Octahedron question?

Question

Octahedron problem?
In a octahedron; all sides equilateral triangles; each length for the sides of equilateral triangles are 12 cm. How would you find the total surface area and volume?

Answer 1

Hi,

The area A and volume V formulas for an octahedron with edge a are:

A = 2radical(3)* side^2

V = 1/3 radical(2) a^3

So if a = 12,

A = 2radical(3)* 12^2 = 2radical(3)*144 = 288 radical(3) sq cm.

V = 1/3 radical(2) 12^3= 1/3 radical(2)*1728 = 576 radical(2) cu. cm.

I hope that helps. :-)

Answer 2

The surface area is just 8 times the area of an equilateral triangle.

For the volume, the key is to note an octahedron cut in half is two square pyramids. The volume is therefore 2bh/3.

The base is just 12². To get the height, note a right triangle can be formed from a side (12 cm), half the diagonal of a square (6√2 cm), and the altitude. So, the altitude is &radic(144 - 72) = √72 = 6√2. Putting all this together, you get the volume is:

2 * 12² * 6 *√2 / 3 = 576√2

Answer 3

well, you have to break up the octahedron and you get 8 equilateral triangles. to find the surface area, you add up the areas of all the triangles. all the triangles are exactly the same remember. to find A of tri is half base * height, so the A is 6*12=72. then theres 8 tri.s. 72*8=576cm^2 is the surface area. volume....i don't think you gave enough information. i mean i could ASSUME the added dimension is 12 cm but if when we break up the polygon we get triangular prisms instead of a polygon with all triangles on each side, then theres a problem.

Answer 4

For surface area, consider one equilateral triangular face. The altitude from one vertex to the other side is 6 sqrt(3). So the area of each face is 36 sqrt(3) cm2. For 8 faces, 288 sqrt(3) cm2.

For volume, consider octahedron as two square pyramids put together. There is a formula for the volume of a pyramid somewhere, that V=bh/3, where b is the pyramid base and h is the pyramid height. The height of the pyramid is equal to 6 sqrt(2) cm^2. The base is 144 cm^2.
So each pyramid is 288 sqrt(2) cm^3.

Answer 5

WIKIPEDIA HAS NICE DIAGRAM AND FORMULA UNDER
OCTAHEDRON