GMAT Math : Circles

Study concepts, example questions & explanations for GMAT Math

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Example Questions

Example Question #26 : Dsq: Calculating The Length Of A Radius

Rectangle  is inscribed inside a circle. What is the radius of the circle?

Statement 1: Rectangle  has area 200.

Statement 2: Rectangle  has perimeter 60.

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question. 

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

The diameter of a circle with an inscribed rectangle is equal to the length of a diagonal of the rectangle , which, given the length  and width , can be found using the Pythagorean Theorem:

Statement 1 alone gives insufficient information. For example, a 20 by 10 rectangle has area , and a 40 by 5 rectangle has area 

The first rectangle has diagonals of length

.

The second rectangle has diagonals of length

,

Since the diagonals of the rectangles differ, so do the diameters, and, consequently, the radii, of the circles.

Statement 2 alone gives insufficient information for a similar reason. For example, a 20 by 10 rectangle has perimeter , and a 25 by 5 rectangle has perimeter . Again, the first rectangle has  diagonals of length . The second has diagonals of length

.

Now, assume both statements to be true. We are looking for two numbers whose product is 200 and whose sum is 30 (since the perimeter is twice the sum, or 60). The only such pair of numbers can be found by trial and error to be 20 and 10, so these are the length and width of the rectangle. As shown before, a rectangle with these dimensions has diagonals of length . This is the diameter of the circle in which it is inscribed, so half this, or , is the radius.

Example Question #27 : Dsq: Calculating The Length Of A Radius

Give the radius of a circle on the coordinate plane.

Statement 1: Of the three intercepts of the circle, exactly two are -intercepts, one of which is at the point .

Statement 2: Of the three intercepts of the circle, exactly two are -intercepts, one of which is at the point .

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question. 

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

Statement 1 alone only gives one point through which the circle passes, so no information can be determined about the other points or about the size or location of the circle. A similar argument holds for the insufficiency of Statement 2.

Now assume both statements are true. The circle has exactly three intercepts, but it is given that there are two -intercepts -  and one other point - and two -intercepts -  and one other point. The unidentified -intercept and the unidentified -intercept must be one and the same, and the only possible way this can happen is for this common point to be the origin . Since three points define a circle, we can now identify the unique circle through the points , and , and we can figure out its radius.

Example Question #31 : Circles

Sector

Give the radius of the above circle with center .

Statement 1:  is a  angle.

Statement 2:  is a  angle.

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question. 

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question. 

Explanation:

The two statements together only give information about angle measures; arc degree measures can be deduced from this information but not any arc lengths or side lengths. Without this information, we cannot obtain the radius of this circle.

Example Question #1 : Dsq: Calculating Circumference

Square S is inscribed in circle C as in the figure above. What is the circumference of C?

(1) The perimeter of S is 16.

(2) The area of S is 36.

Possible Answers:

B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

E. Statements (1) and (2) TOGETHER are NOT sufficient.

D. EACH statement ALONE is sufficient.

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Correct answer:

D. EACH statement ALONE is sufficient.

Explanation:

From statement (1), we know that the side of S is 4, and then we can calculate the diagonal of S using the Pythagorean theorem: . The diagonal of S is the diameter of C. Therefore, we can calculate the circumference by using . From statement (2), we know that the side of S is 6, and then we can calculate the diagonal of S using the Pythagorean theorem:

The diagonal of S is the diameter of C. Therefore, we can calculate the circumference by using .

Example Question #31 : Radius

What is the circumference of circle J?

I) Circle J has an area of .

II) Circle J has a diameter of .

Possible Answers:

Statement 2 is sufficient to solve the question, but statement 1 is not sufficient to solve the question.

Statement 1 is sufficient to solve the question, but statement 2 is not sufficient to solve the question.

Each statement alone is enough to solve the question.

Neither statement is sufficient to solve the question. More information is needed.

Both statements taken together are sufficient to solve the question.

Correct answer:

Each statement alone is enough to solve the question.

Explanation:

We are given the area and diameter of a circle and asked to find the circumference. We know that diameter is twice the length of a radius, so we also have our radius.

Given the following equations:

We can see that knowing either diameter or area will allow us to find the circumference.

Thus: Each statement alone is enough to solve the question.

Example Question #31 : Circles

What is the circumference of the circle given by: 

I) .

II) The slope of the tangent to the circle at  is undefined.

Possible Answers:

Statement II is sufficient to answer the question, but Statement I is not sufficient to answer the question.

Both statements together are needed to answer the question.

Neither statement is sufficient to answer the question. More information is needed.

Statement I is sufficient to answer the question, but Statement II is not sufficient to answer the question.

Either statement alone is sufficient to answer the question.

Correct answer:

Either statement alone is sufficient to answer the question.

Explanation:

All we need to find circumference is the radius. 

I) Gives us the radius squared, so we could find circumference with I.

II) Tells us the slope of the tangent line at a given point is undefined. Only vertical lines have undefined slope. The tangent line is perpendicular to the radius, so we can find our radius by drawing a picture and comparing the location of the center to the location of the tangent line.

So either statement will be sufficient.

Example Question #1 : Dsq: Calculating Circumference

What is the circumference of Circle ?

1.) The radius of the circle is .

2.) The circle is inside another circle of area .

Possible Answers:

Statement 1 is sufficient to solve the question, but Statement 2 is not sufficient to solve the question.

Both statements taken together are sufficient to solve the question.

Neither statement is sufficient to solve the question. More information is needed. 

Each statement alone is sufficient to solve the question.

Statement 2 is sufficient to solve the question, but Statement 1 is not sufficient to solve the question.

Correct answer:

Statement 1 is sufficient to solve the question, but Statement 2 is not sufficient to solve the question.

Explanation:

We are asked to find the circumference  of Circle  and are given the diameter and the area. We also know that .  Taking each statement individually:

1.) The radius  is  and we know that . Therefore, Statement 1 is sufficient to solve for the circumference of the circle by itself.

 

2.) The area  of the outside circle is , but we cannot use this to determine the circumference of Circle  because we don't know where  is inside the larger outside circle.  

Example Question #461 : Geometry

What is the circumference of Circle ?

1.) The diameter of the circle is .

2.) The area of the circle is .

Possible Answers:

Both statements taken together are sufficient to solve the question.

Statement 2 is sufficient to solve the question, but Statement 1 is not sufficient to solve the question.

Each statement alone is sufficient to solve the question.

Neither statement is sufficient to solve the question. More information is needed. 

Statement 1 is sufficient to solve the question, but Statement 2 is not sufficient to solve the question.

Correct answer:

Each statement alone is sufficient to solve the question.

Explanation:

We are asked to find the circumference  of Circle  and are given the diameter and the area. We also know that .  Taking each statement individually:

1.) The diameter  is  and we know that the radius , so . Therefore, Statement 1 is sufficient to solve for the circumference of the circle by itself.

 

2.) The area  of Circle  is , so we can determine that the radius . Since the circumference , Statement 2 is is sufficient to solve for the circumference of the circle by itself.

Example Question #1 : Sectors

Brendan's girlfriend made him a cheescake for his birthday. He eats one slice a day. What is the measure of the central angle of each slice?

I) The diameter of the cake is .

II) Each slice is  of the total cake.

Possible Answers:

Neither statement is sufficient to answer the question. More information is needed.

Statement II is sufficient to answer the question, but Statement I is not sufficient to answer the question.

Either statement alone is sufficient to answer the question.

Statement I is sufficient to answer the question, but Statement II is not sufficient to answer the question.

Both statements together are needed to answer the question.

Correct answer:

Statement II is sufficient to answer the question, but Statement I is not sufficient to answer the question.

Explanation:

In this case we are given a circle and asked to find the angle of a portion of it.

The diameter would allow us to find many things related to the circle, but not an individual slice. 

However, knowing that each slice is 1/12 of the total allows us to multiply 360 by 1/12 and find out that each slice is 30 degrees.

Therefore statement II alone is sufficient in answering the question.

Example Question #2 : Sectors

One slice of a pizza is . What is the central angle of one slice?

I) Each slice is  of the whole pizza.

II) Each straight edge the slice is  inches.

Possible Answers:

Neither statement is sufficient to answer the question. More information is needed.

Both statements together are needed to answer the question.

Statement II is sufficient to answer the question, but Statement I is not sufficient to answer the question.

Statement I is sufficient to answer the question, but Statement II is not sufficient to answer the question.

Either statement alone is sufficient to answer the question.

Correct answer:

Either statement alone is sufficient to answer the question.

Explanation:

I) Gives us the percentage of one slice of the whole pizza. We can take 15% of 360 to find the central angle.

II) Gives us the radius of the pizza. We can use the radius to find the area of the pizza. With the total area and the area of one slice we can find the percentage of the whole and from there, the angle of one slice.

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