Circles - Geometry

Card 0 of 1632

Question

Find the length of the radius of a circle given the circumference is $6\pi$ .

Answer

To solve, simply use the formula for the circumference of a circle and solve for r. Thus,

$C=2\pi{r}\Rightarrow r=\frac{C}{2\pi}$

$r=\frac{6\pi}{2\pi}=3$

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Question

Find the radius of a circle given the circumference is $8\pi$ .

Answer

To solve, simply use the formula for the circumference of a circle and solve for r. Thus,

$C=2\pi{r}\Rightarrow r=\frac{C}{2\pi}$

$r=\frac{8\pi}{2\pi}=4$

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Question

Find the length of the radius of a circle given the diameter is 3.

Answer

To solve, simply use the formula for the diameter and solve for r. Thus,

Therefore, when solved for r,

$r=\frac{d}{2}$

Plug in d and:

$r=\frac{3}{2}$

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Question

Find the ratio of the diameter to the circumference in a circle with radius r.

Answer

To find a ratio, simply divide your two quantities. Remember,

$C=2\pi{r}$

Thus,

$ratio=\frac{D}{C}=\frac{2r}{2\pi{r}}=\frac{1}{\pi}$

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Question

A circle with an area of 13_π_ in2 is centered at point C. What is the circumference of this circle?

Answer

The formula for the area of a circle is A = _πr_2.

We are given the area, and by substitution we know that 13_π_ = _πr_2.

We divide out the π and are left with 13 = _r_2.

We take the square root of r to find that r = √13.

We find the circumference of the circle with the formula C = 2_πr_.

We then plug in our values to find C = 2√13_π_.

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Question

A 6 by 8 rectangle is inscribed in a circle. What is the circumference of the circle?

Answer

First you must draw the diagram. The diagonal of the rectangle is also the diameter of the circle. The diagonal is the hypotenuse of a multiple of 2 of a 3,4,5 triangle, and therefore is 10.
Circumference = π * diameter = 10_π_.

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Question

Ashley has a square room in her apartment that measures 81 square feet. What is the circumference of the largest circular area rug that she can fit in the space?

Answer

In order to solve this question, first calculate the length of each side of the room.

The length of each side of the room is also equal to the length of the diameter of the largest circular rug that can fit in the room. Since $C=\pi d$ , the circumference is simply

$C=9\pi$

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Question

A gardener wants to build a fence around their garden shown below. How many feet of fencing will they need, if the length of the rectangular side is 12 and the width is 8?

Answer

The shape of the garden consists of a rectangle and two semi-circles. Since they are building a fence we need to find the perimeter. The perimeter of the length of the rectangle is 24. The perimeter or circumference of the circle can be found using the equation C=2π(r), where r= the radius of the circle. Since we have two semi-circles we can find the circumference of one whole circle with a radius of 4, which would be 8π.

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Question

If a circle has an area of $81\pi$ , what is the circumference of the circle?

Answer

The formula for the area of a circle is πr2. For this particular circle, the area is 81π, so 81π = πr2. Divide both sides by π and we are left with r2=81. Take the square root of both sides to find r=9. The formula for the circumference of the circle is 2πr = 2π(9) = 18π. The correct answer is 18π.

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Question

A car tire has a radius of 18 inches. When the tire has made 200 revolutions, how far has the car gone in feet?

Answer

If the radius is 18 inches, the diameter is 3 feet. The circumference of the tire is therefore 3π by C=d(π). After 200 revolutions, the tire and car have gone 3π x 200 = 600π feet.

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Question

A circle has the equation below. What is the circumference of the circle?

(x – 2)2 + (y + 3)2 = 9

Answer

The radius is 3. Yielding a circumference of $6\pi$ .

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Question

The diameter of a circle is defined by the two points (2,5) and (4,6). What is the circumference of this circle?

Answer

We first must calculate the distance between these two points. Recall that the distance formula is:√((x2 - x1)2 + (y2 - y1)2)

For us, it is therefore: √((4 - 2)2 + (6 - 5)2) = √((2)2 + (1)2) = √(4 + 1) = √5

If d = √5, the circumference of our circle is πd, or π√5.

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Question

What is the circumference of a circle with a radius of $5.2\textup{ in}$ ?

(Round your answer to the nearest tenth.)

Answer

The circumference is given by the formula:

$C=2\pi r$

where is the radius.

$2\pi (5.2)=32.7\ in$

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Question

If this circle has a diameter of 12 inches, what is its circumference?

Answer

Know that the formula for circumference is $C=D\pi$ , where C is the circumference and D is the diameter. It is given that the diameter is 12 inches. Therefore, the circumference is $C=12\pi \hspace\ \text{inches}$

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Question

If a circle has an area of $16\pi ; in^{2}$ , what is the circumference?

Answer

For a circle, the formula for area is $A = \pi r^{2}$ and the formula for circumference is $C=\pi d$ , where is the radius and is the diameter.

Plug the known quantities into the area formula and solve for the radius:

$16\pi =\pi r^{2}$

Now plug this value into the circumference formula to solve:

$C = \pi d = \pi 2r = \pi 2(4) = 8\pi; in$

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Question

A circle is circumscribed around a square that has sides of 5cm. What is the perimeter of the circle?

Answer

Because the circle is circumscribed around the square, the diagonal of the square is the same as the diameter of the circle. So to find the diagonal we use the Pythagorean Theorem:

$side^{2}+side^{2}=diagonal^{2}$

So the equation to solve becomes $5^{2}+5^{2}=d^{2}$ or $d= 5\sqrt{2}$

The perimeter, or circumference, of the circle is given by $P=\pi d$ or $P=5\sqrt{2}\pi ; cm$

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Question

Line is the radius of the circle shown and has a length of 7.5. What is the circumference of the circle?

Answer

Plug in the given value for the radius to find the circumference:

$C=2\pi r$

$C=2(7.5)\pi=15\pi$

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Question

Find the perimeter of the ice rink formed by the rectangle and semi-circles with the dimensions shown.

Answer

The perimeter of the rink runs along the outside of the combined shapes. The two outer edges of the rectangle are each, so of the perimeter consists of those two sides.

$60\ ft2=120\ ft$

The two semi-circles that form the remainder of the perimeter, and can be considered two halves of one circle. The diameter of the circle would be $25\ ft$ . The circumference of the circle would account for the outer edges of the two semi-circles in the ice rink. Find the circumference of the circle with the formula $c = \pi d$ as follows.

$c = \pi25$

$c = 25\pi\ ft$

Finally, add this circumference that accounts for the semi-circles to the length of the two sides formed by the rectangle to find the total perimeter of the rink.

$120 + 25\pi$

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Question

The diameter of a circle is 16 centimeters. What is the circumference of the circle?

Answer

To find the circumference of a circle, multiply the circle's diameter by pi (3.14).

$16\cdot 3.14=50.24$

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Question

Find the circumference of a circle that has a radius of $\frac{1}{2}$ .

Answer

Recall the formula for finding the circumference of a circle:

$\text{Circumference}=2\times\text{radius}\times\pi$

We can substitute in the value for the radius in order to find the circumference of the circle in question.

$\text{Circumference}=2\times \frac{1}{2} \times\pi$

Solve.

$\text{Circumference}=\frac{2}{2} \times\pi$

Simplify.

$\text{Circumference}= \pi$

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