GMAT Math : Geometry

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Example Questions

Example Question #3 : Dsq: Calculating The Perimeter Of A Right Triangle

Find the perimeter of the right triangle.

The product of the base and height measures .
The hypotenuse measures .

Possible Answers:

Each statement alone is sufficient to answer the question.

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

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

Statements 1 and 2 are not sufficient, and additional data is needed to answer the question.

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

Correct answer:

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

Explanation:

Statement 1: We need additional information.

$b \times h =48$

But this can mean our base and height measure 2 and 24, 4 and 12, or 6 and 8.

We cannot determine which one based solely on this statement.

Statement 2: We're given the length of the hypotenuse so we can narrow down the possible base and height values.

c^2=a^2+b^2

We have to see which pair of values makes the statement 100=a^2 + b^2 true.

The only pair that does is 6 and 8.

We can now find the perimeter of the right triangle:

a+b+c=6+8+10=24

or, if you're more familiar with the equation $perimeter=a+b+\sqrt{a^2+b^2}$ , then:

$6+8+\sqrt{6^2+8^2}=24$

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Example Question #4 : Dsq: Calculating The Perimeter Of A Right Triangle

Calculate the perimeter of the triangle.

The hypotenuse of the right triangle is .
The legs of the right triangle measure and .

Possible Answers:

Each statement alone is sufficient to answer the question.

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

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

Statements 1 and 2 are not sufficient, and additional data is needed to answer the question.

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

Correct answer:

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

Explanation:

Statement 1: In order to find the perimeter of a right triangle, we need to know the lengths of the legs, not the hypotenuse.

Statement 2: Since we have the values to both of the legs' lengths, we can just plug it into the equation for the perimeter:

$p=a+b+\sqrt{a^2+b^2}$

$p=3+4+\sqrt{9+16}=3+4+\sqrt{25}=12cm$

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Example Question #431 : Geometry

You are given a circle and a square. Which one has the larger area?

Statement 1: The radius of the circle is two-thirds the sidelength of the square.

Statement 2: The circumference of the circle is $\frac{\pi }{3}$ times the perimeter of the square.

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER 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

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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Let be the sidelength of the square. Then its area is $s^{2}$ .

From Statement 1, it follows that the radius of the circle is $r = \frac{2}{3}s$ .

From Statement 2, it follows that , since the perimeter of the square is , the circumference of the circle is $\frac{\pi }{3} \cdot 4s = \frac{4\pi }{3} \cdot s$ , and the radius is $\frac{ \frac{4\pi }{3} \cdot s}{2\pi } = \frac{2}{3} s$ - the same fact given in Statement 1.

Either way, it follows that the area of the circle in terms of is

$A = \pi r ^{2} =\pi \cdot \left ( \frac{2}{3} s \right )^{2} = \frac{\pi 4}{9} s ^{2}$ ,

so all we have to do is compare $\frac{\pi 4}{9}$ to 1 in order to determine whether the square or the circle is larger in area.

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Example Question #432 : Geometry

You are given a circle and an equilateral triangle. Which one has the greater area?

Statement 1: The sidelength of the triangle is three times the radius of the circle.

Statement 2: The perimeter of the triangle is 99 inches.

Possible Answers:

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.

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

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

Explanation:

From Statement 2, we can calculate the area of the triangle, but we are given no clues about the area of the circle, actual or relative.

From Statement 1, we know that if we call the radius of the circle , we know the sidelength of the triangle is s = 3r .

The area of the circle is $A = \pi r^{2}$ .

The area of the triangle is $A =\frac{ s^{2} \sqrt{ 3}}{4} = \frac{ (3r)^{2} \sqrt{ 3}}{4} = \frac{ 9 \sqrt{ 3}}{4} r^{2}$ .

All we have to do is compare $\pi$ to $\frac{ 9 \sqrt{ 3}}{4}$ to determine whether the circle or the triangle has the greater area.

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Example Question #2 : Dsq: Calculating The Area Of A Circle

Data Sufficiency Question

Calculate the area of a circle.

1. The radius of the circle is 4.

2. The circumference of the circle is 24.

Possible Answers:

Statements 1 and 2 together are not sufficient, and additional data is needed to answer the question

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient

Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question

Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question

Each statement alone is sufficient

Correct answer:

Each statement alone is sufficient

Explanation:

The area of a circle can be calcuated using the equation:

$A=r^2\pi$

and the circumference calculated using:

$C=2r\pi$

The radius is the only information required for calculating the area of a circle and that can be obtained from the circumference, therefore, either statement is sufficient.

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Example Question #4 : Radius

What is the area of the gray region in the above figure?

Statement 1: The diameter of the larger circle is one mile.

Statement 2: The radius of the smaller circle is 1,320 feet.

Possible Answers:

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

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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

The radius of the larger circle is equal to the diameter of the smaller, and, subsequently, twice the radius of the smaller. If one radius is known, then the other can be calculated, and the areas of the two circles can be as well; the difference between their areas is the area of the gray region. Statement 1 tells us the diameter of the large circle, from which its radius can be determined by dividing by 2; Statement 2 tells us the radius of the radius of the smaller circle. From either, the radius of the other circle can be calculated.

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Example Question #3 : Circles

Two circles are constructed; one is inscribed inside a given regular hexagon, and the other is circumscribed about the same hexagon.

What is the area of the inscribed circle?

Statement 1: The area of the circumscribed circle is $25 \pi$

Statement 2: The perimeter of the hexagon is 30.

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.

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.

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

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Examine the diagram below, which shows the hexagon, segments from its center to two vertices and the midpoint of a side, and the two circles.

Thingy

From Statement 1 alone, the radius of the circumscribed circle can be calculated using the area formula. From Statement 2 alone, the perimeter can be divided by 6 to obtain ; since two consecutive radii and one side of a hexagon can be proved to form an equilateral triangle, OA = AB . As a consequence, from either statement alone, can be calculated.

$\Delta ACO$ can be proved a 30-60-90 triangle, so the 30-60-90 Theorem can be used to calculate , the radius of the inscribed circle. From this, the area of the inscribed circle can be calculated.

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Example Question #4 : Circles

The equation of a given circle is

$(x-A) ^{2} + (y - B) ^{2} = C ^{2}$ .

What is the radius of the circle?

Statement 1: A = B = 5

Statement 2: The circle passes through the origin.

Possible Answers:

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.

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

EITHER 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:

The standard form of the equation of a circle is

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

where the radius is and the center is (h,k ) .

The equation given is this same form, with replacing , replacing , and replacing , so to find the radius, we need to find .

Statement 1 alone tells us that the center is $\left (5,5 \right )$ but it tells us nothing about the radius. Statement 2 alone tells us only that the circle passes through (0,0) .

The two together, however, reveal enough information to give the radius. The radius is the distance from the center to a point on the circle, so we can use the distance formula to find the distance between (0,0) and $\left (5,5 \right )$ . This is the radius.

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Example Question #5 : Circles

Find the radius of circle B

I) Circle B has a circumference of $\small 8\pi m$ .

II) Circle B has an area of $\small 16\pi m^2$ .

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 circumference of a circle and asked to find the radius.

Given the following equations:

$\small a=\pi r^2$

$\small c=2\pi r$

We can use either equation to work backwards and find our radius, therefore; Each statement alone is enough to solve the question.

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Example Question #5 : Radius

Calculate the length of the radius of a circle.

Statement 1): The circumference of the circle is .

Statement 2): x^2+y^2 = 1

Possible Answers:

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

EACH statement ALONE is sufficient.

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

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

BOTH statements TOGETHER are NOT sufficient, and additional data is needed to answer the question.

Correct answer:

EACH statement ALONE is sufficient.

Explanation:

Statement 1) gives the circumference of a circle. The formula for finding the circumference of a circle is $C=2\pi r$ . The radius can be solved by using this formula.

Statement 2) gives the standard form of a circle, where $\left ( h,k\right )$ is the center of the circle:

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

The radius is also given in the equation.

Therefore, either statement alone is enough to solve for the radius of a circle.

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GMAT Math : Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #3 : Dsq: Calculating The Perimeter Of A Right Triangle

Example Question #4 : Dsq: Calculating The Perimeter Of A Right Triangle

Example Question #431 : Geometry

Example Question #432 : Geometry

Example Question #2 : Dsq: Calculating The Area Of A Circle

Example Question #4 : Radius

Example Question #3 : Circles

Example Question #4 : Circles

Example Question #5 : Circles

Example Question #5 : Radius

All GMAT Math Resources

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