GMAT Math : Calculating the length of a radius

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Example Question #1 : Calculating The Length Of A Radius

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?

Possible Answers:

$2 \textrm{ to }\sqrt{\pi }$

$\sqrt{\pi } \textrm{ to }1$

$\pi \textrm{ to }1$

$1 \textrm{ to }\sqrt{\pi }$

$1 \textrm{ to }\pi$

Correct answer:

$\sqrt{\pi } \textrm{ to }1$

Explanation:

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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Example Question #2 : Calculating The Length Of A Radius

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?

Possible Answers:

$5 \frac{1}{2}$

$4 \frac{1}{2}$

Correct answer:

$5 \frac{1}{2}$

Explanation:

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 3 (2r) = 6r . 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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Example Question #3 : Calculating The Length Of A Radius

The points L (-4,5) and K (9,3) 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?

Possible Answers:

13.15

15.00

13.2

6.58

6.6

Correct answer:

6.58

Explanation:

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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Example Question #4 : Calculating The Length Of A Radius

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?

Possible Answers:

96 ft

45 ft

8.5 ft

34 ft

17 ft

Correct answer:

17 ft

Explanation:

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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Example Question #5 : Calculating The Length Of A Radius

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

Possible Answers:

Correct answer:

Explanation:

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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Example Question #6 : Calculating The Length Of A Radius

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

Possible Answers:

Correct answer:

Explanation:

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:

$A=\pi r^2$

$25\pi=\pi r^2$

r^2=25

r=5

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Example Question #1 : Calculating The Length Of A Radius

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

Possible Answers:

$16\pi$

$8\pi$

Correct answer:

Explanation:

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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Example Question #8 : Calculating The Length Of A Radius

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

Possible Answers:

$21\pi$

6.48

Correct answer:

Explanation:

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

42 = 2r

r=21

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Example Question #9 : Calculating The Length Of A Radius

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

Possible Answers:

5.2

8.6

13.5

Correct answer:

13.5

Explanation:

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

27=2r

r=13.5

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Example Question #10 : Calculating The Length Of A Radius

The arc $\widehat{AB}$ of a circle measures $120^{\circ }$ . The chord of the arc, $\overline{AB}$ , has length x + 2 . Give the length of the radius of the circle.

Possible Answers:

$\frac{\sqrt{3}}{4} x+ \frac{ \sqrt{3}}{2}$

$\frac{\sqrt{3}}{12} x+ \frac{ \sqrt{3}}{6}$

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

x + 2

$x \sqrt{3} + 2 \sqrt{3}$

Correct answer:

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

Explanation:

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:

Circle and triangle

Since AB = x + 2 , 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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GMAT Math : Calculating the length of a radius

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