GMAT Math : Calculating the length of a radius

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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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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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GMAT Math : Calculating the length of a radius

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

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

Example Question #15 : Calculating The Length Of A Radius

Example Question #71 : Circles

All GMAT Math Resources

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