A tetrahedron has three mutually perpendicular faces and three mutually perpendicular edges of lengths 2, 3, and 4 cm, respectively. Find its volume.
To find the volume of a tetrahedron with mutually perpendicular edges, we can use the formula:
V = (a^3) / (6√2),
where a is the length of one of the edges of the tetrahedron.
In this case, the given lengths of the edges are 2, 3, and 4 cm. Let's take the smallest edge, which is 2 cm, and substitute it into the formula:
V = (2^3) / (6√2).
Calculating:
V = 8 / (6√2),
Simplifying:
V = 4 / (3√2).
To express the volume in a simplified form, we can rationalize the denominator by multiplying both the numerator and denominator by √2:
V = (4 / (3√2)) * (√2 / √2),
V = (4√2) / (3 * 2),
V = (4√2) / 6,
Simplifying:
V = (2√2) / 3.
Therefore, the volume of the tetrahedron is (2√2) / 3 cubic units.
To find the volume of the tetrahedron, we can use the formula:
Volume = (1/3) * Base Area * Height
First, let's find the base area and height of the tetrahedron.
Since the tetrahedron has three mutually perpendicular faces, we can deduce that it is a right tetrahedron. In a right tetrahedron, the height is the perpendicular distance from the apex (top vertex) to the base.
To find the base area, we can consider the two mutually perpendicular edges of lengths 3 cm and 4 cm as the dimensions of a right triangle. The base of the triangle would be the edge of length 3 cm, and the height of the triangle would be the edge of length 4 cm. Therefore, the base area of the tetrahedron is:
Base Area = (1/2) * base * height
= (1/2) * 3 cm * 4 cm
= 6 cm²
Now, let's find the height. Since the edge of length 2 cm is perpendicular to the base, it is the height of the tetrahedron.
Height = 2 cm
Finally, we can calculate the volume using the formula:
Volume = (1/3) * Base Area * Height
= (1/3) * 6 cm² * 2 cm
= 4 cm³
Therefore, the volume of the tetrahedron is 4 cubic centimeters.
The base is just a right triangle, so
as always, V = 1/3 Bh = 1/3 * 1/2 (2*3) * 4 = 4 cm^3