GMAT Math : Problem-Solving Questions

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Example Questions

Example Question #4 : 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 #31 : 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:

$8\pi$

$16\pi$

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 #7 : Calculating The Length Of A Radius

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

Possible Answers:

6.48

$21\pi$

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 #2 : Calculating The Length Of A Radius

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

Possible Answers:

8.6

5.2

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 #9 : 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:

x + 2

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

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

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

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

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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Example Question #11 : Calculating The Length Of A Radius

The arc $\widehat{AB}$ of a circle measures $90^{\circ }$ . The chord of the arc, $\overline{AB}$ , has length x + 2 . Give the radius of the circle.

Possible Answers:

2x+4

$\pi x+ 2 \pi$

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

$2 \pi x+ 4 \pi$

$\frac{1}{2} x \sqrt{2} + \sqrt{2}$

Correct answer:

$\frac{1}{2} x \sqrt{2} + \sqrt{2}$

Explanation:

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 x + 2 . 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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Example Question #301 : Problem Solving Questions

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?

Possible Answers:

$5 \sqrt{6}$

$5 \sqrt{2}$

$10 \sqrt{2}$

Correct answer:

$5 \sqrt{2}$

Explanation:

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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Example Question #13 : Calculating The Length Of A Radius

A $60^{\circ }$ arc of a circle measures x+ 4 . Give the radius of this circle.

Possible Answers:

$\frac{6}{\pi}x+\frac{24}{\pi}$

$9 \pi x^{2}+ 72 \pi x+ 144 \pi$

$6 \pi x+24 \pi$

$\frac{3}{\pi}x+\frac{12}{\pi}$

$3\pi x+12\pi$

Correct answer:

$\frac{3}{\pi}x+\frac{12}{\pi}$

Explanation:

A $60^{\circ }$ arc of a circle is $\frac{60}{360 } = \frac{1}{6}$ of the circle. Since the length of this arc is x+ 4 , the circumference is $\textup{six times }$ this, or

C = 6( x+ 4) = 6x+ 24

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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Example Question #14 : Calculating The Length Of A Radius

The arc $\widehat{AB}$ of a circle measures $60^{\circ }$ . The chord of the arc, $\overline{AB}$ , has length x + 2 . Give the length of the radius of the circle.

Possible Answers:

x+2

$2 \pi x + 4 \pi$

2x+4

$\textup{None of the other responses gives a correct answer.}$

$\pi x +2 \pi$

Correct answer:

x+2

Explanation:

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 x+2 .

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GMAT Math : Problem-Solving Questions

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #4 : Calculating The Length Of A Radius

Example Question #31 : Radius

Example Question #1 : Calculating The Length Of A Radius

Example Question #7 : Calculating The Length Of A Radius

Example Question #2 : Calculating The Length Of A Radius

Example Question #9 : Calculating The Length Of A Radius

Example Question #11 : Calculating The Length Of A Radius

Example Question #301 : Problem Solving Questions

Example Question #13 : Calculating The Length Of A Radius

Example Question #14 : Calculating The Length Of A Radius

All GMAT Math Resources

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