GMAT Math : DSQ: Calculating the volume of a tetrahedron

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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.

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.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT 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 #2 : Dsq: Calculating The Volume Of A Tetrahedron

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 : Dsq: Calculating The Volume Of A Tetrahedron

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:

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.

EITHER statement ALONE is 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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Example Question #8 : Dsq: Calculating The Volume Of A Tetrahedron

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: Isosceles right triangle $\bigtriangleup AEF$ has area 32.

Statement 2: AH = 16

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.

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.

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,

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

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

From Statement 1 alone, we know $\bigtriangleup AEF$ is isosceles and has area 32; therefore, its common leg length can be determined using the area formula:

$\frac{1}{2} s^{2}= 32$

$s^{2}= 64$

s = 8

Therefore, AE=EF = 8 . However, nothing can be determined about .

Statement 2 alone does not give any of the three desired lengths or any information necessary to find them.

However, Statement 2, along with the information from Statement 1, can be used to find . From Statement 2, AH = 16 , and from Statement 1, AE = 8 ; the Pythagorean Theorem can be used to find . Therefore, all three of , , and can be found, and the volume of the pyramid can be calculated.

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Example Question #2 : Rectangular Solids & Cylinders

A regular tetrahedron is a solid with four faces, each of which is an equilateral triangle.

Give the volume of a regular tetrahedron.

Statement 1: Each edge has length 8.

Statement 2: Each face has area $16 \sqrt{3}$ .

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.

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.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

The formula for the volume of a regular tetrahedron given the length of each edge is

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

Statement 1 gives information explicitly. Statement 2 gives the means to find , since, if $16 \sqrt{3}$ is substituted for in the formula for an equilateral triangle:

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

the value of can be determined.

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Example Question #11 : 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 200.

Statement 2: Rectangle ABFE has area 120.

Possible Answers:

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.

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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

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

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

We need to know the values of , , and to find the volume of of the pyramid. We show that the two statements give insufficient information by examining two scenarios.

Case 1: AE = 10, EH = 20, EF = 12

Rectangle ADHE has area $AE \cdot EH = 10 \cdot 20 = 200$ .

Rectangle ABFE has area $AE \cdot EF = 10 \cdot 12= 120$ .

The volume of the pyramid is

$V = \frac{1}{6} \cdot AE \cdot EF \cdot EH= V = \frac{1}{6} \cdot 10 \cdot 12 \cdot 20 = 400$

Case 2: AE = 8, EH = 25, EF = 15

Rectangle ADHE has area $AE \cdot EH = 8 \cdot 25 = 200$ .

Rectangle ABFE has area $AE \cdot EF = 8 \cdot 15= 120$ .

The volume of the pyramid is

$V = \frac{1}{6} \cdot AE \cdot EF \cdot EH= V = \frac{1}{6} \cdot 8 \cdot 25 \cdot 15 =500$

In each case, the conditions of both statements are met, but the volumes of the pyramids differ.

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GMAT Math : DSQ: Calculating the volume of a tetrahedron

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

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

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

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

Example Question #171 : Geometry

Example Question #172 : Geometry

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

Example Question #8 : Dsq: Calculating The Volume Of A Tetrahedron

Example Question #2 : Rectangular Solids & Cylinders

Example Question #11 : Tetrahedrons

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