GMAT Math : Geometry

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Example Questions

Example Question #13 : Tetrahedrons

A tetrahedron is a solid with four triangular faces.

Give the volume of a tetrahedron.

Statement 1: The tetrahedron has four equilateral faces.

Statement 2: The surface area of the tetrahedron is $120 \sqrt{3}$ .

Possible Answers:

EITHER 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.

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.

Correct answer:

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

Explanation:

Neither statement is enough to determine the volume of the tetrahedron; Statement 1 alone gives no actual measurements, and Statement 2 gives only the surface area, which can apply to infinitely many tetrahedrons.

Assume both statements to be true. A tetrahedron with four equilateral faces is a regular tetrahedron, whose surface area, relative to the common length of its edges, is defined by the formula

$A = s^{2}\sqrt{3}$ .

By substituting $120 \sqrt{3}$ for , it is possible to calculate . Consequently, the volume of the tetrahedron can be calculated using the volume formula

$V = \frac{s^{3} }{6\sqrt{2}}$ .

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Example Question #14 : Tetrahedrons

Tetra_3

Note: Figure NOT drawn to scale.

The above figure shows a rectangular prism with an inscribed tetrahedron, or triangular pyramid, with vertices A,F,E,H . What is the volume of the tetrahedron?

Statement 1: Rectangle ADHE has area 240.

Statement 2: Square ABFE has area 144.

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.

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.

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 the height, which is , and the area of the base; this base, being a right triangle, is equal to one half the product of the lengths of its legs, or and . Therefore,

$V = \frac{1}{3} \cdot AE \cdot \frac{1}{2} EF \cdot EH$

$V = \frac{1}{6} \cdot AE \cdot EH \cdot EF$

From Statement 1, it can be determined that $AE \cdot EH = 240$ , but without knowing anything about , the volume of the pyramid cannot be determined. Similarly, from Statement 2, it can be determined that $AE \cdot EF = 144$ , but nothing is given about .

Now assume both statements are true. Statement 2 gives that Quadrilateral ABFE is a square with area 144, so $AE = EF = \sqrt{144} = 12$ . From Statement 1, we can tell $AE \cdot EH = 240$ . The volume of the pyramid can be calculated as

$V = \frac{1}{6} \cdot AE \cdot EH \cdot EF = \frac{1}{6}\cdot240 \cdot 12= 480$ .

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Example Question #11 : Tetrahedrons

Two right regular pyramids, Pyramid 1 and Pyramid 2, have the same height. Do the pyramids have the same volume?

Statement 1: The bases of the two pyramids have the same area.

Statement 2: The base of Pyramid 1 has seven sides; the base of Pyramid 2 has nine sides.

Possible Answers:

EITHER 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.

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.

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 any pyramid is one third the product of its height and the area of its base. Since the two pyramids have the same height, their volume will be the same if and only if their bases have the same area. Statement 1 gives us this information explicity. Statement 2 is irrelevant, since the number of sides of a polygon by itself does not give a clue as to its area.

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Example Question #16 : Tetrahedrons

Tetra_3

Note: Figure NOT drawn to scale.

The above figure shows a rectangular prism with an inscribed tetrahedron, or triangular pyramid, with vertices A,F,E,H . What is the volume of the tetrahedron?

Statement 1: AE = EF

Statement 2: EH = 40

Possible Answers:

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.

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:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

We show that the two statements together provide insufficient information by assuming them both to be true.

$V = \frac{1}{3} \cdot AE \cdot \frac{1}{2} EF \cdot EH$

$V = \frac{1}{6} \cdot AE \cdot EF \cdot EH$

By Statement 1, AE=EF , and by Statement 2, EH = 40 , so by substitution,

$V = \frac{1}{6} \cdot AE \cdot AE \cdot 40$

$V = \frac{20}{3} \cdot (AE )^{2}$

Without any further information, however, the volume cannot be determined.

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Example Question #17 : Tetrahedrons

A solid on the three-dimensional coordinate plane has four vertices, at points (0,0,0) , (10,0,0) , (0,14,0) , and (A,B,C) for some positive values of A,B,C . What is the volume of the solid?

Statement 1: A=C

Statement 2: B = 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 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 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 insufficient 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) , (10,0,0) , and (0,14,0) , and the area of its base is half the product of the lengths of its legs, which is

$b = \frac{1}{2} \cdot 10 \cdot 14 = 70$ .

The volume of the pyramid is one third the product of the area of its base, which is 70, 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, which is . Therefore, the only thing that is needed to determine the volume of the pyramid is . However, the two statements together only yield , and therefore do not give sufficient information to solve the problem.

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Example Question #18 : Tetrahedrons

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

Statement 1: D+E+F > G+H+J

Statement 2: D+E= G+H

Possible Answers:

BOTH statements TOGETHER are insufficient 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.

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 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, . Pyramid 2 has the same base and apex (G,H,J) , so its height is the -coordinate of its apex, .

Therefore, whichever is greater - or - determines which pyramid has the greater volume.

Neither statement alone gives a clue as to which is greater. However, if we assume both, then, by the subtraction property of inequality,

D+E+F > G+H+J

and

D+E= G+H

together imply that

$D+E+F - ( D+E) > G+H+J - \left ( G+H \right )$

and

F > J .

This means that Pyramid 1 has the greater height and, consequently, the greater volume.

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Example Question #19 : Tetrahedrons

Do right regular pyramids Pyramid 1 and Pyramid 2 have the same volume?

Statement 1: The bases of the two pyramids have the same perimeter.

Statement 2: The two pyramids have the same height.

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.

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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

The volume of any pyramid is one third the product of its height and the area of its base. Also, the base of a right regular pyramid must be a regular polygon.

Assume both statements are true. Since, from Statement 2, both pyramids have the same height, their volumes are equal if and only if the areas of their bases are equal as well.

However, from Statement 1, the perimeters of the bases are what are given as being equal. Two regular poygons with the same perimeter and the same number of sides have the same area. But as exemplified below, two regular polygons with different numbers of sides and the same perimeter have different areas.

For example, an equilateral triangle with perimeter 60 has sidelength 20 and, consequently, area

$A = \frac{s^{2}\sqrt{3}}{4} = \frac{20^{2}\sqrt{3}}{4} = 100\sqrt{3}$

A square with perimeter 60 has sidelength 15, and, consequently, area equal to the square of this, 225.

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Example Question #21 : Rectangular Solids & Cylinders

A solid on the three-dimensional coordinate plane has four vertices, at points (0,0,0) , (10,0,0) , (0,14,0) , and (A,B,C) for some positive values of A,B,C . What is the volume of the solid?

Statement 1: A+B+C= 50

Statement 2: A+B-C= 30

Possible Answers:

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.

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.

Correct answer:

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

Explanation:

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

Tetrahedron

$b = \frac{1}{2} \cdot 10 \cdot 14 = 70$ .

Neither statement alone is enough to gain this information. However, if both statements are assumed true, we can subtract each side of the latter equation from the former as follows:

A+B+C= 50

$\underline{A+B-C= 30}$

2C = 20

C= 10

The value of is obtained and the volume of the pyramid can be calculated.

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Example Question #22 : Rectangular Solids & Cylinders

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:

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 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.

Correct answer:

BOTH statements TOGETHER are insufficient 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

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.

We show that the two statements together provide insufficient information by examining two scenarios.

Case 1: F =12, J = 10

$V_{1} = \left ( 33 \frac{1}{3}\right )F = \left ( 33 \frac{1}{3}\right ) \cdot 12 = 400$

$V_{2} = 36 J = 36 \cdot 10 = 360$

$V_{1} > V_{2}$

Case 2: $F =10 \frac{1}{6}, J = 10$

$V_{1} = \left ( 33 \frac{1}{3}\right )F = \left ( 33 \frac{1}{3}\right ) \cdot 10 \frac{1}{6} = 338 \frac{8}{9}$

$V_{2} = 36 J = 36 \cdot 10 = 360$

$V_{1}< V_{2}$

Each case fits the conditions of the two statements and the main body of the question; in one case, Pyramid 1 has the greater volume and in the other case, Pyramid 2 does.

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Example Question #22 : Rectangular Solids & Cylinders

Two right regular pyramids, Pyramid 1 and Pyramid 2, have the same height. Do the pyramids have the same volume?

Statement 1: The bases of the two pyramids have the same perimeter.

Statement 2: The base of Pyramid 1 is a hexagon; the base of Pyramid 2 is an octagon.

Possible Answers:

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.

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 any pyramid is one third the product of its height and the area of its base. Since both pyramids have the same height, their volumes are equal if and only if the areas of their bases are equal as well. Also, since both pyramids are regular, their bases are regular polygons.

Statement 1 alone is insufficient. It is possible for two regular polygons to have the same perimeter but different areas. For example, an equilateral triangle with perimeter 60 has sidelength 20 and, consequently, area

$A = \frac{s^{2}\sqrt{3}}{4} = \frac{20^{2}\sqrt{3}}{4} = 100\sqrt{3}$

A square with perimeter 60 has sidelength 15, and, consequently, area equal to the square of this, 225.

Statement 2 only gives the number of sides, and no information about their measures.

Now assume both statements are true. If the common perimeter is , then the length of one side of the base of Pyramid 1, it being a regular hexagon, is $\frac{1}{6}P$ , likewise, the length of one side of the base of Pyramid 2 is $\frac{1}{8}P$ . The area of each in terms of can be calculated, and the two can be compared.

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GMAT Math : Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #13 : Tetrahedrons

Example Question #14 : Tetrahedrons

Example Question #11 : Tetrahedrons

Example Question #16 : Tetrahedrons

Example Question #17 : Tetrahedrons

Example Question #18 : Tetrahedrons

Example Question #19 : Tetrahedrons

Example Question #21 : Rectangular Solids & Cylinders

Example Question #22 : Rectangular Solids & Cylinders

Example Question #22 : Rectangular Solids & Cylinders

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