GMAT Math : Data-Sufficiency Questions

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Example Questions

Example Question #32 : Polygons

What is the area of a regular hexagon?

Statement 1: The area of the circle inscribed inside the hexagon is $108 \pi$ .

Statement 2: The circumference of the circle that is circumscribed about the hexagon is $24 \pi$ .

Possible Answers:

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.

BOTH statements TOGETHER are insufficient 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:

A regular hexagon can be viewed as a composite of six equilateral triangles, each of whose sidelength is the radius - the distance from its center to a vertex - of the hexagon. If the radius of the hexagon is known, then the area of the hexagon can be calculated to be $A = 6 \cdot \frac{r^{2}\sqrt{3}}{4} = \frac{3r^{2}\sqrt{3}}{2}$ .

From Statement 1 alone, the radius of the inscribed circle, or incircle, can be calculated from the area formula (by dividing the area by $\pi$ and extracting the square root). This length coincides with the height of each equilateral triangle. From there, the 30-60-90 Theorem can be used to find the sidelength of each triangle, and the area of the hexagon follows.

From Statement 2 alone, the radius of the circumscribed circle, or circumcircle, can be found by dividing its circumference by $2 \pi$ . This radius coincides with the radius of the hexagon, and the area can be calculated from there.

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

Octagon_1

Give the area of the above regular octagon.

Statement 1: The circle that circumscribes Quadrilateral WXYZ has area $32 \pi$ .

Statement 2: $\Delta BXC$ has area 16.

Possible Answers:

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.

BOTH statements TOGETHER are insufficient 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:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Quadrilateral WXYZ is a square; each of its diagonals is a diameter of its circumscribed circle, or circumcircle. Therefore, if we know the area of its circumcircle from Statement 1 to be $32 \pi$ , we can calculate the radius from the area formula (divide by $\pi$ , extract the square root of the quotient). Twice this is the diameter, which is also the length of a diagonal of this square. Divide this by $\sqrt{2}$ to get . This is also equal to , the length of one side; this is sufficient to get the area of the octagon.

From Statement 2 alone, since the area of isosceles triangle $\Delta BXC$ is known to be 8, the length of each leg can be found using the formula

$\frac{1}{2} s^{2} = 16$

Since $\Delta BXC$ is a 45-45-90 triangle, multiply this leg length by $\sqrt{2}$ to get , the length of one side; this is sufficient to get the area of the octagon.

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Example Question #1 : Rectangular Solids & Cylinders

Find the length of the edge of a tetrahedron.

Statement 1: The volume is 6.

Statement 2: The surface area is 6.

Possible Answers:

$\textup{Statement 1) ALONE is sufficient, but Statement 2) ALONE is not sufficient to answer the question.}$

$\textup{BOTH statements TOGETHER are NOT sufficient, and additional data is needed to answer the question. }$

$\textup{EACH statement ALONE is sufficient.}$

$\textup{BOTH statements taken TOGETHER are sufficient to answer the question, but neither statement ALONE is sufficient.}$

$\textup{Statement 2) ALONE is sufficient, but Statement 1) ALONE is not sufficient to answer the question.}$

Correct answer:

$\textup{EACH statement ALONE is sufficient.}$

Explanation:

Statement 1:) The volume is 6.

Write the formula to find the edge of the tetrahedron given the volume.

$V=\frac{e^3}{6\sqrt2}$

Given the volume, it is possible to find the edge of the tetrahedron.

Statement 2:) The surface area is 6.

Write the formula to find the edge of the tetrahedron given the surface.

$SA=\sqrt3 e^2$

Substitute the surface area to find the edge.

Therefore:

$\textup{EACH statement ALONE is sufficient.}$

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Example Question #1 : Dsq: Calculating The Volume Of A Tetrahedron

Pyramid 1 in three-dimensional coordinate space has as its base the square with vertices at the origin, (10,0,0) , (10,10,0) , and (0,10,0) , and its apex at the point (D,E,F) ; Pyramid 2 has as its base the square with vertices at the origin, (12,0,0) , (12,12,0) , and (0,12,0) , and its apex at the point (G,H,J) . All six variables represent positive quantities. Which pyramid has the greater volume?

Statement 1: $10 \le F \le 12$ and $10 \le J \le 12$

Statement 2: F< J

Possible Answers:

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.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

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

Explanation:

The volume of a pyramid is one third the product of its height and the area its base. The two pyramids have the same base, so the pyramid with the greater height will have the greater volume (and if their heights are equal, their volumes are equal).

Pyramid 1 is shown below:

Pyramid

The base of the pyramid is on the -plane, so the height of the pyramid is the perpendicular distance from apex (D,E,F) to this plane; this is the -coordinate, . The base of the pyramid is a square of sidelength 10, so its area is the square of 10, or 100. This makes the volume of Pyramid 1

$V_{1} = \frac{1}{3} \cdot 100 \cdot F = \frac{100}{3}F = \left ( 33 \frac{1}{3}\right )F$

Similarly, the volume of Pyramid 2 is

$V_{2} = \frac{1}{3} \cdot 12 ^{2}\cdot J= \frac{1}{3} \cdot 144 \cdot J = 36 J$

Therefore, the problem asks us to determine which of $\left ( 33 \frac{1}{3}\right )F$ and 36J is the greater.

Assume Statement 1 alone. Since $10 \le F \le 12$ , we can multiply all expressions by $33 \frac{1}{3}$ to get a range for the volume of Pyramid 1:

$10 \le F \le 12$

$33 \frac{1}{3} \cdot 10 \le 33 \frac{1}{3} \cdot F \le 33 \frac{1}{3} \cdot 12$

$333 \frac{1}{3} \le V_{1} \le 400$

Similarly, since $10 \le J \le 12$ , we can multiply all expressions by 36 to get a range of values for the volume of Pyramid 2:

$10 \le J \le 12$

$36 \cdot 10 \le 36 \cdot J \le 36 \cdot 12$

$360 \le V_{2} \le 432$

Since the two ranges share values, it cannot be determined for certain which pyramid has the greater volume.

Assume Statement 2 alone. Then, since $33 \frac{1}{3} < 36$ and F < J , it easily follows that

$33 \frac{1}{3} \cdot F < 36 \cdot J$ ,

and, subsequently, Pyramid 2 has the greater volume.

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Example Question #2 : Dsq: Calculating The Volume Of A Tetrahedron

Pyramid_2

Note: Figure NOT drawn to scale, but you may assume AE < EF and EH < EF .

In the above figure, a pyramid with a rectangular base is inscribed inside a rectangular prism; its vertices are A,E,F,G,H . What is the volume of the pyramid?

Statement 1: 30-60-90 triangle $\bigtriangleup AEF$ has area $50 \sqrt{3}$ .

Statement 2: Isosceles right triangle $\bigtriangleup AEH$ has area 50.

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.

EITHER 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.

Correct answer:

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

Explanation:

The volume of the pyramid is one third the product of height and the area of its rectangular base, which is $EF \cdot EH$ ; that is,

$V = \frac{1}{3} \cdot AE \cdot EF \cdot EH$

Assume Statement 1 alone. $\bigtriangleup AEF$ has area $50 \sqrt{3}$ , which is half the product of the length of shorter leg $\overline{AE}$ and longer leg $\overline{EF}$ . Also, by the 30-60-90 Theorem, $EF = AE \cdot \sqrt{3}$ , so, combining these statements,

$\frac{1}{2} \cdot AE \cdot EF = 50 \sqrt{3}$

$\frac{1}{2} \cdot AE \cdot AE \cdot \sqrt{3} = 50 \sqrt{3}$

$\frac{\sqrt{3}}{2} \cdot (AE) ^{2} = 50 \sqrt{3}$

$\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{2} \cdot (AE) ^{2} = \frac{2}{\sqrt{3}} \cdot 50 \sqrt{3}$

$(AE) ^{2} =100$

AE = 10 , and $EF = 10 \sqrt{3}$ .

However, we do not have any way of finding out , so the volume cannot be calculated.

Assume Statement 2 alone. $\bigtriangleup AEH$ is isosceles, so AE = EH ; again, since the area of a right triangle is half the product of the lengths of its legs,

$\frac{1}{2} \cdot AE \cdot EH = 50$

$\frac{1}{2} \cdot AE \cdot AE = 50$

$\frac{1}{2} \cdot (AE )^{2}= 50$

$\frac{1}{2} \cdot (AE )^{2} \cdot 2 = 50 \cdot 2$

$(AE )^{2} =100$

AE = 10

EH = 10

However, we have no way of finding out .

The two statements together give all three of , , and , so the volume can be calculated as

$V = \frac{1}{3} \cdot 10 \cdot 10 \sqrt{3}\cdot 10 = \frac{1,000 \sqrt{3}}{3}$

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Example Question #1 : Tetrahedrons

A solid in three-dimensional coordinate space has four vertices, at points (0,0,0) , (12,0,0) , (0,8,0) , and (A,B,C) for some positive values of A,B,C . What is the volume of the solid?

Statement 1: A=B

Statement 2: C= 30

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

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

Correct answer:

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

Explanation:

The figure described is the triangular pyramid, or tetrahedron, in the coordinate three-space below.

Tetrahedron

The base of the pyramid can be seen as a triangle with the three known coordinates (0,0,0) , (12,0,0) , and (0,8,0) , and the area of its base is half the product of the lengths of its legs, which is

$b = \frac{1}{2} \cdot 8 \cdot 12 = 48$ .

The volume of the pyramid is one third the product of the area of its base, which is 48, and its height, which is the perpendicular distance from the unknown point to the base. Since the base is entirely within the -plane, this distance is the -coordinate of the apex, which is . Therefore, the only thing that is needed to determine the volume of the pyramid is ; this information is provided in Statement 2, but not Statement 1.

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Example Question #1 : Tetrahedrons

Tetra_1

In the above diagram, a tetrahedron - a triangular pyramid - with vertices D,E,G,H is shown inside a cube. Give the volume of the tetrahedron.

Statement 1: The perimeter of Square ABCD is 16.

Statement 2: The area of $\bigtriangleup EGH$ is 8.

Possible Answers:

EITHER 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 sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

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

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

The volume of the pyramid is one third the product of height and the area of its base, which in turn, since here it is a right triangle, is half the product of the lengths and of its legs. Since the prism in the figure is a cube, the three lengths are equal, so we can set each to . The volume of the pyramid is

$V = \frac{1}{3} \cdot s\cdot \frac{1}{2} s\cdot s$

$V = \frac{1}{6} s^{3}$

Therefore, knowing the length of one edge of the cube is sufficient to determine the volume of the pyramid.

Assume Statement 1 alone. Since the perimeter of Square ABCD is 16, each side of the square, and each edge of the cube has one fourth this measure, or 4.

Assume Statement 2 alone. $\bigtriangleup EGH$ has congruent legs, each of measure ; since its area is 8, can be found as follows:

$\frac{1}{2} s^{2} = A$

$\frac{1}{2} s^{2} = 8$

$s^{2}= 16$

s = 4

From either statement alone, the length of each side of the cube, and, subsequently, the volume of the pyramid, can be calculated.

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Example Question #171 : Geometry

Tetra_1

Note: Figure NOT drawn to scale.

Refer to the above figure, which shows a tetrahedron, or triangular pyramid. What is the volume of the tetrahedron?

Statement 1: $\bigtriangleup ABD$ is an isosceles triangle with area 64.

Statement 2: $\bigtriangleup ABC$ is an equilateral triangle with perimeter 48.

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.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 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:

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

Explanation:

Each statement gives enough information about one triangle to determine its area, its angles, and its sidelengths, but no information about the other three triangles is given except for one side.

Assume both statements are known. $\bigtriangleup ABD$ is an isosceles triangle with area 64. Since AD = BD , we can find this common sidelength using the area formula for a triangle, with these lengths as height and base:

$\frac{1}{2} \cdot AD \cdot BD = 64$

$AD \cdot AD = 128$

$(AD)^{2} = 128$

$AD = \sqrt{128} =8 \sqrt{2}$ .

This is the length of both $\overline{AD}$ and $\overline{BD}$ .

By the 45-45-90 Theorem, $\overline{AB}$ has length $\sqrt{2}$ times this, or $8 \sqrt{2} \cdot \sqrt{2} = 8 \cdot 2 = 16$ .

Since $\bigtriangleup ABC$ is an equilateral triangle, AB = BC = AC = 16 . Since $\bigtriangleup ADC$ is a right triangle, AC = 16 , and $AD =8 \sqrt{2}$ , the triangle is also isosceles, and $CD= 8\sqrt{2}$ ; by a similar argument, $BD= 8\sqrt{2}$ .

The volume of the pyramid can be calculated. Its base, which is congruent to $\bigtriangleup ABD$ , has area 64, and its height is $AD = 8 \sqrt{2}$ ; multiply one third by their product to get the volume.

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Example Question #172 : Geometry

Pyramid 1 in three-dimensional coordinate space has as its base the square with vertices at the origin, (10,0,0) , (10,10,0) , and (0,10,0) , and its apex at the point (D,E,F) ; Pyramid 2 has as its base the square with vertices at the origin, (15,0,0) , (15,15,0) , and (0,15,0) , and its apex at the point (G,H,J) . All six variables represent positive quantities. Which pyramid has the greater volume?

Statement 1: 2F = 3J

Statement 2: F + J =25

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.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 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.

BOTH statements TOGETHER are insufficient 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:

The volume of a pyramid is one third the product of its height and the area its base.

Pyramid 1 is shown below:

Pyramid

$V_{1} = \frac{1}{3} \cdot 100 \cdot F = \frac{100}{3}F = \left ( 33 \frac{1}{3}\right )F$

Similarly, the volume of Pyramid 2 is

$V_{2} = \frac{1}{3} \cdot 15 ^{2}\cdot J= \frac{1}{3} \cdot 225 \cdot J = 75J$

The problem therefore asks us which, if either, of $\left ( 33 \frac{1}{3}\right )F$ to 75J is the greater quantity.

Assume Statement 1 alone. If 2F = 3J , then $J = \frac{2}{3} F$ , and

$V_{2}= 75J = 75 \cdot \frac{2}{3} F = 50 F$

Since $50 > 33\frac{1}{3}$ , it follows that $50F > 33\frac{1}{3} F$ , and $V_{2} > V_{1}$ - that is, Pyramid 2 has the greater volume.

Statement 2 alone gives insufficient information. We take two sets of values of and that add up to 25:

Case 1: F = 1,J = 24

In this case, Pyramid 2 has the greater height and the greater base area, so it easily follows that Pyramid 2 has the greater volume.

Case 2: F = 24,J = 1

Then the volume of Pyramid 1 is

$V_{1} = \left ( 33 \frac{1}{3}\right )F = \left ( 33 \frac{1}{3}\right ) \cdot 24 = 800$

and that of Pyramid 2 is

$V_{2} = 50 F = 50 \cdot 1 = 50$

This makes Pyramid 1 the greater in volume.

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Example Question #7 : Dsq: Calculating The Volume Of A Tetrahedron

Pyramid_2

Note: Figure NOT drawn to scale, but you may assume AE < EF .

In the above figure, a pyramid with a rectangular base is inscribed inside a rectangular prism; its vertices are A,E,F,G,H . What is the volume of the pyramid?

Statement 1: The hypotenuse $\overline{AF}$ of 30-60-90 triangle $\bigtriangleup AEF$ has length 16.

Statement 2: The hypotenuse $\overline{AH}$ of 45-45-90 right triangle $\bigtriangleup AEH$ has length $8\sqrt{2}$ .

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.

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.

Correct answer:

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

Explanation:

The volume of the pyramid is one third the product of height and the area of its rectangular base, which is $EF \cdot EH$ ; that is,

$V = \frac{1}{3} \cdot AE \cdot EF \cdot EH$

Assume Statement 1 alone. $\bigtriangleup AEF$ is a 30-60-90 triangle with a hypotenuse of length 16. By the 30-60-90 Triangle Theorem, short leg $\overline{AE}$ has length half this, or 8, and long leg $\overline{EF}$ has length $\sqrt{3}$ times that of $\overline{AE}$ , or $8 \sqrt{3}$ . However, the length of $\overline{EH}$ cannot be determined.

Assume Statement 2 alone. $\bigtriangleup AEH$ is a 45-45-90 right triangle with a hypotenuse of length $8\sqrt{2}$ . By the 45-45-90 Theorem, its legs $\overline{AE}$ and $\overline{EH}$ each have length $8\sqrt{2}$ divided by $\sqrt{2}$ , which is $\frac{8\sqrt{2}}{\sqrt{2}}= 8$ ; however, the length of $\overline{EF}$ cannot be determined.

From the two statements together, we can determine that AE=EH = 8 and $EF = 8 \sqrt{3}$ , and calculate the volume:

$V = \frac{1}{3} \cdot 8 \cdot 8 \sqrt{3} \cdot 8 = \frac{512\sqrt{3}}{3}$ .

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GMAT Math : Data-Sufficiency Questions

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #32 : Polygons

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

Example Question #1 : Rectangular Solids & Cylinders

Example Question #1 : Dsq: Calculating The Volume Of A Tetrahedron

Example Question #2 : Dsq: Calculating The Volume Of A Tetrahedron

Example Question #1 : Tetrahedrons

Example Question #1 : Tetrahedrons

Example Question #171 : Geometry

Example Question #172 : Geometry

Example Question #7 : Dsq: Calculating The Volume Of A Tetrahedron

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