GMAT Math : Geometry

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Example Questions

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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Example Question #191 : Geometry

Jiminy wants to paint one of his silos. One gallon of this paint covers about 300 square feet. How many gallons will he need?

I) The radius of the silo is $\frac{14}{\pi}$ feet.

II) The height is $12\pi$ times longer the radius.

Possible Answers:

Neither I nor II is sufficient to answer the question. More information is needed.

Both statements are necessary to answer the question.

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Either statement alone is sufficient to answer the question.

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Correct answer:

Both statements are necessary to answer the question.

Explanation:

Review our statements:

I) The radius of the silo is $\frac{14}{\pi}$ feet.

II) The height is $12\pi$ times longer the radius

We need to find our surface area in order to find how many gallons we need. Surface area is given by:

$\small SA=2\pi r^2+2\pi r h$

So to find the surface area, we need the radius and the height, so both statments are needed here.

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Example Question #2 : Dsq: Calculating The Surface Area Of A Cylinder

A tin can has a volume of $\small 375\pi in^3$ .

I) The height of the can is inches.

II) The radius of the base of the can is inches.

What is the surface area of the can? (Assume it is a perfect cylinder)

Possible Answers:

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Both statements are needed to answer the question.

Either statement is sufficient to answer the question.

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Neither statement is sufficient to answer the question. More information is needed.

Correct answer:

Either statement is sufficient to answer the question.

Explanation:

To find surface area of a cylinder we need the radius and the height.

If we are given the volume, and either the radius or the height, we can work backwards to find the other dimension.

Since I and II give us the height and the radius, either statement can be used to find the surface area.

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Example Question #2 : Dsq: Calculating The Surface Area Of A Cylinder

The tank of a tanker truck is made by bending sheet metal and then welding on the ends. If the length of the tank is meters, what is its radius?

I) The volume of the tank is 250 m^3 .

II) It takes $150\pi$ square meters of metal to build the tank.

Possible Answers:

Both statements are needed to answer the question.

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Neither statement is sufficient to answer the question. More information is needed.

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Either statement is sufficient to answer the question.

Correct answer:

Either statement is sufficient to answer the question.

Explanation:

To find the radius of a cylinder from either volume or surface area we need the height.

We are given the height in the question.

We are given volume and surface area in the two statements.

Thus, either statement is sufficient.

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Example Question #192 : Geometry

Of Cylinder 1 and Cylinder 2, which, if either, has the greater surface area?

Statement 1: The sum of the height of Cylinder 1 and the radius of one of its bases is equal to the sum of the height of Cylinder 2 and the radius of one of its bases.

Statement 2: The bases of Cylinder 1 and Cylinder 2 have the same cicumference.

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.

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.

Correct answer:

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

Explanation:

We will let and represent the radii of Cylinder 1 and Cylinder 2, respectively, and and represent the heights of Cylinder 1 and Cylinder 2, respectively.

The surface area of Cylinder 1 is

$a = 2 \pi r(r+h)$ ,

and the surface area of Cylinder 2 is

$A = 2 \pi R(R+H)$ .

Statement 1 alone is insufficient, as can be seen by examining these two cases.

Case 1: H = 4, R= 4, h = 1, r= 7

For each cylinder, the sum of the radius and the height is 8 - that is, r+h = R+H = 8 .

The surface area of Cylinder 1 is

$a = 2 \pi r(r+h)$

$= 2 \pi \cdot 4 \cdot 8$

$= 64 \pi$

The surface area of Cylinder 2 is

$A = 2 \pi R(R+H)$ ,

$= 2 \pi \cdot 7 \cdot 8$

$= 112 \pi$

Therefore, Cylinder 2 has the greater area.

Case 2: H = 1, R= 7, h = 4, r= 4

This simply switches the dimensions of the cylinders, and consequently, it switches the surface areas. Cylinder 1 has the greater surface area.

Each scenario satisfies the condition of Statement 1.

Assume Statement 2 alone. The circumferences of the bases are the same, so, subsequently, the radii are as well. But the heights are also needed, and Statement 2 does not clue us in to the heights.

Assume both statements are true.

By Statement 1, R + H = r + h .

By Statement 2, since the circumferences of the bases are equal, so are their radii, so R = r .

By subtraction, it follows that R + H -R= r + h - r , and H = h . Since the cylinders have the same height and their bases have the same radius, it follows that their surface areas are equal.

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Example Question #193 : Geometry

Give the surface area of a cylinder.

Statement 1: The circumference of each base is $14 \pi$ .

Statement 2: Each base has radius 7.

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.

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:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

The surface area of the cylinder can be calculated from radius and height using the formula:

$A = 2 \pi rh + 2 \pi r^{2}$

Statement 1 gives the circumference of the bases, which can be divided by $2 \pi$ to yield the radius; Statement 2 gives the radius outright. However, neither statement yields information about the height, so the surface area cannot be calculated.

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Example Question #6 : Dsq: Calculating The Surface Area Of A Cylinder

Of Cylinder 1 and Cylinder 2, which, if either, has the greater lateral area?

Statement 1: The product of the height of Cylinder 1 and the radius of one of its bases is less than the product of the height of Cylinder 2 and the radius of one of its bases.

Statement 2: The product of the height of Cylinder 2 and the radius of one of its bases is equal to the product of the height of Cylinder 1 and the diameter of one of its bases.

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:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

The lateral area of the cylinder can be calculated from radius and height using the formula:

$a = 2 \pi rh$ .

In this problem we will use and as the dimensions of Cylinder 1 and and as those of Cylinder 2. Therefore, the lateral area of Cylinder 2 will be

$A = 2 \pi RH$

Assume Statement 1 alone. This means

rh < RH ;

multiplying both sides of the inequality by $2 \pi$ , we get

$2 \pi rh <2 \pi RH$ ,

a < A ,

Therefore, Cylinder 2 has the greater lateral area.

Assume Statement 2 alone. Since the diameter of a base of Cylinder 1 is twice its radius, or , this means

2 rh = RH

$rh =\frac{1}{2} RH$

It follows that rh < RH , and, again, $2 \pi rh <2 \pi RH$ , or a < A . Cylinder 2 has the greater lateral area.

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GMAT Math : Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #23 : Tetrahedrons

Example Question #24 : Tetrahedrons

Example Question #25 : Tetrahedrons

Example Question #26 : Tetrahedrons

Example Question #191 : Geometry

Example Question #2 : Dsq: Calculating The Surface Area Of A Cylinder

Example Question #2 : Dsq: Calculating The Surface Area Of A Cylinder

Example Question #192 : Geometry

Example Question #193 : Geometry

Example Question #6 : Dsq: Calculating The Surface Area Of A Cylinder

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