GMAT Math : Calculating the angle of a sector

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Example Question #1 : Calculating The Angle Of A Sector

Chords

Note: Figure NOT drawn to scale.

$m \angle A = 65 ^{\circ }; m \angle B = 55 ^{\circ }$ .

Order the degree measures of the arcs $\widehat{AB}, \widehat{BC}, \widehat{CA}$ from least to greatest.

Possible Answers:

$\widehat{ AC} , \widehat{ AB } , \widehat{ BC}$

$\widehat{ BC}, \widehat{ AB} , \widehat{ AC }$

$\widehat{ AB} , \widehat{ AC } , \widehat{ BC}$

$\widehat{ AB} , \widehat{ BC}, \widehat{ AC }$

$\widehat{ AC} , \widehat{ BC } , \widehat{ AB}$

Correct answer:

$\widehat{ AC} , \widehat{ AB } , \widehat{ BC}$

Explanation:

$m \angle C = 180 ^{\circ } - m \angle A - m \angle B = 180 ^{\circ } - 65^{\circ } - 55^{\circ } = 60^{\circ }$

$m \angle B < m \angle C < m \angle A$ , so, by the Multiplication Property of Inequality,

$2 \cdot m \angle B < 2 \cdot m \angle C < 2 \cdot m \angle A$ .

The degree measure of an arc is twice that of the inscribed angle that intercepts it, so the above can be rewritten as

$m \widehat{ AC} < m\widehat{ AB } < m\widehat{ BC}$ .

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Example Question #21 : Sectors

In the figure shown below, line segment passes through the center of the circle and has a length of $8\:cm$ . Points , , and are on the circle. Sector COB covers $\frac{1}{6}th$ of the total area of the circle. Answer the following questions regarding this shape.

Circle1

Find the value of central angle AOC .

Possible Answers:

$60^{\circ}$

$240^{\circ}$

$120^{\circ}$

$150^{\circ}$

$180^{\circ}$

Correct answer:

$120^{\circ}$

Explanation:

Here we need to recall the total degree measure of a circle. A circle always has exactly 360 degrees.

Knowing this, we need to utilize two other clues to find the degree measure of AOC .

1) Angle AOB measures 180 degrees, because it is made up of line segment , which is a straight line.

2) Angle BOC can be found by using the following equation. Because we are given the fractional value of its area, we can construct a ratio to solve for angle BOC :

$\frac{\Theta }{360}=\frac{1}{6}$

$\Theta=\frac{1}{6}*360^{\circ}=60^{\circ}$

So, to find angle AOC , we just need to subtract our other values from 360 :

$\angle AOC= 360^{\circ}-\angle AOB - \angle BOC= 360^{\circ}-180^{\circ}-60^{\circ}=120^{\circ}$

So, $\angle AOC=120^{\circ}$ .

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Example Question #3 : Calculating The Angle Of A Sector

The radius of Circle A is equal to the perimeter of Square B. A sector of Circle A has the same area as Square B. Which of the following is the degree measure of this sector?

Possible Answers:

$\left (\frac{90}{\pi} \right )^{\circ }$

$45^{\circ }$

$22 \frac{1}{2}^{\circ }$

$\left (\frac{45}{ \pi} \right )^{\circ }$

$\left (\frac{45}{2\pi} \right )^{\circ }$

Correct answer:

$\left (\frac{45}{2\pi} \right )^{\circ }$

Explanation:

Call the length of a side of Square B . Its perimeter is , which is the radius of Circle A.

The area of the circle is $A_{a}= \pi \left ( 4r\right )^{2} = 16 \pi r^{2}$ ; that of the square is $A_{b}= r^{2}$ . Therefore, a sector of the circle with area $r^{2}$ will be $\frac{1}{16 \pi}$ of the circle, which is a sector of measure

$\frac{1}{16 \pi} \times 360 ^{\circ } = \left (\frac{45}{2\pi} \right )^{\circ }$

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Example Question #4 : Calculating The Angle Of A Sector

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Angle $\widehat{DCB}$ is $35^{\circ}$ . What is angle $\widehat{DAB}$ ?

Possible Answers:

$70^{\circ}$

$75^{\circ}$

$55^{\circ}$

$65^{\circ}$

$60^{\circ}$

Correct answer:

$70^{\circ}$

Explanation:

This is the kind of question we can't get right if we don't know the trick. In a circle, the size of an angle at the center of the circle, formed by two segments intercepting an arc, is twice the size of the angle formed by two lines intercepting the same arc, provided one of these lines is the diameter of the circle. in other words, $\widehat{DAB}$ is twice $\widehat{DCB}$ .

Thus,

$2(35^\circ)=70^\circ$

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Example Question #5 : Calculating The Angle Of A Sector

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A,B,C,D,E are evenly spaced points on the circle. What is angle $\widehat{DFC}$ ?

Possible Answers:

$72^{\circ}$

$80^{\circ}$

$144^{\circ}$

$78^{\circ}$

$36^{\circ}$

Correct answer:

$72^{\circ}$

Explanation:

We can see that the points devide the $360^{\circ}$ of the circle in 5 equal portions.

The final answer is given simply by $\frac{360}{5}$ which is $72^{\circ}$ , this is the angle of a slice of a pizza cut in 5 parts if you will!

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Example Question #6 : Calculating The Angle Of A Sector

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The points A,B,C,D,E and are evenly spaced on the circle of center . What is the size of angle $\widehat{FBE}$ ?

Possible Answers:

$35^{\circ}$

$45^{\circ}$

$30^{\circ}$

$32^{\circ}$

$60^{\circ}$

Correct answer:

$30^{\circ}$

Explanation:

As we have seen previously, the 6 points divide the $360^{\circ}$ of the circle in 6 portion of same angle. Each portion form an angle of $\frac{360}{6}$ or 60 degrees. As we also have previously seen, the angle formed by the lines intercepting an arc is twice more at the center of the circle than at the intersection of the lines intercepting the same arc with the circle, provided one of these lines is the diameter. In other words, $\widehat{FGE}=2\widehat{FBE}$ . Since $\widehat{FGE}$ is 60 degrees, than, $\widehat{FBE}$ must be 30 degrees, this is our final answer.

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GMAT Math : Calculating the angle of a sector

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1 : Calculating The Angle Of A Sector

Example Question #21 : Sectors

Example Question #3 : Calculating The Angle Of A Sector

Example Question #4 : Calculating The Angle Of A Sector

Example Question #5 : Calculating The Angle Of A Sector

Example Question #6 : Calculating The Angle Of A Sector

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