GMAT Math : DSQ: Calculating the surface area of a tetrahedron

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Example Question #1 : Dsq: Calculating The Surface Area Of A Tetrahedron

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:

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.

EITHER 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:

$\bigtriangleup ADC$ , $\bigtriangleup ADB$ , 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 #191 : Geometry

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:

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

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

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

Each statement alone is sufficient to answer the question.

Statements 1 and 2 are not sufficient, and additional data is needed 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 : DSQ: Calculating the surface area of a tetrahedron

Study concepts, example questions & explanations for GMAT Math

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Example Question #1 : Dsq: Calculating The Surface Area Of A Tetrahedron

Example Question #191 : Geometry

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