# GMAT Math : Calculating the surface area of a tetrahedron

## Example Questions

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

The slant height of a pyramid is one and one-half times the perimeter of its square base. The base has sides of length 15 inches. What is the surface area of the pyramid?

Explanation:

The square base of the pyramid has four sides with length 15 inches, making its perimeter four times that, or 60 inches. The slant height is

inches.

Therefore, the area of the base is  square inches.

Each of the four lateral triangles has area  square inches.

The total surface area is  square inches.

### Example Question #2 : Calculating The Surface Area Of A Tetrahedron

In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates .

Give the surface area of the tetrahedron.

Explanation:

The tetrahedron looks like this:

is the origin and  are the other three points, which are 60 units away from the origin on each of the three (perpendicular) axes.

The bottom, front, and left faces are each right triangles whose legs each measure 60. Each face has area

.

The remaining face has three edges each a hypotenuse of one of three congruent right triangles, so its sides are congruent, and it is an equilateral triangle. Its sidelength can be found via the 45-45-90 Theorem to be , so its area is

The total area is

### Example Question #2 : Calculating The Surface Area Of A Tetrahedron

A regular tetrahedron comprises four faces, each of which is an equilateral triangle. If the sum of the lengths of its edges is 120, what is its surface area?

Explanation:

As shown in the diagram below, a regular tetrahedron has six congruent edges, so each has length :

The area of one face is the area of an equilateral triangle with sidelength 20, which is

The total surface area is four times this, or .

### Example Question #3 : Calculating The Surface Area Of A Tetrahedron

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

Each edge of a regular tetrahedron has length . What is the surface area of the tetrahedron?

Explanation:

The area of one face of the tetrahedron, it being an equilateral triangle, can be calulated using the formula

There are four congruent faces, so the total surface area is

Since , the surface area of the tetrahedron is

### Example Question #4 : Calculating The Surface Area Of A Tetrahedron

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

If the lengths of all of the edges of a regular tetrahedron are added, the total length is 120. What is the surface area of the tetrahedron?

Explanation:

A tetrahedron looks like this:

The tetrahedron has six edges, and in a regular tetrahedron, they are congruent, so each edge has length .

The area of one face of the tetrahedron, it being an equilateral triangle, can be calulated using the formula

There are four congruent faces, so the total surface area is

Since , the surface area of the tetrahedron is

### Example Question #6 : Calculating The Surface Area Of A Tetrahedron

Evaluate the surface area of the above tetrahedron.

Explanation:

Three of the faces of the tetrahedron are isosceles right triangles with legs of length 8, and, subsequently, by the 45-45-90 Theorem, hypotenuses of length . The fourth face, consequently, is an equilateral triangle with three sides of length .

Each of the three right triangle faces has area equal to half the product of its legs , which is

.

The equilateral face has as its area

The sum of the areas of the faces is

### Example Question #5 : Calculating The Surface Area Of A Tetrahedron

Refer to the above diagram, which shows a tetrahedron.

, and . Give the surface area of the tetrahedron.

Explanation:

Three of the surfaces of the tetrahedron - , and   - are isosceles right triangles with hypotenuse 30, so by the 45-45-90 Theorem, each leg measures this length divided by , or

.

The area of each of these triangles is half the product of its legs, so each area is

Also, the legs are of the same measure among the triangles, the hypotenuses are as well, so the fourth surface  is an equilateral triangle. Its sidelength is , so we use the equilateral triangle area formula to calculate its area:

Add the areas of the faces:

### Example Question #8 : Calculating The Surface Area Of A Tetrahedron

The above diagram shows a regular right triangular pyramid. Its base  is an equilateral triangle; the other three faces are congruent isosceles triangles, with  an altitude of . Give the surface area of the pyramid.

Explanation:

The base is an equilateral triangle with sidelength 12, so its area can be calculated as follows:

.

Each of the three other faces is congruent, with base 12. The area of each is the product of its base and its height. To find the common height, we examine , which, since  is an altitude of isosceles , is a right triangle with hypotenuse of length 18 and one leg of length . We can find  using the Pythagorean Theorem:

The area of  is half the product of this height and the base:

All three lateral faces have this area.

Now add the areas of the four faces:

### Example Question #21 : Rectangular Solids & Cylinders

The cube in the above figure has surface area 384. Give the surface area of the tetrahedron with vertices , shown in red.

Explanation:

The surface area formula can be used to find the length of each edge of the cube:

Three faces of the tetrahedron - ,   - are right triangles with legs of length 8, so the area of each is half the product of the lengths of their legs:

.

Each triangle is isosceles, so, by the 45-45-90 Theorem, each of their hypotentuses measures  times a leg, or .  is therefore an equilateral triangle with sidelenghth . Its area can be found as follows:

The total surface area is