theorem, the diameter of the circle (also the diagonal of the square) can be found to be 4√2. Thus, the radius of the circle is half of the diameter, or 2√2. The area of the circle is then π(2√2)2, which equals 8π. Next, the area of the square must be subtracted from the entire circle, yielding an area of 8π-16 square inches.

Find the area of a circle that has a radius of $\sqrt{11}$ .

$11\pi$

$\sqrt{11}\pi$

$2\sqrt{11}\pi$

$\sqrt{22}\pi$

Explanation

Use the following formula to find the area of a circle:

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

For the circle in question, plug in the given radius to find the area.

We know the radius is $\sqrt{11}$ therefore, the area equation becomes,

$\text{Area}=\pi \times (\sqrt{11})^2=11\pi$ .

Recall that when a square root is squared you are left with the number under the square root sign. This happens because when you square a number you are multiplying it by itself. In our case this is,

$(\sqrt{11})^2=\sqrt{11}\cdot \sqrt{11}$ .

From here we can use the property of multiplication and radicals to rewrite our expression as follows,

$\sqrt{11}\cdot \sqrt{11}=\sqrt{11\cdot 11}$

and when there are two numbers that are the same under a square root sign you bring out one and the other number and square root sign go away.

$\sqrt{11\cdot 11}=11$

If the diameter of the circle below is , what is the area of the shaded region?

Explanation

From the given figure, you should notice that the base of the triangle is the same as the diameter of the circle.

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 triangle.

Recall how to find the area of a circle:

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

Now recall the relationship between the radius and the diameter.

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

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

$\text{Radius}=\frac{8}{2}=4$

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

$\text{Area of Circle}=\pi\times 4^2=16\pi$

Next, recall how to find the area of a triangle.

$\text{Area of Triangle}=\frac{\text{base}\times\text{height}}{2}$

The height is already given by the question, and remember that the base is the same as the diameter of the circle.

Plug in these values to find the area of the triangle.

$\text{Area of Triangle}=\frac{4 \times 8}{2}=16$

We are now ready to find the area of the shaded region.

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

$\text{Area of Shaded Region}=16\pi-16=34.27$

Remember to round to decimal places.

A sector in a circle with a radius of has an area of $\frac{1}{4}\pi$ . In degrees, what is the measurement of the central angle of the sector?

$5.625^\circ$

$2.775^\circ$

$6.225^\circ$

$8.555^\circ$

Explanation

Recall how to find the area of a sector:

$\text{Area of Sector}=\frac{\text{Measurement of Central Angle}}{360}\times\pi\times r^2$

Since the question asks for the measurement of the central angle, rearrange the equation like thus:

$\text{Measurement of Central Angle}=\frac{360\times\text{Area of Sector}}{\pi\times r^2}$

Plug in the given information to find the measurement of the central angle.

$\text{Measurement of Central Angle}=\frac{360\times \frac{1}{4}\pi}{\pi\times 4^2}=\frac{90\pi}{16\pi}=5.625$

The central angle is degrees.

In the figure below, $\text{chord AB}\parallel\text{chord CD}$ . If $\text{Arc AC}$ is degrees, in degrees, what is the measure of $\text{Arc BD}$ ?

The measurement of $\text{Arc BD}$ cannot be determined with the information given.

Explanation

Recall that when chords are parallel, the arcs that are intercepted are congruent. Thus, $\text{Arc AC}=\text{Arc BD}$ .

Then, $\text{Arc BD}$ must also be degrees.

What is the sector angle, in degrees, if the area of the sector is $4\pi$ with a given radius of ?

$90^{\circ}$

$0^{\circ}$

$45^{\circ}$

$360^{\circ}$

$60^{\circ}$

Explanation

Write the formula for the area of a circular sector.

$A= \frac{\Theta }{360}\pi r^2$

Substitute the given information and solve for theta:

$4\pi= \frac{\Theta }{360}\pi (4)^2$

$\frac{1}{4}= \frac{\Theta }{360}$

$\Theta= \frac{360}{4} = 90^{\circ}$

A sector in a circle with a radius of has an area of $12\pi$ . In degrees, what is the measurement of the central angle for this sector?

$30^\circ$

$45^\circ$

$60^\circ$

$15^\circ$

Explanation

Recall how to find the area of a sector:

$\text{Area of Sector}=\frac{\text{Measurement of Central Angle}}{360}\times\pi\times r^2$

Since the question asks for the measurement of the central angle, rearrange the equation like thus:

$\text{Measurement of Central Angle}=\frac{360\times\text{Area of Sector}}{\pi\times r^2}$

Plug in the given information to find the measurement of the central angle.

$\text{Measurement of Central Angle}=\frac{360\times12\pi}{\pi\times 12^2}=\frac{4320\pi}{144\pi}=30$

The central angle is degrees.

Find the length of the arc if the radius of a circle is and the measure of the central angle is degrees.

$\frac{11}{3}\pi$

$4\pi$

$\frac{13}{3}\pi$

$5\pi$

Explanation

An arc is just a piece—or a fraction—of a circle's circumference. Use the following formula to find the length of an arc:

$\text{Length of Arc}=\frac{\text{Measure of central angle}}{360}\times2\pi\times \text{radius}$

Substitute in the given values for the central angle and the radius.

$\text{Length of Arc}=\frac{55}{360}\times 2\pi \times 12$

Solve.

$\text{Length of Arc}=\frac{11}{3}\pi$

Find the area of a sector if it has an arc length of and a radius of .

$\frac{3}{2}$

The area of the sector cannot be determined.

$\frac{5}{2}$

$\frac{7}{2}$

Explanation

The length of the arc of the sector is just a fraction of the arc of the circumference. The area of the sector will be the same fraction of the area as the length of the arc is of the circumference.

We can then write the following equation to find the area of the sector:

$\text{Area}=\frac{\text{Arc Length}}{2\pi r}\pi r^2$