GRE Math : How to find the length of an arc

Study concepts, example questions & explanations for GRE Math

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Example Questions

Example Question #4 : Sectors

What is the perimeter of a pie piece if the pie is sliced into 40 degree pieces and its area is 361π?

Possible Answers:

38π

38π/9

40.1π

2.1

38 + 38π/9

Correct answer:

38 + 38π/9

Explanation:

The perimeter of a given pie-piece will be equal to 2 radii plus the outer arc (which is a percentage of the circumference). For our piece, this arc will be 40/360 or 1/9 of the circumference. If the area is 361π, this means πr2 = 361 and that the radius is 19. The total circumference is therefore 2 * 19 * π or 38π.

The total perimeter of the pie piece is therefore 2 * r + (1 / 9) * c = 2 * 19 + (1/9) * 38π = 38 + 38π/9

Example Question #5 : Sectors

What is the length of the arc of a circle with radius 10 that traces a 50 degree angle?

Possible Answers:

25π/9

25π

20π/9

25π/7

25π/29

Correct answer:

25π/9

Explanation:

length of an arc = (degrees * 2πr)/360 = (50 * 2π * 10)/360 = 25π/9

Example Question #2 : Sectors

An ant walks around the edge of circular pizzas left on the counter of a pizza shop. On most days, it is shaken off the pizza before it manages to walk the complete distance.

Quantity A: The distance covered by the ant when walking over four slices of a pizza with a diameter of  and  equally-sized pieces.

Quantity B: The distance covered by the ant when walking over a complete personal pizza with a diameter of  inches.

What can we say about the two quantities?

Possible Answers:

The relationship between the two quantities cannot be determined.

The two quantities are equal.

Quantity A is larger.

Quantity B is larger.

Correct answer:

Quantity B is larger.

Explanation:

Let's compute each quantity.

Quantity A

This is a little bit more difficult than quantity B.  It requires us to compute an arc length, for the ant does not walk around the whole pizza; however, this is not very hard. What we know is that the ant walks around , or , of the pizza.

Now, we know the circumference of a circle is  or 

For our example, let's use the latter. Since , we know:

However, our ant walks around only part of this, namely:

Quantity B

This is really just a matter of computing the circumference of the circle.  For our value, we know this to be:

 

Now, we can compare these by taking quantity A and reducing the fraction to be . Thus, we know that Quantity B is larger than Quantity A.

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