What is the volume of a triangular pyramid with each edge measuring 1?

Question

Please show your work.

Answer 1

V = 1/12 a^3 √2 according to Wikipedia, so here that would be (√2)/12.

If side edge is a, base is equilateral triangle with height (a/2)√3, so base area is (a²/4)√3.

Height of pyramid, h, satisfies a² - h² = {[2(a/2)√3]/3}², so
h² = a² - a² • 3 / 9
h² = a² - a²/3 = 2a²/3
h = a√6 / 3

so volume is 1/3 (a√6 / 3)(a²/4 √3) =
a^3 /36 √18 =
a^3 /12 √2

Answer 2

V=base area * height/3

To find the base area, you have to find the height of the triangle--which is not 1. 1 is an edge of the triangle. The height of the triangle is the altitude, (a segment goes from a vertex of the triangle and is perpendicular to the opposite side). Since the triangle is regular, (all sides are equal length), the altitude bisects both an angle and its opposite side. So, the original triangle is divided into two right triangles. We can assume the angle measures of these small triangles are 30, 60 and 90. The ninety is given because the altitude is perpendicular to the side, the sixty must be because the original triangle is regular, and the angle measures of that triangle are all 60. The 30, we know because the angle is bisected by the altitude. The side opposite of the 30 measures .5, since it is just half the original side. 30,60,90 triangles have special ratios for their sides, so the length of the altitude, (the height), = .5*sqrt(3)

Base area=base*height
Base area=.5*sqrt(3)*1
(approx. .866)

Next step is to find the height of the pyramid...which is also NOT equal to 1. It is the line that is drawn from the top vertex of the pyramid to the base triangle--and is perpendicular to the base triangle. To get this, you have to use the pythagorean theorem. There is a triangle that is formed between one side of the pyramid, (the hypotenuse), the height of the pyramid, and half of the altitude of the base triangle. (it is a triangle that is perpendicular to the base triangle). The altitude of the base triangle we already know is .5*sqrt(3), so we divide that by 2 and get .25sqrt(3)
To find the height of the pyramid--

1^2- .25*sqrt(3)^2=height^2

the height is approximately .901

Now plug all the numbers into the formula

.5*sqrt(3) * .901/3= .26 units^3

remember, some of these numbers are rounded

Answer 3

height = 1 / sqr(4) = 1/2
area = 1/2 ( 1/2 ) [ sqr(3) / 2 ] = (1/8 ) sqr( 3 )
vol = 1/3 Area x height
. . . = 1/3 ( 1/8 ) sqr(3) / 2
. . . = sqr(3) /48
. . . = 0.036