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

Study concepts, example questions & explanations for GMAT Math

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Example Questions

Example Question #22 : Tetrahedrons

Tetra_1

Note: Figure NOT drawn to scale.

Refer to the above tetrahedron or triangular pyramid. .

Calculate the surface area of the tetrahedron.

Statement 1:  has perimeter 60.

Statement 2:  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 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. 

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:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

 , and , all being right triangles with the same leg lengths, are congruent, and, consequently, their diagonals are congruent, making  equilateral. 

Assume Statement 1 alone. The sidelength of equilateral  is one third of its perimeter of 60, or 20. This is also the common length of the hypotenuses of isoseles right triangles  , and , so by the 45-45-90 Theorem, each leg length can be computed by dividing 20 by :

 

Assume Statement 2 alone.  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:

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  - is  times this, or .

 

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.

Example Question #23 : Tetrahedrons

What is the surface area of the tetrahedron? 

  1. The length of an edge measures .
  2. The volume of the tetrahedron is .
Possible Answers:

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

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.

Statements 1 and 2 are not sufficient, and additional data is needed to answer the question.

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

Correct answer:

Each statement alone is sufficient to answer the question.

Explanation:

The surface area of a tetrahedron is found by  where  represents the edge value.

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

 

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

 

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

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

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