How to find circumference

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Math › How to find circumference

Questions 1 - 10

The circumferences of eight circles form an arithmetic sequence. The smallest circle has radius two inches; the second smallest circle has radius five inches. Give the radius of the largest circle.

1 foot, 11 inches

2 feet

2 feet, 1 inch

4 feet 2 inches

3 feet 10 inches

Explanation

The circumference of a circle can be determined by multiplying its radius by $2 \pi$ , so the circumferences of the two smallest circles are

$C_{1} = 2 \cdot 2 \pi = 4 \pi$

and

$C_{2} = 2 \cdot 5 \pi = 10 \pi$

The circumferences form an arithmetic sequence with common difference

$C_{2}- C_{1}= 10 \pi - 4 \pi = 6 \pi$

The circumference of a circle can therefore be found using the formula

$C_{n} = C_{1} + (n-1)d$

where $C_{1} = 4 \pi$ and $d = 6 \pi$ ; we are looking for that of the th smallest circle, so

$C_{8} = 4 \pi + (8-1)6 \pi$

$= 4 \pi + 7 \cdot 6 \pi$

$= 4 \pi + 42 \pi$

$= 46 \pi$

Since the radius of a circle is the circumference of the circle divided by $2 \pi$ , the radius of this eighth circle is

$r = \frac{46 \pi}{2 \pi} = 23$ inches, or 1 foot 11 inches.

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

2√13_π_

√13_π_

26_π_

√26_π_

13_π_

Explanation

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_.

Track

The track at Simon Bolivar High School is a perfect circle of radius 500 feet, and is shown in the above figure. Manuel starts at point C, runs around the track counterclockwise three times, and continues to run clockwise until he makes it to point D. Which of the following comes closest to the number of miles Manuel has run?

$2\frac{1}{4} \textup{ mi}$

$2\frac{1}{2} \textup{ mi}$

$2\frac{3}{4} \textup{ mi}$

$2 \textup{ mi}$

$3 \textup{ mi}$

Explanation

The circumference of a circle with radius 500 feet is

$C = 2 \pi r \approx 2 \cdot 3.14 \cdot 500 = 3,140$ feet.

Manuel runs this distance three times, then he runs from Point C to D, which is about four-fifths of this distance. Therefore, Manuel's run will be about

$3,140 \times 3 \frac{4}{5} = 11,932$ feet.

Divide by 5,280 to convert to miles:

$11,932 \div 5,280 \approx 2.26$ ,

making $2\frac{1}{4}$ miles the response closest to the actual running distance.

Find the circumference of a circle that is inscribed in a square that has side lengths of .

$3\pi$

$\frac{3\pi}{2}$

$\frac{9\pi}{4}$

$6\pi$

Explanation

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

Recall how to find the circumference of a circle:

$\text{Circumference}=\text{Diameter}\times\pi$

Plug in the given diameter to find the circumference.

$\text{Circumference}=3\pi$

Find the circumference of a circle inscribed in a square that has a diagonal of $64\sqrt2$ .

$64\pi$

$32\pi$

$4096\pi$

$1024\pi$

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.

$\text{side}=\frac{64\sqrt2(\sqrt2)}{2}$

Simplify.

$\text{side}=64$

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

$\text{diameter}=\text{side}=64$

Now, recall how to find the circumference of a circle.

$\text{Circumference}=\pi\times\text{diameter}$

Substitute in the diameter you just found to find the circumference.

$\text{Circumference}=64\pi$

What is the circumference of a circle with a radius of 12?

$144\pi$

$13\pi$

$12\pi$

$24\pi$

Explanation

To find the circumference of a circle given the radius we must first know the equation for the circumference of a circle which is $Cicumference=2r\pi$

We then plug in the number for the radius into the equation yielding $Cicumference=2(12)\pi$

We multiply to find the value for the circumference is $24\pi$ .

The answer is $24\pi$ .

Track

The track at James Buchanan High School is shown above; it is comprised of a square and a semicircle.

Diane wants to run two miles. If she begins at Point A and begins running counterclockwise, when she is finished, which of the five points will she be closest to?

Explanation

First, it is necessary to know the length of the semicircle connecting Points B and D, which has diameter 400 feet; this length is

$\frac{1}{2} C = \frac{1}{2} \cdot \pi d \approx \frac{1}{2} \cdot 3.14 \cdot 400 = 628$ feet.

The distance around the track is about

feet.

Diane wants to run two miles, or

$5,280 \times 2 = 10, 560$ feet.

She will make about

$10,560 \div 1,828 \approx 5.78$

circuits around the track.

Equivalently, she will run the track 5 complete times for a total of about

$5 \times 1,828 = 9,140$ feet,

so she will have

feet to go.

She is running counterclockwise, so she will proceed from Point A to Point D, running another 800 feet, leaving

feet.

She will almost, but not quite, finish the 628 feet from Point D to Point B.

The correct response is Point B.

Track

The track at Truman High School is shown above; it is comprised of a square and a semicircle.

Veronica begins at Point A, runs three times around the track counterclockwise, and continues until she reaches Point B. Which of the following comes closest to the distance Veronica runs?

$1\frac{1}{2} \textup{ mi}$

$1 \textup{ mi}$

$1\frac{1}{4} \textup{ mi}$

$1\frac{3}{4} \textup{ mi}$

$2 \textup{ mi}$

Explanation

First, it is necessary to know the length of the semicircle connecting Points B and D, which has diameter 500 feet; this length is about

$\frac{1}{2} C = \frac{1}{2} \cdot \pi d \approx \frac{1}{2} \cdot 3.14 \cdot 500 = 785$ feet.

The distance around the track is about

feet.

Veronica runs around the track three complete times, for a distance of about

$3 \times 2,285 = 6,855$ feet.

She then runs from Point A to Point E, which is another 500 feet; Point E to Point D, which is yet another 500 feet, and, finally Point D to Point B, for a final 785 feet. The total distance Veronica runs is about

feet.

Divide by 5,280 to convert to miles:

$8,640 \div5,280 \approx 1.6$

The closest answer is $1 \frac{1}{2}$ miles.

Track

The track at Monroe High School is a perfect circle of radius 600 feet, and is shown in the above figure. Quinnella wants to run around the track for one and a half miles. If Quinnella starts at point C and runs counterclockwise, which of the following is closest to the point at which she will stop running?

(Assume the five points are evenly spaced)

Between Points B and C

Between Points C and D

Between Points D and E

Between Points E and A

Between Points A and B

Explanation

A circle of radius 600 feet will have a circumference of

$C = 2 \pi r \approx 2 \cdot 3.14 \cdot 600 = 3,768$ feet.

Quinnella will run one and a half miles, or

$5,280 \times 1\frac{1}{2} = 7,920$ feet,

which is about times the circumference of the circle.

Quinnella will run around the track twice, returning to Point C; she will not quite make it to Point B a third time, since that is one-fifth of the track, or 0.2. The correct response is that she will be between Points B and C.

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

2√13_π_

√13_π_

26_π_

√26_π_

13_π_