GMAT Math : Tetrahedrons

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Example Questions

Example Question #21 : 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+B+C= 50

Statement 2: A+B-C= 30

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 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.

EITHER statement ALONE is 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 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 .

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 #21 : Tetrahedrons

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.

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.

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 : 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 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.

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 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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Example Question #23 : 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 2 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 > G

Statement 2: E < H

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.

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.

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. 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. However, the two statements to not give this information.

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Example Question #24 : 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 cube can be inscribed inside a sphere with volume $972 \pi$ .

Statement 2: A sphere with surface area $108 \pi$ can be inscribed inside the cube.

Possible Answers:

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.

BOTH statements TOGETHER are insufficient 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. If the volume of the circumscribing sphere is known to be $972 \pi$ , the radius can be calculated as follows:

$\frac{4}{3} \pi r^{3} = V$

$\frac{4}{3} \pi r^{3} =972 \pi$

$\frac{3}{4\pi} \cdot \frac{4}{3} \pi r^{3} =\frac{3}{4\pi} \cdot 972 \pi$

$r^{3} =729$

$r = \sqrt[3]{729}= 9$

The diameter, which is twice this, or 18, is the length of a diagonal of the cube. By the three-dimensional extension of the Pythagorean Theorem, the relationship of this length to the side length of the cube is

$s^{2}+s^{2}+s^{2} = d^{2}$ , or

$3s^{2} = d^{2}$

$3s^{2} = 18^{2}$

$s^{2} = \frac{18^{2}}{3} = 108$

$s = \sqrt{108} = 6 \sqrt{3}$ ,

Assume Statement 2 alone. If the surface area of the inscribed sphere is known to be $108 \pi$ , then its radius can be calculated as follows:

$4 \pi r^{2} = A$

$4 \pi r^{2} = 108 \pi$

$\frac{4 \pi r^{2} }{4 \pi }=\frac{ 108 \pi}{4 \pi }$

$r^{2} = 27$

$r = \sqrt{27} = 3\sqrt{3}$ .

The diameter of the inscribed sphere, which is twice this, or $6 \sqrt{3}$ , is equal to the length of one edge of the cube.

Either statement alone gives us the length of one side of the cube, which is enough to allow the volume of the pyramid to be calculated.

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Example Question #25 : Tetrahedrons

Tetra_1

Note: Figure NOT drawn to scale.

Refer to the above tetrahedron or triangular pyramid. AD = BD = CD .

Calculate the surface area of the tetrahedron.

Statement 1: $\bigtriangleup ABC$ has perimeter 60.

Statement 2: $\bigtriangleup ADC$ has area 100.

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:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

$\bigtriangleup ADC$ , $\bigtriangleup ADB$ , and $\bigtriangleup BDC$ , all being right triangles with the same leg lengths, are congruent, and, consequently, their diagonals are congruent, making $\bigtriangleup ABC$ equilateral.

Assume Statement 1 alone. The sidelength of equilateral $\bigtriangleup ABC$ is one third of its perimeter of 60, or 20. This is also the common length of the hypotenuses of isoseles right triangles $\bigtriangleup ADC$ , $\bigtriangleup ADB$ , and $\bigtriangleup BDC$ , so by the 45-45-90 Theorem, each leg length can be computed by dividing 20 by $\sqrt{2}$ :

$\frac{20}{\sqrt{2}} = \frac{20\cdot \sqrt{2}}{\sqrt{2}\cdot \sqrt{2}} = \frac{20 \sqrt{2}}{2} = 10 \sqrt{2}$

Assume Statement 2 alone. $\bigtriangleup ADC$ has area 100; since the area of a right triangle is half the product of the lengths of its legs, we can find the common leg length using this formula:

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

$\frac{1}{2} s^{2} = 100$

$s^{2} = 200$

$s =\sqrt{200} = \sqrt{100}\cdot \sqrt{2}= 10 \sqrt{2}$

This is the common length of the legs of the three right triangles; by the 45-45-90 Theorem, each hypotenuse - and each side of equilateral $\bigtriangleup ADC$ - is $\sqrt{2}$ times this, or $10 \sqrt{2} \cdot \sqrt{2} = 10 \cdot 2 = 20$ .

Therefore, from either statement alone, the lengths of all sides of the tetrahedron can be found, and the area formulas for a right triangle and an equilateral can be applied to find the areas of all four faces.

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Example Question #26 : Tetrahedrons

What is the surface area of the tetrahedron?

The length of an edge measures $3\sqrt{2}cm$ .
The volume of the tetrahedron is .

Possible Answers:

Each statement alone is sufficient to answer the question.

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

Correct answer:

Each statement alone is sufficient to answer the question.

Explanation:

The surface area of a tetrahedron is found by $SA=a^2 \sqrt{3}$ where represents the edge value.

Situation 1: We're given our value so we just need to plug it into our equation.

$SA=(3\sqrt{2})^2 \sqrt{3}$

$SA=18\sqrt{3}$

Situation 2: We use the given volume to solve for the length of the edge.

$V=\frac{a^3}{6\sqrt{2}}=9$

$a=3\sqrt{2}$

Now that we have a length, we can plug it into the equation for the surface area:

$SA=(3\sqrt{2})^2 \sqrt{3}=18\sqrt{2}$

Thus, each statement alone is sufficient to answer the question.

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GMAT Math : Tetrahedrons

Study concepts, example questions & explanations for GMAT Math

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Example Questions

Example Question #21 : Tetrahedrons

Example Question #21 : Tetrahedrons

Example Question #22 : Tetrahedrons

Example Question #23 : Tetrahedrons

Example Question #24 : Tetrahedrons

Example Question #25 : Tetrahedrons

Example Question #26 : Tetrahedrons

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