GMAT Math : Geometry

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Example Questions

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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Example Question #15 : Calculating The Length Of A Radius

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

Possible Answers:

3ft

2.5ft

5ft

10ft

Correct answer:

5ft

Explanation:

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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Example Question #71 : Circles

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?

Possible Answers:

$5 \sqrt{2}$

$6 \sqrt{5}$

$5 \sqrt{6}$

Correct answer:

$5 \sqrt{6}$

Explanation:

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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Example Question #1 : Calculating Circumference

A circle on the coordinate plane has equation

$x^{2} + y ^{2} = 40$

Which of the following represents its circumference?

Possible Answers:

$80 \pi$

$16 \pi \sqrt{10}$

$4 \pi \sqrt{10}$

$40 \pi$

$8 \pi \sqrt{10}$

Correct answer:

$4 \pi \sqrt{10}$

Explanation:

The equation of a circle centered at the origin is

$x^{2} + y ^{2} = r ^{2}$

where is the radius of the circle.

In this equation, $r ^{2} = 40$ , so $r = \sqrt{40}$ ; this simplifies to

$r = \sqrt{40} = \sqrt{4} \cdot \sqrt{10} = 2 \sqrt{10}$

The circumference of a circle is $C = 2 \pi r$ , so substitute $r = 2 \sqrt{10}$ :

$C = 2 \pi r = 2 \pi \cdot 2 \sqrt{10} = 4 \pi \sqrt{10}$

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Example Question #1 : Calculating Circumference

A circle on the coordinate plane has equation

$x ^{2} - 8x + y ^{2} - 10x = 7$

What is its circumference?

Possible Answers:

$4 \pi \sqrt{ 3}$

$48 \pi$

$16 \pi \sqrt{ 3}$

$8 \pi \sqrt{ 3}$

$96 \pi$

Correct answer:

$8 \pi \sqrt{ 3}$

Explanation:

The standard form of the area of a circle with radius and center (h,k ) is

$(x-h)^{2} + (y-k)^{2} = r^{2}$

Once we get the equation in standard form, we can find radius , and multiply it by $2 \pi$ to get the circumference.

Complete the squares:

$\left (\frac{-8}{2} \right )^{2} = 16,\left (\frac{-10}{2} \right )^{2} = 25$

so $x ^{2} - 8x + y ^{2} - 10x = 7$ can be rewritten as follows:

$x ^{2} - 8x + 16 + y ^{2} - 10x + 25 = 7 + 16 + 25$

$\left ( x -4 \right )^{2} + \left ( y -5 \right ) ^{2} = 48$

$r ^{2} = 48$ ,

so $r = \sqrt{48} = \sqrt{16} \cdot \sqrt{3} = 4 \sqrt{3}$

And $C = 2\pi \cdot 4 \sqrt{ 3} = 8 \pi \sqrt{ 3}$

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Example Question #71 : Geometry

On average, Stephanie walks $2\pi$ feet every seconds. If Stephanie walks at her usual pace, how long will it take her to walk around a circular track with a radius of 200 feet, in seconds?

Possible Answers:

None of the other answers are correct.

800 seconds

400 seconds

$800 \pi$ seconds

$400 \pi$ seconds

Correct answer:

800 seconds

Explanation:

The length of the track equals the circumference of the circle.

$C = 2\pi r$

$C=2\pi (200)=400\pi$

$Speed = \frac{distance}{time}$

Therefore, $time = \frac{distance}{speed}$ .

$time = 400\pi\: ft \div \frac{2\pi \: ft}{4\: sec} = 400\pi\: ft \times\frac{4\:sec}{2\pi\: feet}$

$time= 400\pi\: ft \times\frac{2\:sec}{1\pi\: feet} = 800\:seconds$

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Example Question #2 : Calculating Circumference

A circle on the coordinate plane has equation $x^{2} + y ^{2} = 64$ .

What is its circumference?

Possible Answers:

$8 \pi$

$64 \pi$

$16 \pi ^2$

$8 \pi ^2$

$16 \pi$

Correct answer:

$16 \pi$

Explanation:

The equation of a circle centered at the origin is

$x^{2} + y ^{2} = r ^{2}$ ,

where is the radius of the circle.

In the equation given in the question stem, $r ^{2} = 64$ , so $r = \sqrt{64} = 8$ .

The circumference of a circle is $C = 2 \pi r$ , so substitute r = 8 :

$C = 2 \pi r = 2 \pi \cdot 8 = 16 \pi$

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Example Question #2 : Calculating Circumference

Let A, B be concentric circles. Circle has a radius of , and the shortest distance from the edge of circle to the edge of circle is . What is the circumference of circle ?

Possible Answers:

$10\pi$

$6\pi$

$40\pi$

$\pi$

$14\pi$

Correct answer:

$40\pi$

Explanation:

Since A,B are concentric circles, they share a common center, like sections of a bulls-eye target. Since the radius of is less than half the distance from the edge of to the edge of , we must have circle is inside of circle . (It's helpful to draw a picture to see what's going on!)

Now we can find the radius of by adding and , which is 20. And the equation for finding circumfrence is $C =2r\pi$ . Plugging in for gives $40\pi$ .

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Example Question #3 : Calculating Circumference

Consider the Circle :

Circle3

(Figure not drawn to scale.)

Suppose Circle represents a circular pen for Frank's mules. How many meters of fencing does Frank need to build this pen?

Possible Answers:

$45 \pi\:m$

$225 \pi\:m$

$30 \pi\:m$

$60 \pi\:m$

$15 \pi\:m$

Correct answer:

$30 \pi\:m$

Explanation:

We need to figure out the length of fencing needed to surround a circular enclosure, or in other words, the circumference of the circle.

Circumference equation:

$C=2 \pi r$

Where is our radius, which is $15\:m$ in this case. Plug it in and simplify:

$C=2 \pi *15=30 \pi\:m$

And we have our answer!

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GMAT Math : Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #13 : Calculating The Length Of A Radius

Example Question #14 : Calculating The Length Of A Radius

Example Question #15 : Calculating The Length Of A Radius

Example Question #71 : Circles

Example Question #1 : Calculating Circumference

Example Question #1 : Calculating Circumference

Example Question #71 : Geometry

Example Question #2 : Calculating Circumference

Example Question #2 : Calculating Circumference

Example Question #3 : Calculating Circumference

All GMAT Math Resources

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