Calculating the length of a radius - GMAT Quantitative Reasoning

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Question

A square and a circle have the same area. What is the ratio of the length of one side of the square to the radius of the circle?

Answer

Let be the sidelength of the square is the square and be the radius of the circle. Then since the areas of the circle and the square are equal, we can set up this equation:

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

We find the ratio of to - that is, $\frac{s}{r}$ - as follows:

$s^{2 }\div r^{2}= \pi r^{2} \div r^{2}$

$\frac{s^{2 } }{r^{2}} = \pi$

$\sqrt{\frac{s^{2 } }{r^{2}}} =\sqrt{ \pi}$

$\frac{s}{r} =\sqrt{\pi } = \frac{\sqrt{\pi }}{1}$

The correct ratio is $\sqrt{\pi } \textrm{ to }1$ .

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Question

The radius of Circle A is three times the radius of Circle B, which is, in turn, equal to the diameter of Circle C. The sum of their circumferences is $99\pi$ . What is the radius of Circle C?

Answer

Let be the radius of circle C. Then Circle B has radius equal to twice this, or , and Circle A has radius equal to three times that, or . The circumference of a circle is equal to $2 \pi$ times the radius, so the circumferences of the three circles are:

A: $2 \pi \cdot 6r = 12 \pi r$

B: $2 \pi \cdot 2r = 4 \pi r$

C: $2 \pi r$

These circumferences add up to $99\pi$ , so set up and solve the equation:

$12 \pi r + 4\pi r + 2 \pi r =99 \pi$

$18 \pi r =99 \pi$

$18 \pi r \div 18 \pi =99 \pi \div 18 \pi$

$r =5 \frac{1}{2}$

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Question

If the circumference of a circle is $42\pi$ , what is its radius?

Answer

Using the formula for the circumference of a circle, we can solve for its radius. Plugging in the given value for the circumference, we have:

$C=2\pi r$

$42\pi=2\pi r$

$r=\frac{42\pi }{2\pi }=21$

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Question

If the area of a circle is $25\pi$ , what is its radius?

Answer

Using the formula for the area of a circle, we can solve for its radius. Plugging in the given value for the area of the circle, we have:

$A=\pi r^2$

$25\pi=\pi r^2$

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Question

The points and form a line which passes through the center of circle Q. Both points are on circle Q.

To the nearest hundreth, what is the length of the radius of circle Q?

Answer

To begin this problem, we need to recognize that the distance between points L and K is our diameter. Segment LK passes from one point on circle Q through the center, to another point on circle Q. Sounds like a diameter to me! Use distance formula to find the length of LK.

$d=\sqrt{(x-x')^2+(y-y')^2}$

Plug in our points and simplify:

$d=\sqrt{(9--4)^2+(3-5)^2}=\sqrt{13^2+2^2}=\sqrt{173}\approx13.15$

Now, don't be fooled into choosing 13.15. That is our diameter, so our radius will be half of 13.15, or 6.575. This rounds to 6.58

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Question

If the trunk of a particular tree is $34\pi$ feet around at chest height, what is the radius of the tree at the same height?

Answer

A close reading of the question reveals that we are given the circumference and asked to find the radius.

Circumference formula:

$C=2 \pi r$

So,

$34 \pi=2 \pi r$

$r=34\pi\div2 \pi=17$

So 17 feet

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Question

Given a circumferene of $42 \pi$ , find the circle's radius.

Answer

The circumference of a circle, in terms of radius, can be found by: $C=2r \pi$

We are told the circumference with respect to $\pi$ so we can easily solve for the radius:

$42 \pi = 2 r \pi$

Notice how the $\pi$ cancels out

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Question

The circumference of a circle measures $27\pi$ . Find the radius.

Answer

Solving this problem is rather straightforward, we just need to remember the circumference of circle is found by $C=2r\pi$ and in this case, we're given the circumference in terms of $\pi$ .

$27\pi = 2 r \pi$

Notice how the $\pi$ cancel out

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Question

Given that the area of a circle is $64\pi$ , determine the radius.

Answer

To solve, use the formula for the area of a circle, $A={\pi}r^2$ , and solve for .

$64\pi=\pi{r^2}\Rightarrow 64=r^2\Rightarrow r=8$

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Question

A $60^{\circ }$ arc of a circle measures . Give the radius of this circle.

Answer

A $60^{\circ }$ arc of a circle is $\frac{60}{360 } = \frac{1}{6}$ of the circle. Since the length of this arc is , the circumference is $\textup{six times }$ this, or

The radius of a circle is its circumference divided by $2 \pi$ ; therefore, the radius is

$r = \frac{C}{2 \pi} = \frac{6x+24}{2\pi} = \frac{3}{\pi}x+\frac{12}{\pi}$

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Question

The arc $\widehat{AB}$ of a circle measures $120^{\circ }$ . The chord of the arc, $\overline{AB}$ , has length . Give the length of the radius of the circle.

Answer

A circle can be divided into three congruent arcs that measure

$\frac{1}{3} \cdot 360 ^{\circ } = 120 ^{\circ }$ .

If the three (congruent) chords are constructed, the figure will be an equilateral triangle. The figure is below, along with the altitudes of the triangle:

Since , it follows by way of the 30-60-90 Triangle Theorem that

$BM = \frac{1}{2} AB = \frac{1}{2} (x+2) = \frac{1}{2} x+1$

and

$AM = \frac{ \sqrt{3}}{2} \cdot BM = \frac{ \sqrt{3}}{2} \cdot \left ( \frac{1}{2} x+1 \right ) = \frac{\sqrt{3}}{4} x+ \frac{ \sqrt{3}}{2}$

The three altitudes of an equilateral triangle split each other into segments that have ratio 2:1. Therefore,

$OA = \frac{2}{3} \cdot AM$

$=\frac{2}{3} \cdot \left ( \frac{\sqrt{3}}{4} x+ \frac{ \sqrt{3}}{2} \right )$

$= \frac{\sqrt{3}}{6} x+ \frac{ \sqrt{3}}{3}$

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Question

The arc $\widehat{AB}$ of a circle measures $60^{\circ }$ . The chord of the arc, $\overline{AB}$ , has length . Give the length of the radius of the circle.

Answer

A circle can be divided into congruent arcs that measure

$\frac{1}{6} \cdot 360 ^{\circ } = 60 ^{\circ }$ .

If the (congruent) chords are constructed, the figure will be a regular hexagon. The radius of this hexagon will be equal to the length of one side - one $60 ^{\circ }$ chord of the circle; this radius will coincide with the radius of the circle. Therefore, the radius of the circle is the length of chord $\overline{AB}$ , or .

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Question

The arc $\widehat{AB}$ of a circle measures $90^{\circ }$ . The chord of the arc, $\overline{AB}$ , has length . Give the radius of the circle.

Answer

A circle can be divided into four congruent arcs that measure

$\frac{1}{4} \cdot 360 ^{\circ } = 90 ^{\circ }$ .

If the four (congruent) chords of the arcs are constructed, they will form a square with sides of length . The diagonal of a square has length $\sqrt{2}$ times that of a side, which will be

$(x + 2) \sqrt{2} = x \sqrt{2} + 2 \sqrt{2}$

A diagonal of the square is also a diameter of the circle; the circle will have radius half this length, or

$\frac{1}{2} \left ( x \sqrt{2} + 2 \sqrt{2} \right ) =\frac{1}{2} x \sqrt{2} + \sqrt{2}$

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Question

Two circles in the same plane have the same center. The larger circle has radius 10; the area of the region between the circles is $50 \pi$ . What is the radius of the smaller circle?

Answer

The area of a circle with radius is $A = \pi r^{2}$ .

Let be the radius of the smaller circle. Its area is $A_{1} = \pi R^{2}$ . The area of the larger circle is $A_{2} = \pi \cdot 10^{2} = 100 \pi$ . Since the area of the region between the circles is $50 \pi$ , and is the difference of these areas, we have

$100 \pi - \pi R^{2} = 50 \pi$

$100 \pi - \pi R^{2} - 50 \pi + \pi R^{2} = 50 \pi - 50 \pi + \pi R^{2}$

$50 \pi = \pi R^{2}$

$R^{2} = 50$

$R = \sqrt{50} = \sqrt {25 } \cdot \sqrt{2} = 5 \sqrt{2}$

The smaller circle has radius $5 \sqrt{2}$ .

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Question

Two circles in the same plane have the same center. The smaller circle has radius 10; the area of the region between the circles is $50 \pi$ . What is the radius of the larger circle?

Answer

The area of a circle with radius is $A = \pi r^{2}$ .

Let be the radius of the larger circle. Its area is $A_{1} = \pi R^{2}$ . The area of the smaller circle is $A_{2} = \pi \cdot 10^{2} = 100 \pi$ . Since the area of the region between the circles is $50 \pi$ , and is the difference of these areas, we have

$\pi R^{2} - 100 \pi = 50 \pi$

$\pi R^{2} - 100 \pi+ 100 \pi = 50 \pi + 100 \pi$

$\pi R^{2} = 150 \pi$

$R^{2} = 150$

$R = \sqrt{150} = \sqrt {25 } \cdot \sqrt{6} = 5 \sqrt{6}$

The smaller circle has radius $5 \sqrt{6}$ .

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Question

If a monster truck's wheels have circumference of , what is the distance from the ground to the center of the wheel?

Answer

If a monster truck's wheels have circumference of , what is the distance from the ground to the center of the wheel?

This question is asking us to find the radius of a circle. the distance from the outside of the circle to the center is the radius. We are given the circumference, so use the following formula:

$Circ=2 \pi r$

Then, plug in what we know and solve for r

$31.4=2 \pi r$

$r=\frac{31.4}{2 \pi}=4.997\approx5ft$

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