A tetrahedron is a triangular pyramid and can be looked at as such.

Three of the vertices -  - are on the horizontal plane , and can be seen as the vertices of the triangular base. This triangle, as seen below, is isosceles:

Its base is 12 and its height is 15, so its area is

The fourth vertex is off this plane; its perpendicular (vertical) distance to the aforementioned face is the difference between the -coordinates, , so this is the height of the pyramid. The volume of the pyramid is 

A tetrahedron is a triangular pyramid and can be looked at as such.

Three of the vertices -  - are on the horizontal plane , and can be seen as the vertices of the triangular base. This triangle, as seen below, is isosceles (drawing not to scale):

Its base is 20 and its height is 9, so its area is

The fourth vertex is off this plane; its perpendicular (vertical) distance to the aforementioned face is the difference between the -coordinates, , so this is the height of the pyramid. The volume of the pyramid is 

The tetrahedron looks like this:

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

This is a triangular pyramid, so look at  as its base; the area  of the base is half the product of its legs, or

.

The volume of the tetrahedron, it being essentially a pyramid, is one third the product of its base and its height, the latter of which is 12. Therefore,

.

Altitude  divides  into two 30-60-90 triangles.

By the 30-60-90 Theorem, , or

 is the midpoint of , so 

The area of the triangular base is half the product of its base and its height:

The volume of the pyramid is one third the product of this area and the height of the pyramid:

A regular tetrahedron has four faces, each of which is an equilateral triangle. Therefore, its surface area, given sidelength , is 

.

Substitute :

The area of each face of a regular tetrahedron, that face being an equilateral triangle, is 

Substitute edge length 16 for :

The tetrahedron has four faces, so the total surface area is 

The tetrahedron looks like this:

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

Three of the surfaces are right triangles with two legs of length 12, so the area of each is 

.

The fourth surface, , has three edges each of which is the hypotenuse of an isosceles right triangle with legs 12, so each has length  by the 45-45-90 Theorem. That makes this triangle equilateral, so its area is'

The surface area is therefore

.

The tetrahedron looks like this:

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

Three of the surfaces are right triangles with two legs of length 12, so the area of each is 

.

The fourth surface, , has three edges each of which is the hypotenuse of an isosceles right triangle with legs , so each has length  by the 45-45-90 Theorem. That makes this triangle equilateral, so its area is'

The surface area is therefore

.

There is a perfectly spherical weather balloon with a surface area of  , what is its diameter?

Begin with the formula for surface area of a sphere:

Now, set it equal to the given surface area and solve for r:

First divide both sides by .

Then square root both sides to get our radius:

Now, because the question is asking for our diameter and not our radius, we need to double our radius to get our answer:

A wooden ball has a surface area of .

What is its radius?

Begin with the formula for surface area of a sphere:

Now, plug in our surface area and solve with algebra:

Get rid of the pi

Divide by 4

Square root both sides to get our answer: